Which branch of science has four fundamental principles: addition, subtraction, multiplication, and division.
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Science Subtraction Multiplication Division (mathematics) Arithmetic
ScienceFrom Latin scientia, meaning "knowledge". 
*  is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe."... modern science is a discovery as well as an invention. It was a discovery that nature generally acts regularly enough to be described by laws and even by mathematics; and required invention to devise the techniques, abstractions, apparatus, and organization for exhibiting the regularities and securing their law-like descriptions."— 

Contemporary science is typically subdivided into the natural sciences, which study the material universe; the social sciences, which study people and societies; and the formal sciences, such as mathematics. The formal sciences are often excluded as they do not depend on empirical observations.  Disciplines which use science like engineering and medicine may also be considered to be applied sciences. 

During the Middle Ages in the Middle East, foundations for the scientific method were laid by Alhazen in his Book of Optics.    From classical antiquity through the 19th century, science as a type of knowledge was more closely linked to philosophy than it is now and, in fact, in the Western world, the term "natural philosophy" encompassed fields of study that are today associated with science, such as  astronomy, medicine, and physics.Isaac Newton's Philosophiae Naturalis Principia Mathematica (1687), for example, is translated "Mathematical Principles of Natural Philosophy", and reflects the then-current use of the words "natural philosophy", akin to "systematic study of nature" While the classification of the material world by the ancient Indians and Greeks into air, earth, fire and water was more philosophical, medieval Middle Eastern scientists used practical, experimental observation to classify materials.Science and Islam, Jim Al-Khalili. BBC, 2009

In the 17th and 18th centuries, scientists increasingly sought to formulate knowledge in terms of laws of nature. Over the course of the 19th century, the word "science" became increasingly associated with the scientific method itself, as a disciplined way to study the natural world. It was in the 19th century that scientific disciplines such as biology, chemistry, and physics reached their modern shapes. The same time period also included the origin of the terms "scientist" and "scientific community," the founding of scientific institutions, and increasing significance of the interactions with society and other aspects of culture.   Nowadays, science is counted among the ten function systems of modern societies.  

History 

Science in a broad sense existed before the modern era, and in many historical civilizations."The historian ... requires a very broad definition of "science" — one that ... will help us to understand the modern scientific enterprise. We need to be broad and inclusive, rather than narrow and exclusive ... and we should expect that the farther back we go [in time] the broader we will need to be." — David Pingree (1992), "Hellenophilia versus the History of Science" Isis 83 554–63, as cited in , The beginnings of Western science: the European Scientific tradition in philosophical, religious, and institutional context,  Second ed. Chicago: Univ. of Chicago Press ISBN 978-0-226-48205-7
*See Edward Grant (1997) [http://www.jstor.org/stable/41212592?Searchyes&resultItemClick
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2#page_scan_tab_contents "When did modern science begin?"] The American Scholar pp.105-113 in JSTOR:
** History of science#Early cultures
** History of science#Ancient Near East, Mesopotamia
** History of science#Ancient Near East, Egypt
** History of Science in China
** History of science#India
  Modern science is distinct in its approach and successful in its results:  'modern science' now defines what science is in the strictest sense of the term. 

Science in its original sense is a word for a type of knowledge, rather than a specialized word for the pursuit of such knowledge. In particular it is one of the types of knowledge which people can communicate to each other and share. For example, knowledge about the working of natural things was gathered long before recorded history and led to the development of complex abstract thinking. This is shown by the construction of complex calendars, techniques for making poisonous plants edible, and buildings such as the pyramids. However no consistent conscientious distinction was made between knowledge of such things which are true in every community and other types of communal knowledge, such as mythologies and legal systems.

Antiquity

Before the invention or discovery of the concept of "nature" (Ancient Greek phusis), by the Pre-Socratic philosophers, the same words tend to be used to describe the natural "way" in which a plant grows,  and the "way" in which, for example, one tribe worships a particular god. For this reason it is claimed these men were the first philosophers in the strict sense, and also the first people to clearly distinguish "nature" and "convention".  Science was therefore distinguished as the knowledge of nature, and the things which are true for every community, and the name of the specialized pursuit of such knowledge was philosophy — the realm of the first philosopher-physicists. They were mainly speculators or theorists, particularly interested in astronomy. In contrast, trying to use knowledge of nature to imitate nature (artifice or technology, Greek technē) was seen by classical scientists as a more appropriate interest for lower class artisans. 
A clear-cut distinction between formal (eon) and empirical science (doxa) was made by pre-Socratic philosopher Parmenides (fl. late sixth or early fifth century BCE). Although his work peri physeos is a poem, it may be viewed as an epistemological essay, an essay on method in natural science.  Parmenides' ἐὸν may refer to a formal system, a calculus which can describe nature more precisely than natural languages. 'Physis' may be identical to ἐὸν. 

A major turning point in the history of early philosophical science was the controversial but successful attempt by Socrates to apply philosophy to the study of human things, including human nature, the nature of political communities, and human knowledge itself. He criticized the older type of study of physics as too purely speculative, and lacking in self-criticism. He was particularly concerned that some of the early physicists treated nature as if it could be assumed that it had no intelligent order, explaining things merely in terms of motion and matter. The study of human things had been the realm of mythology and tradition, and Socrates was executed.  Aristotle later created a less controversial systematic programme of Socratic philosophy, which was teleological, and human-centred. He rejected many of the conclusions of earlier scientists. For example, in his physics the sun goes around the earth, and many things have it as part of their nature that they are for humans. Each thing has a formal cause and final cause and a role in the rational cosmic order. Motion and change is described as the actualization of potentials already in things, according to what types of things they are. While the Socratics insisted that philosophy should be used to consider the practical question of the best way to live for a human being (a study Aristotle divided into ethics and political philosophy), they did not argue for any other types of applied science.

Aristotle maintained the sharp distinction between science and the practical knowledge of artisans, treating theoretical speculation as the highest type of human activity, practical thinking about good living as something less lofty, and the knowledge of artisans as something only suitable for the lower classes. In contrast to modern science, Aristotle's influential emphasis was upon the "theoretical" steps of deducing universal rules from raw data, and did not treat the gathering of experience and raw data as part of science itself."... [A] man knows a thing scientifically when he possesses a conviction arrived at in a certain way, and when the first principles on which that conviction rests are known to him with certainty—for unless he is more certain of his first principles than of the conclusion drawn from them he will only possess the knowledge in question accidentally." — Aristotle, Nicomachean Ethics 6 (H. Rackham, ed.) [http://www.perseus.tufts.edu/hopper/text?doc
Perseus%3Atext%3A1999.01.0054%3Abekker%20page%3D1139b Aristot. Nic. Eth. 1139b]

Medieval science 

During late antiquity and the early Middle Ages, the Aristotelian approach to inquiries on natural phenomena was used. Some ancient knowledge was lost, or in some cases kept in obscurity, during the fall of the Roman Empire and periodic political struggles. However, the general fields of science, or "natural philosophy" as it was called, and much of the general knowledge from the ancient world remained preserved though the works of the early Latin encyclopedists like Isidore of Seville. Also, in the Byzantine empire, many Greek science texts were preserved in Syriac translations done by groups such as Nestorians and Monophysites.  Many of these were translated later on into Arabic under the Caliphate, during which many types of classical learning were preserved and in some cases improved upon.Alhacen had access to the optics books of Euclid and Ptolemy, as is shown by the title of his lost work A Book in which I have Summarized the Science of Optics from the Two Books of Euclid and Ptolemy, to which I have added the Notions of the First Discourse which is Missing from Ptolemy's Book From Ibn Abi Usaibia's catalog, as cited in   The House of Wisdom was established in Abbasid-era Baghdad, Iraq.  It is considered to have been a major intellectual center, during the Islamic Golden Age, where Muslim scholars such as al-Kindi and Ibn Sahl in Baghdad, and Ibn al-Haytham in Cairo, flourished from the ninth to the thirteenth centuries, until the Mongol sack of Baghdad. Ibn al-Haytham, known later to the West as Alhazen, furthered the Aristotelian viewpoint,  by emphasizing  experimental data."[Ibn al-Haytham] followed Ptolemy's bridge building ... into a grand synthesis of light and vision. Part of his effort consisted in devising ranges of experiments, of a kind probed before but now undertaken on larger scale."—   In the later medieval period, as demand for translations grew, for example from the Toledo School of Translators, Western Europeans began collecting texts written not only in Latin, but also Latin translations from Greek, Arabic, and Hebrew. The texts of Aristotle, Ptolemy,The translator, Gerard of Cremona (c. 1114–87), inspired by his love of the Almagest, came to Toledo, where he knew he could find the Almagest in Arabic. There he found Arabic books of every description, and learned Arabic in order to translate these books into Latin, being aware of 'the poverty of the Latins'. —As cited by Charles Burnett (2001) "The Coherence of the Arabic-Latin Translation Program in Toledo in the Twelfth Century", pp. 250, 255, & 257, Science in Context 14(1/2), 249–288 (2001). DOI: 10.1017/0269889701000096 and Euclid, preserved in the Houses of Wisdom, were sought amongst Catholic scholars. In Europe, Alhazen's De Aspectibus directly influenced Roger Bacon (13th century) in England, who argued for more experimental science, as demonstrated by Alhazen. By the late Middle Ages, a synthesis of Catholicism and Aristotelianism known as Scholasticism was flourishing in Western Europe, which had become a new geographic center of science, but all aspects of scholasticism were criticized in the 15th and 16th centuries.

Renaissance, and early modern science 

Medieval science carried on the views of the Hellenist civilization of Socrates, Plato, and Aristotle, as shown by Alhazen's lost work A Book in which I have Summarized the Science of Optics from the Two Books of Euclid and Ptolemy, to which I have added the Notions of the First Discourse which is Missing from Ptolemy's Book from Ibn Abi Usaibia's catalog, as cited in  . Alhazen conclusively disproved Ptolemy's theory of vision. But Alhacen retained Aristotle's ontology; Roger Bacon, Witelo, and John Peckham each built-up a scholastic ontology upon Alhazen's Book of Optics, a causal chain beginning with sensation, perception, and finally apperception of the individual and universal forms of Aristotle.   This model of vision became known as Perspectivism, which was exploited and studied by the artists of the Renaissance.

A. Mark Smith points out the perspectivist theory of vision "is remarkably economical, reasonable, and coherent", which pivots on three of Aristotle's four causes, formal, material, and final. 
Although Alhacen knew that a scene imaged through an aperture is inverted, he argued that vision is about perception. This was overturned by Kepler, who modelled the eye with a water-filled glass sphere, with an aperture in front of it to model the entrance pupil. He found that all the light from a single point of the scene was imaged at a single point at the back of the glass sphere. The optical chain ends on the retina at the back of the eye and the image is inverted.Kepler, Johannes (1604) Ad Vitellionem paralipomena, quibus astronomiae pars opticae traditur (Supplements to Witelo, in which the optical part of astronomy is treated)  as cited in Smith,  A. Mark  (2004) "What is the history of Medieval Optics Really About?" Proceedings of the American Philosophical Society 148(2 — Jun. 2004), pp. 180-194 [http://www.jstor.org/stable/1558283?seq=13#page_scan_tab_contents p.192  via JSTOR]
* The full title translation is from p.60 of James R. Voelkel (2001) Johannes Kepler and the New Astronomy Oxford University Press. Kepler was driven to this experiment after observing the partial solar eclipse at Graz, July 10, 1600. He used Tycho Brahe's method of observation, which was to project the image of the sun on a piece of paper through a pinhole aperture, instead of looking directly at the sun. He disagreed with Brahe's conclusion that total eclipses of the sun were impossible, because there were historical accounts of total eclipses. Instead he deduced that the size of the aperture controls the sharpness of the projected image (the larger the aperture, the more accurate the image — this fact is now fundamental for optical system design). Voelkel, p.61, notes that Kepler's experiments produced the first correct account of vision and the eye, because he realized he could not accurately write about astronomical observation by ignoring the eye.

Copernicus formulated a heliocentric model of the solar system unlike the geocentric model of Ptolemy's Almagest.

Galileo made innovative use of experiment and mathematics. However his persecution began after Pope Urban VIII blessed Galileo to write about the Copernican system. Galileo had used arguments from the Pope and put them in the voice of the simpleton in the work "Dialogue Concerning the Two Chief World Systems" which caused great offense to him. 

In Northern Europe, the new technology of the printing press was widely used to publish many arguments including some that disagreed with church dogma. René Descartes and Francis Bacon published philosophical arguments in favor of a new type of non-Aristotelian science. Descartes argued that mathematics could be used in order to study nature, as Galileo had done, and Bacon emphasized the importance of experiment over contemplation.
Bacon questioned the Aristotelian concepts of formal cause and final cause, and promoted the idea that science should study the laws of "simple" natures, such as heat, rather than assuming that there is any specific nature, or "formal cause", of each complex type of thing. This new modern science began to see itself as describing "laws of nature". This updated approach to studies in nature was seen as mechanistic. Bacon also argued that science should aim for the first time at practical inventions for the improvement of all human life.

Age of Enlightenment 

In the 17th and 18th centuries, the project of modernity, as had been promoted by Bacon and Descartes, led to rapid scientific advance and the successful development of a new type of natural science, mathematical, methodically experimental, and deliberately innovative. Newton and Leibniz succeeded in developing a new physics, now referred to as Newtonian physics, which could be confirmed by experiment and explained using mathematics. Leibniz also incorporated terms from Aristotelian physics, but now being used in a new non-teleological way, for example "energy" and "potential" (modern versions of Aristotelian "energeia and potentia"). In the style of Bacon, he assumed that different types of things all work according to the same general laws of nature, with no special formal or final causes for each type of thing. It is during this period that the word "science" gradually became more commonly used to refer to a type of pursuit of a type of knowledge, especially knowledge of nature — coming close in meaning to the old term "natural philosophy".

19th century 

Both John Herschel and William Whewell systematized methodology: the latter coined the term scientist. When Charles Darwin published On the Origin of Species he established descent with modification as the prevailing evolutionary explanation of biological complexity. His theory of natural selection provided a natural explanation of how species originated, but this only gained wide acceptance a century later. John Dalton developed the idea of atoms. The laws of thermodynamics and the electromagnetic theory were also established in the 19th century, which raised new questions which could not easily be answered using Newton's framework. The phenomena that would allow the deconstruction of the atom were discovered in the last decade of the 19th century: the discovery of X-rays inspired the discovery of radioactivity. In the next year came the discovery of the first subatomic particle, the electron.

20th century and beyond 

Einstein's Theory of Relativity and the development of quantum mechanics led to the replacement of Newtonian physics with a new physics which contains two parts, that describe different types of events in nature.

In the first half of the century the development of artificial fertilizer made possible global human population growth. At the same time, the structure of the atom and its nucleus was elucidated, leading to the release of "atomic energy" (nuclear power). In addition, the extensive use of scientific innovation, stimulated by the wars of this century, led to antibiotics and increased life expectancy, revolutions in transportation (automobiles and aircraft), and the development of ICBMs, a space race, and a nuclear arms race— all giving a widespread public appreciation of the importance of modern science.

Widespread use of integrated circuits in the last quarter of the 20th century, combined with communications satellites, led to a revolution in information technology, and the rise of the global internet and mobile computing, including smartphones.

More recently, it has been argued that the ultimate purpose of science is to make sense of human beings and our nature – for example in his book Consilience, EO Wilson said "The human condition is the most important frontier of the natural sciences." 

The scientific method 

The scientific method seeks to explain the events of nature in a reproducible way.: "The amazing point is that for the first time since the discovery of mathematics, a method has been introduced, the results of which have an intersubjective value!" (Author's punctuation) An explanatory thought experiment or hypothesis is put forward, as explanation, using principles such as parsimony (also known as "Occam's Razor") and are generally expected to seek consilience—fitting well with other accepted facts related to the phenomena. This new explanation is used to make falsifiable predictions that are testable by experiment or observation. The predictions are to be posted before a confirming experiment or observation is sought, as proof that no tampering has occurred. Disproof of a prediction is evidence of progress.: "One learns in a laboratory; one learns how to make experiments only by experimenting, and one learns how to work with his hands only by using them. The first and fundamental form of experimentation in physics is to teach young people to work with their hands. Then they should be taken into a laboratory and taught to work with measuring instruments — each student carrying out real experiments in physics. This form of teaching is indispensable and cannot be read in a book.": "Whatever their discipline, scientists claimed to share a common scientific method that ... distinguished them from non-scientists." This is done partly through observation of natural phenomena, but also through experimentation, that tries to simulate natural events under controlled conditions, as appropriate to the discipline (in the observational sciences, such as astronomy or geology, a predicted observation might take the place of a controlled experiment). Experimentation is especially important in science to help establish causal relationships (to avoid the correlation fallacy).

When a hypothesis proves unsatisfactory, it is either modified or discarded. If the hypothesis survived testing, it may become adopted into the framework of a scientific theory. This is a logically reasoned, self-consistent model or framework for describing the behavior of certain natural phenomena. A theory typically describes the behavior of much broader sets of phenomena than a hypothesis; commonly, a large number of hypotheses can be logically bound together by a single theory. Thus a theory is a hypothesis explaining various other hypotheses. In that vein, theories are formulated according to most of the same scientific principles as hypotheses. In addition to testing hypotheses, scientists may also generate a model based on observed phenomena. This is an attempt to describe or depict the phenomenon in terms of a logical, physical or mathematical representation and to generate new hypotheses that can be tested.

While performing experiments to test hypotheses, scientists may have a preference for one outcome over another, and so it is important to ensure that science as a whole can eliminate this bias.  This can be achieved by careful experimental design, transparency, and a thorough peer review process of the experimental results as well as any conclusions.   After the results of an experiment are announced or published, it is normal practice for independent researchers to double-check how the research was performed, and to follow up by performing similar experiments to determine how dependable the results might be.  Taken in its entirety, the scientific method allows for highly creative problem solving while minimizing any effects of subjective bias on the part of its users (namely the confirmation bias).

Mathematics and formal sciences 

Mathematics is essential to the sciences. One important function of mathematics in science is the role it plays in the expression of scientific models. Observing and collecting measurements, as well as hypothesizing and predicting, often require extensive use of mathematics. Arithmetic, algebra, geometry, trigonometry and calculus, for example, are all essential to physics. Virtually every branch of mathematics has applications in science, including "pure" areas such as number theory and topology.

Statistical methods, which are mathematical techniques for summarizing and analyzing data, allow scientists to assess the level of reliability and the range of variation in experimental results. Statistical analysis plays a fundamental role in many areas of both the natural sciences and social sciences.

Computational science applies computing power to simulate real-world situations, enabling a better understanding of scientific problems than formal mathematics alone can achieve. According to the Society for Industrial and Applied Mathematics, computation is now as important as theory and experiment in advancing scientific knowledge. 

Whether mathematics itself is properly classified as science has been a matter of some debate. Some thinkers see mathematicians as scientists, regarding physical experiments as inessential or mathematical proofs as equivalent to experiments. Others do not see mathematics as a science, since it does not require an experimental test of its theories and hypotheses. Mathematical theorems and formulas are obtained by logical derivations which presume axiomatic systems, rather than the combination of empirical observation and logical reasoning that has come to be known as the scientific method. In general, mathematics is classified as formal science, while natural and social sciences are classified as empirical sciences. 

Scientific community 

The scientific community is the group of all interacting scientists. It includes many sub-communities working on particular scientific fields, and within particular institutions; interdisciplinary and cross-institutional activities are also significant.

Branches and fields 

Scientific fields are commonly divided into two major groups: natural sciences, which study natural phenomena (including biological life), and social sciences, which study human behavior and societies. These groupings are empirical sciences, which means the knowledge must be based on observable phenomena and capable of being tested for its validity by other researchers working under the same conditions. There are also related disciplines that are grouped into interdisciplinary applied sciences, such as engineering and medicine. Within these categories are specialized scientific fields that can include parts of other scientific disciplines but often possess their own nomenclature and expertise. 

Mathematics, which is classified as a formal science,   has both similarities and differences with the empirical sciences (the natural and social sciences). It is similar to empirical sciences in that it involves an objective, careful and systematic study of an area of knowledge; it is different because of its method of verifying its knowledge, using a priori rather than empirical methods. The formal sciences, which also include statistics and logic, are vital to the empirical sciences. Major advances in formal science have often led to major advances in the empirical sciences. The formal sciences are essential in the formation of hypotheses, theories, and laws, both in discovering and describing how things work (natural sciences) and how people think and act (social sciences).

Apart from its broad meaning, the word "Science" sometimes may specifically refer to fundamental sciences (maths and natural sciences) alone. Science schools or faculties within many institutions are separate from those for medicine or engineering, which is an applied science.

Institutions 

Learned societies for the communication and promotion of scientific thought and experimentation have existed since the Renaissance period.  The oldest surviving institution is the Italian  which was established in 1603.  The respective National Academies of Science are distinguished institutions that exist in a number of countries, beginning with the British Royal Society in 1660  and the French  in 1666. 

International scientific organizations, such as the International Council for Science, have since been formed to promote cooperation between the scientific communities of different nations. Many governments have dedicated agencies to support scientific research. Prominent scientific organizations include, the National Science Foundation in the U.S., the National Scientific and Technical Research Council in Argentina, the academies of science of many nations, CSIRO in Australia,  in France, Max Planck Society and  in Germany, and in Spain, CSIC.

Literature 

An enormous range of scientific literature is published.  Scientific journals communicate and document the results of research carried out in universities and various other research institutions, serving as an archival record of science. The first scientific journals, Journal des Sçavans followed by the Philosophical Transactions, began publication in 1665. Since that time the total number of active periodicals has steadily increased. In 1981, one estimate for the number of scientific and technical journals in publication was 11,500.  The United States National Library of Medicine currently indexes 5,516 journals that contain articles on topics related to the life sciences. Although the journals are in 39 languages, 91 percent of the indexed articles are published in English. 

Most scientific journals cover a single scientific field and publish the research within that field; the research is normally expressed in the form of a scientific paper. Science has become so pervasive in modern societies that it is generally considered necessary to communicate the achievements, news, and ambitions of scientists to a wider populace.

Science magazines such as New Scientist, Science & Vie, and Scientific American cater to the needs of a much wider readership and provide a non-technical summary of popular areas of research, including notable discoveries and advances in certain fields of research. Science books engage the interest of many more people. Tangentially, the science fiction genre, primarily fantastic in nature, engages the public imagination and transmits the ideas, if not the methods, of science.

Recent efforts to intensify or develop links between science and non-scientific disciplines such as Literature or, more specifically, Poetry, include the Creative Writing Science resource developed through the Royal Literary Fund. 

Science and society 

Women in science 

Science has traditionally been a male-dominated field, with some notable exceptions.Women in science have included:
* Hypatia (c. 350–415 CE), of the Library of Alexandria.
* Trotula of Salerno, a physician c. 1060 CE.
* Caroline Herschel one of the first professional astronomers of the 18th and 19th centuries.
* Christine Ladd-Franklin, a doctoral student of C. S. Peirce, who published Wittgenstein's proposition 5.101 in her dissertation, 40 years before Wittgenstein's publication of Tractatus Logico-Philosophicus.
* Henrietta Leavitt, a professional human computer and astronomer, who first published the significant relationship between the luminosity of Cepheid variable stars and their distance from Earth. This allowed Hubble to make the discovery of the expanding universe, which led to the Big Bang theory.
* Emmy Noether, who proved the conservation of energy and other constants of motion in 1915.
* Marie Curie, who made discoveries relating to radioactivity along with her husband, and for whom Curium is named.
* Rosalind Franklin, who worked with x-ray diffraction. Women historically faced considerable discrimination in science, much as they did in other areas of male-dominated societies, such as frequently being passed over for job opportunities and denied credit for their work.Nina Byers,[http://cwp.library.ucla.edu/ Contributions of 20th Century Women to Physics] which provides details on 83 female physicists of the 20th century. By 1976, more women were physicists, and the 83 who were detailed were joined by other women in noticeably larger numbers. For example, Christine Ladd (1847–1930) was able to enter a Ph.D. program as 'C. Ladd'; Christine "Kitty" Ladd completed the requirements in 1882, but was awarded her degree only in 1926, after a career which spanned the algebra of logic (see truth table), color vision, and psychology. Her work preceded notable researchers like Ludwig Wittgenstein and Charles Sanders Peirce. The achievements of women in science have been attributed to their defiance of their traditional role as laborers within the domestic sphere. 

In the late 20th century, active recruitment of women and elimination of institutional discrimination on the basis of sex greatly increased the number of women scientists, but large gender disparities remain in some fields; over half of new biologists are female, while 80% of PhDs in physics are given to men. Feminists claim this is the result of culture rather than an innate difference between the sexes, and some experiments have shown that parents challenge and explain more to boys than girls, asking them to reflect more deeply and logically.   In the early part of the 21st century, in America, women earned 50.3% bachelor's degrees, 45.6% master's degrees, and 40.7% of PhDs in science and engineering fields with women earning more than half of the degrees in three fields: Psychology (about 70%), Social Sciences (about 50%), and Biology (about 50-60%). However, when it comes to the Physical Sciences, Geosciences, Math, Engineering, and Computer Science; women earned less than half the degrees.  However, lifestyle choice also plays a major role in female engagement in science; women with young children are 28% less likely to take tenure-track positions due to work-life balance issues,  and female graduate students' interest in careers in research declines dramatically over the course of graduate school, whereas that of their male colleagues remains unchanged. 

Science policy 

Science policy is an area of public policy concerned with the policies that affect the conduct of the scientific enterprise, including research funding, often in pursuance of other national policy goals such as technological innovation to promote commercial product development, weapons development, health care and environmental monitoring.  Science policy also refers to the act of applying scientific knowledge and consensus to the development of public policies.  Science policy thus deals with the entire domain of issues that involve the natural sciences.  In accordance with public policy being concerned about the well-being of its citizens, science policy's goal is to consider how science and technology can best serve the public.

State policy has influenced the funding of public works and science for thousands of years, dating at least from the time of the Mohists, who inspired the study of logic during the period of the Hundred Schools of Thought, and the study of defensive fortifications during the Warring States period in China. In Great Britain, governmental approval of the Royal Society in the 17th century recognized a scientific community which exists to this day.  The professionalization of science, begun in the 19th century, was partly enabled by the creation of scientific organizations such as the National Academy of Sciences, the Kaiser Wilhelm Institute, and State funding of universities of their respective nations.  Public policy can directly affect the funding of capital equipment, intellectual infrastructure for industrial research, by providing tax incentives to those organizations that fund research. Vannevar Bush, director of the Office of Scientific Research and Development for the United States government, the forerunner of the National Science Foundation, wrote in July 1945 that "Science is a proper concern of government". 

Science and technology research is often funded through a competitive process, in which potential research projects are evaluated and only the most promising receive funding. Such processes, which are run by government, corporations or foundations, allocate scarce funds. Total research funding in most developed countries is between 1.5% and 3% of GDP.   In the OECD, around two-thirds of research and development in scientific and technical fields is carried out by industry, and 20% and 10% respectively by universities and government.  The government funding proportion in certain industries is higher, and it dominates research in social science and humanities. Similarly, with some exceptions (e.g. biotechnology) government provides the bulk of the funds for basic scientific research.  In commercial research and development, all but the most research-oriented corporations focus more heavily on near-term commercialisation possibilities rather than "blue-sky" ideas or technologies (such as nuclear fusion).

Media perspectives 

The mass media face a number of pressures that can prevent them from accurately depicting competing scientific claims in terms of their credibility within the scientific community as a whole. Determining how much weight to give different sides in a scientific debate may require considerable expertise regarding the matter.  Few journalists have real scientific knowledge, and even beat reporters who know a great deal about certain scientific issues may be ignorant about other scientific issues that they are suddenly asked to cover.  

Political usage 

Many issues damage the relationship of science to the media and the use of science and scientific arguments by politicians. As a very broad generalisation, many politicians seek certainties and facts whilst scientists typically offer probabilities and caveats. However, politicians' ability to be heard in the mass media frequently distorts the scientific understanding by the public. Examples in the United Kingdom include the controversy over the MMR inoculation, and the 1988 forced resignation of a Government Minister, Edwina Currie for revealing the high probability that battery farmed eggs were contaminated with Salmonella. 

John Horgan, Chris Mooney, and researchers from the US and Canada have described Scientific Certainty Argumentation Methods (SCAMs), where an organization or think tank makes it their only goal to cast doubt on supported science because it conflicts with political agendas. The memo reads "Doubt is our product since it is the best means of competing with the 'body of fact' that exists in the mind of the general public. It is also the means of establishing a controversy." [http://onlinelibrary.wiley.com/doi/10.1111/j.1475-682X.2008.00219.x/abstract William R. Freudenburg, Robert Gramling, Debra J. Davidson (2008)] "Scientific Certainty Argumentation Methods (SCAMs): Science and the politics of doubt". Sociological Inquiry. Vol. 78, No. 1. 2–38 Hank Campbell and microbiologist Alex Berezow have described "feel-good fallacies" used in politics, where politicians frame their positions in a way that makes people feel good about supporting certain policies even when scientific evidence shows there is no need to worry or there is no need for dramatic change on current programs. 

Science and the public 

Various activities are developed to approximate the general public and science/scientists, such as in science outreach,  public awareness of science, science communication, science festivals, citizen science, science journalism, public science, popular science, etc.; see Science and the public for related concepts. Science is represented by the 'S' in STEM fields.

Philosophy of science 

Working scientists usually take for granted a set of basic assumptions that are needed to justify the scientific method: (1) that there is an objective reality shared by all rational observers; (2) that this objective reality is governed by natural laws; (3) that these laws can be discovered by means of systematic observation and experimentation.  Philosophy of science seeks a deep understanding of what these underlying assumptions mean and whether they are valid.

The belief that scientific theories should and do represent metaphysical reality is known as realism.  It can be contrasted with anti-realism, the view that the success of science does not depend on it being accurate about unobservable entities such as electrons.  One form of anti-realism is idealism, the belief that the mind or consciousness is the most basic essence, and that each mind generates its own reality.This realization is the topic of intersubjective verifiability, as recounted, for example, by Max Born (1949, 1965) [https://archive.org/stream/naturalphilosoph032159mbp/naturalphilosoph032159mbp_djvu.txt Natural Philosophy of Cause and Chance], who points out that all knowledge, including natural or social science, is also subjective. p. 162: "Thus it dawned upon me that fundamentally everything is subjective, everything without exception. That was a shock."  In an idealistic world view, what is true for one mind need not be true for other minds.

There are different schools of thought in philosophy of science. The most popular position is empiricism,In his investigation of the law of falling bodies, Galileo (1638) serves as example for scientific investigation: Two New Sciences "A piece of wooden moulding or scantling, about 12 cubits long, half a cubit wide, and three finger-breadths thick, was taken; on its edge was cut a channel a little more than one finger in breadth; having made this groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball. Having placed this board in a sloping position, by lifting one
end some one or two cubits above the other, we rolled the ball, as I was just saying, along the channel, noting, in a manner presently to be described, the time required to make the descent. We . . . now rolled the ball only one-quarter the length of the channel; and having measured the time of its descent, we found it precisely one-half of the former. Next we tried other distances, comparing the time for the whole length with that for the half, or with that for two-thirds, or three-fourths, or indeed for any fraction; in such experiments, repeated many, many, times."  Galileo solved the problem of time measurement by weighing a jet of water collected during the descent of the bronze ball, as stated in his Two New Sciences. which holds that knowledge is created by a process involving observation and that scientific theories are the result of generalizations from such observations.  Empiricism generally encompasses inductivism, a position that tries to explain the way general theories can be justified by the finite number of observations humans can make and hence the finite amount of empirical evidence available to confirm scientific theories. This is necessary because the number of predictions those theories make is infinite, which means that they cannot be known from the finite amount of evidence using deductive logic only. Many versions of empiricism exist, with the predominant ones being bayesianism   and the hypothetico-deductive method.

Empiricism has stood in contrast to rationalism, the position originally associated with Descartes, which holds that knowledge is created by the human intellect, not by observation. Critical rationalism is a contrasting 20th-century approach to science, first defined by Austrian-British philosopher Karl Popper. Popper rejected the way that empiricism describes the connection between theory and observation. He claimed that theories are not generated by observation, but that observation is made in the light of theories and that the only way a theory can be affected by observation is when it comes in conflict with it. Popper proposed replacing verifiability with falsifiability as the landmark of scientific theories, and replacing induction with falsification as the empirical method. Popper further claimed that there is actually only one universal method, not specific to science: the negative method of criticism, trial and error.  It covers all products of the human mind, including science, mathematics, philosophy, and art. 

Another approach, instrumentalism, colloquially termed "shut up and calculate", emphasizes the utility of theories as instruments for explaining and predicting phenomena.  It views scientific theories as black boxes with only their input (initial conditions) and output (predictions) being relevant. Consequences, theoretical entities and logical structure are claimed to be something that should simply be ignored and that scientists shouldn't make a fuss about (see interpretations of quantum mechanics). Close to instrumentalism is constructive empiricism, according to which the main criterion for the success of a scientific theory is whether what it says about observable entities is true.

Paul K Feyerabend advanced the idea of epistemological anarchism, which holds that there are no useful and exception-free methodological rules governing the progress of science or the growth of knowledge, and that the idea that science can or should operate according to universal and fixed rules is unrealistic, pernicious and detrimental to science itself. Feyerabend advocates treating science as an ideology alongside others such as religion, magic and mythology, and considers the dominance of science in society authoritarian and unjustified. He also contended (along with Imre Lakatos) that the demarcation problem of distinguishing science from pseudoscience on objective grounds is not possible and thus fatal to the notion of science running according to fixed, universal rules. Feyerabend also stated that science does not have evidence for its philosophical precepts, particularly the notion of Uniformity of Law and the Uniformity of Process across time and space. 

Finally, another approach often cited in debates of scientific skepticism against controversial movements like "scientific creationism", is methodological naturalism. Its main point is that a difference between natural and supernatural explanations should be made, and that science should be restricted methodologically to natural explanations. credits Willard Van Orman Quine (1969) "Epistemology Naturalized" Ontological Relativity and Other Essays New York: Columbia University Press, as well as John Dewey, with the basic ideas of naturalism — Naturalized Epistemology, but Godfrey-Smith diverges from Quine's position: according to Godfrey-Smith, "A naturalist can think that science can contribute to answers to philosophical questions, without thinking that philosophical questions can be replaced by science questions.". That the restriction is merely methodological (rather than ontological) means that science should not consider supernatural explanations itself, but should not claim them to be wrong either. Instead, supernatural explanations should be left a matter of personal belief outside the scope of science. Methodological naturalism maintains that proper science requires strict adherence to empirical study and independent verification as a process for properly developing and evaluating explanations for observable phenomena.  The absence of these standards, arguments from authority, biased observational studies and other common fallacies are frequently cited by supporters of methodological naturalism as characteristic of the non-science they criticize.

Certainty and science 

A scientific theory is empirical,  and is always open to falsification if new evidence is presented. That is, no theory is ever considered strictly certain as science accepts the concept of fallibilism."No amount of experimentation can ever prove me right; a single experiment can prove me wrong." —Albert Einstein, noted by Alice Calaprice (ed. 2005) The New Quotable Einstein Princeton University Press and Hebrew University of Jerusalem, ISBN 0-691-12074-9 p. 291. Calaprice denotes this not as an exact quotation, but as a paraphrase of a translation of A. Einstein's "Induction and Deduction". Collected Papers of Albert Einstein 7 Document 28. Volume 7 is The Berlin Years: Writings, 1918–1921. A. Einstein; M. Janssen, R. Schulmann, et al., eds. The philosopher of science Karl Popper sharply distinguishes truth from certainty. He writes that scientific knowledge "consists in the search for truth", but it "is not the search for certainty ... All human knowledge is fallible and therefore uncertain."

New scientific knowledge rarely results in vast changes in our understanding. According to psychologist Keith Stanovich, it may be the media's overuse of words like "breakthrough" that leads the public to imagine that science is constantly proving everything it thought was true to be false. While there are such famous cases as the theory of relativity that required a complete reconceptualization, these are extreme exceptions. Knowledge in science is gained by a gradual synthesis of information from different experiments, by various researchers, across different branches of science; it is more like a climb than a leap. Theories vary in the extent to which they have been tested and verified, as well as their acceptance in the scientific community. Claims that before a specific fact "existed", it had to be created as part of a social agreement within a community. Steven Shapin (1980) "A view of scientific thought" Science ccvii (Mar 7, 1980) 1065–66 states "[To Fleck,] facts are invented, not discovered. Moreover, the appearance of scientific facts as discovered things is itself a social construction: a made thing. " For example, heliocentric theory, the theory of evolution, relativity theory, and germ theory still bear the name "theory" even though, in practice, they are considered factual. 
Philosopher Barry Stroud adds that, although the best definition for "knowledge" is contested, being skeptical and entertaining the possibility that one is incorrect is compatible with being correct. Ironically then, the scientist adhering to proper scientific approaches will doubt themselves even once they possess the truth.  The fallibilist C. S. Peirce argued that inquiry is the struggle to resolve actual doubt and that merely quarrelsome, verbal, or hyperbolic doubt is fruitless —but also that the inquirer should try to attain genuine doubt rather than resting uncritically on common sense.  He held that the successful sciences trust, not to any single chain of inference (no stronger than its weakest link), but to the cable of multiple and various arguments intimately connected. 

Stanovich also asserts that science avoids searching for a "magic bullet"; it avoids the single-cause fallacy. This means a scientist would not ask merely "What is the cause of ...", but rather "What are the most significant causes of ...". This is especially the case in the more macroscopic fields of science (e.g. psychology, cosmology).  Of course, research often analyzes few factors at once, but these are always added to the long list of factors that are most important to consider. For example: knowing the details of only a person's genetics, or their history and upbringing, or the current situation may not explain a behavior, but a deep understanding of all these variables combined can be very predictive.

Fringe science, pseudoscience and junk science 

An area of study or speculation that masquerades as science in an attempt to claim a legitimacy that it would not otherwise be able to achieve is sometimes referred to as pseudoscience, fringe science, or junk science."Pseudoscientific – pretending to be scientific, falsely represented as being scientific", from the Oxford American Dictionary, published by the Oxford English Dictionary; Hansson, Sven Ove (1996)."Defining Pseudoscience", Philosophia Naturalis, 33: 169–176, as cited in [http://plato.stanford.edu/entries/pseudo-science/#NonSciPosSci "Science and Pseudo-science"] (2008) in Stanford Encyclopedia of Philosophy. The Stanford article states: "Many writers on pseudoscience have emphasized that pseudoscience is non-science posing as science. The foremost modern classic on the subject (Gardner 1957) bears the title Fads and Fallacies in the Name of Science. According to Brian Baigrie (1988, 438), "[w]hat is objectionable about these beliefs is that they masquerade as genuinely scientific ones." These and many other authors assume that to be pseudoscientific, an activity or a teaching has to satisfy the following two criteria (Hansson 1996): (1) it is not scientific, and (2) its major proponents try to create the impression that it is scientific".
* For example, Hewitt et al. Conceptual Physical Science Addison Wesley; 3 edition (July 18, 2003) ISBN 0-321-05173-4, Bennett et al. The Cosmic Perspective 3e Addison Wesley; 3 edition (July 25, 2003) ISBN 0-8053-8738-2; See also, e.g., Gauch HG Jr. Scientific Method in Practice (2003).
* A 2006 National Science Foundation report on Science and engineering indicators quoted Michael Shermer's (1997) definition of pseudoscience: '"claims presented so that they appear [to be] scientific even though they lack supporting evidence and plausibility"(p. 33). In contrast, science is "a set of methods designed to describe and interpret observed and inferred phenomena, past or present, and aimed at building a testable body of knowledge open to rejection or confirmation"(p. 17)'. as cited by 
* "A pretended or spurious science; a collection of related beliefs about the world mistakenly regarded as being based on scientific method or as having the status that scientific truths now have," from the Oxford English Dictionary, second edition 1989. Physicist Richard Feynman coined the term "cargo cult science" for cases in which researchers believe they are doing science because their activities have the outward appearance of science but actually lack the "kind of utter honesty" that allows their results to be rigorously evaluated.[http://neurotheory.columbia.edu/~ken/cargo_cult.html Cargo Cult Science] by Feynman, Richard. Retrieved July 21, 2011. Various types of commercial advertising, ranging from hype to fraud, may fall into these categories.

There also can be an element of political or ideological bias on all sides of scientific debates. Sometimes, research may be characterized as "bad science", research that may be well-intended but is actually incorrect, obsolete, incomplete, or over-simplified expositions of scientific ideas. The term "scientific misconduct" refers to situations such as where researchers have intentionally misrepresented their published data or have purposely given credit for a discovery to the wrong person.

Scientific practice 

Although encyclopedias such as Pliny (fl. 77 AD) Natural History offered purported fact, they proved unreliable. A skeptical point of view, demanding a method of proof, was the practical position taken to deal with unreliable knowledge. As early as 1000 years ago, scholars such as Alhazen (Doubts Concerning Ptolemy), Roger Bacon, Witelo, John Pecham, Francis Bacon (1605), and C. S. Peirce (1839–1914) provided the  community  to address these points of uncertainty. In particular, fallacious reasoning can be exposed, such as 'affirming the consequent'.
 The methods of inquiry into a problem have been known for thousands of years,  and extend beyond theory to practice. The use of measurements, for example, is a practical approach to settle disputes in the community.

John Ziman points out that intersubjective pattern recognition is fundamental to the creation of all scientific knowledge.  Ziman shows how scientists can identify  patterns to each other across centuries: Ziman refers to this ability as 'perceptual consensibility'. Ziman then makes consensibility, leading to consensus, the touchstone of reliable knowledge.

Basic and applied research 

Although some scientific research is applied research into specific problems, a great deal of our understanding comes from the curiosity-driven undertaking of basic research. This leads to options for technological advance that were not planned or sometimes even imaginable. This point was made by Michael Faraday when, allegedly in response to the question "what is the use of basic research?" he responded "Sir, what is the use of a new-born child?".  For example, research into the effects of red light on the human eye's rod cells did not seem to have any practical purpose; eventually, the discovery that our night vision is not troubled by red light would lead search and rescue teams (among others) to adopt red light in the cockpits of jets and helicopters.
In a nutshell: Basic research is the search for knowledge. Applied research is the search for solutions to practical problems using this knowledge. Finally, even basic research can take unexpected turns, and there is some sense in which the scientific method is built to harness luck.

Research in practice 

Due to the increasing complexity of information and specialization of scientists, most of the cutting-edge research today is done by well funded groups of scientists, rather than individuals. D.K. Simonton notes that due to the breadth of very precise and far reaching tools already used by researchers today and the amount of research generated so far, creation of new disciplines or revolutions within a discipline may no longer be possible as it is unlikely that some phenomenon that merits its own discipline has been overlooked. Hybridizing of disciplines and finessing knowledge is, in his view, the future of science.

Practical impacts of scientific research 

Discoveries in fundamental science can be world-changing. For example: Subtraction is a mathematical operation that represents the operation of removing objects from a collection. It is signified by the minus sign (−). For example, in the picture on the right, there are 5 − 2 apples—meaning 5 apples with 2 taken away, which is a total of 3 apples. Therefore, 5 − 2 = 3. Besides counting fruits, subtraction can also represent combining other physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices.

Subtraction follows several important patterns. It is anticommutative, meaning that changing the order changes the sign of the answer. It is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Subtraction of 0 does not change a number. Subtraction also obeys predictable rules concerning related operations such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond. General binary operations that continue these patterns are studied in abstract algebra.

Performing subtraction is one of the simplest numerical tasks. Subtraction of very small numbers is accessible to young children. In primary education, students are taught to subtract numbers in the decimal system, starting with single digits and progressively tackling more difficult problems.

Notation and terminology

Subtraction is written using the minus sign "−" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example,
2 - 1 = 1  (verbally, "two minus one equals one")
4 - 2 = 2  (verbally, "four minus two equals two")
6 - 3 = 3  (verbally, "six minus three equals three")
4 - 6 = -2  (verbally, "four minus six equals negative two")

There are also situations where subtraction is "understood" even though no symbol appears:
*A column of two numbers, with the lower number in red, usually indicates that the lower number in the column is to be subtracted, with the difference written below, under a line. This is most common in accounting.

Formally, the number being subtracted is known as the subtrahend, while the number it is subtracted from is the minuend. The result is the difference.

All of this terminology derives from Latin. "Subtraction" is an English word derived from the Latin verb subtrahere, which is in turn a compound of sub "from under" and trahere "to pull"; thus to subtract is to draw from below, take away.  Using the gerundive suffix -nd results in "subtrahend", "thing to be subtracted".  Likewise from minuere "to reduce or diminish", one gets "minuend", "thing to be diminished".

Of integers and real numbers

Integers

Imagine a line segment of length b with the left end labeled a and the right end labeled c. 
Starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition:
a + b = c.

From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction:
c − b = a.

Now, a line segment labeled with the numbers 1, 2, and 3. 
From position 3, it takes no steps to the left to stay at 3, so 3 − 0 3. It takes 2 steps to the left to get to position 1, so 3 − 2 
 1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3.
To represent such an operation, the line must be extended.

To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, 5, 6, ...).  
From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0.
But 3 − 4 is still invalid since it again leaves the line.
The natural numbers are not a useful context for subtraction.

The solution is to consider the integer number line (..., −3, −2, −1, 0, 1, 2, 3, ...). From 3, it takes 4 steps to the left to get to −1:
3 − 4 = −1.

Natural numbers

Subtraction of natural numbers is not closed. The difference is not a natural number unless the minuend is greater than or equal to the subtrahend. For example, 26 cannot be subtracted from 11 to give a natural number. Such a case uses one of two approaches:
# Say that 26 cannot be subtracted from 11; subtraction becomes a partial function.
# Give the answer as an integer representing a negative number, so the result of subtracting 26 from 11 is -15.

Real numbers

Subtraction of real numbers is defined as addition of signed numbers.  Specifically, a number is subtracted by adding its additive inverse. Then we have 3 − π = 3 + (−π). This helps to keep the ring of real numbers "simple" by avoiding the introduction of "new" operators such as subtraction. Ordinarily a ring only has two operations defined on it; in the case of the integers, these are addition and multiplication. A ring already has the concept of additive inverses, but it does not have any notion of a separate subtraction operation, so the use of signed addition as subtraction allows us to apply the ring axioms to subtraction without needing to prove anything.

Properties

Anticommutativity

Subtraction is anti-commutative, meaning that if one reverses the terms in a difference left-to-right, the result is the negative of the original result. Symbolically, if a and b are any two numbers, then
a − b = −(b − a).

Non-associativity

Subtraction is non-associative, which comes up when one tries to define repeated subtraction. Should the expression
"a − b − c"
be defined to mean (a − b) − c or a − (b − c)? These two possibilities give different answers. To resolve this issue, one must establish an order of operations, with different orders giving different results.

Predecessor

In the context of integers, subtraction of one also plays a special role: for any integer a, the integer (a − 1) is the largest integer less than a, also known as the predecessor of a.

Units of measurement

When subtracting two numbers with units of measurement such as kilograms or pounds, they must have the same unit. In most cases the difference will have the same unit as the original numbers.

Percentages

Changes in percentages can be reported in at least two forms, percentage change and percentage point change. Percentage change represents the relative change between the two quantities as a percentage, while percentage point change is simply the number obtained by subtracting the two percentages.   

As an example, suppose that 30% of widgets made in a factory are defective. Six months later, 20% of widgets are defective. The percentage change is -33 1/3%, while the percentage point change is -10 percentage points.

In computing

The method of complements is a technique used to subtract one number from another using only addition of positive numbers. This method was commonly used in mechanical calculators and is still used in modern computers.

To subtract a binary number y (the subtrahend) from another number x (the minuend), the ones' complement of y is added to x and one is added to the sum.  The leading digit  '1' of the result is then discarded.

The method of complements is especially useful in binary (radix 2) since the ones' complement is very easily obtained by inverting each bit (changing '0' to '1' and vice versa). And adding 1 to get the two's complement can be done by simulating a carry into the least significant bit. For example:

   01100100  (x, equals decimal 100)
 - 00010110  (y, equals decimal 22)

becomes the sum:

   01100100  (x)
 + 11101001  (ones' complement of y)
 +        1  (to get the two's complement)
 ——————————
  101001110

Dropping the initial "1" gives the answer: 01001110 (equals decimal 78)

The teaching of subtraction in schools

Methods used to teach subtraction to elementary school vary from country to country, and within a country, different methods are in fashion at different times. In what is, in the United States, called traditional mathematics, a specific process is taught to students at the end of the 1st year or during the 2nd year for use with multi-digit whole numbers, and is extended in either the fourth or fifth grade to include decimal representations of fractional numbers.

In America

Almost all American schools currently teach a method of subtraction using borrowing or regrouping  (the decomposition algorithm) and a system of markings called crutches.   Although a method of borrowing had been known and published in textbooks previously, the use of crutches in American schools spread after William A. Brownell published a study claiming that crutches were beneficial to students using this method.  This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.

In Europe

Some European schools employ a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid memory), which vary by country.  

Comparing the two main methods

Both these methods break up the subtraction as a process of one digit subtractions by place value. Starting with a least significant digit, a subtraction of subtrahend:
sj sj−1 ... s1
from minuend 
mk mk−1 ... m1,
where each si and mi is a digit, proceeds by writing down m1 − s1, m2 − s2, and so forth, as long as si does not exceed mi. Otherwise, mi is increased by 10 and some other digit is modified to correct for this increase. The American method corrects by attempting to decrease the minuend digit mi+1 by one (or continuing the borrow leftwards until there is a non-zero digit from which to borrow). The European method corrects by increasing the subtrahend digit si+1 by one.

Example: 704 − 512.

The minuend is 704, the subtrahend is 512. The minuend digits are m3 7, m2 
 0
and m1 4. The subtrahend digits are s3 
 5, s2 1 and s1 
 2. Beginning at the one's place, 4 is not less than 2 so the difference 2 is written down in the result's one place. In the ten's place, 0 is less than 1, so the 0 is increased by 10, and the difference with 1, which is 9, is written down in the ten's place. The American method corrects for the increase of ten by reducing the digit in the minuend's hundreds place by one. That is, the 7 is struck through and replaced by a 6. The subtraction then proceeds in the hundreds place, where 6 is not less than 5, so the difference is written down in the result's hundred's place. We are now done, the result is 192.

The Austrian method does not reduce the 7 to 6. Rather it increases the subtrahend hundred's digit by one. A small mark is made near or below this digit (depending on the school). Then the subtraction proceeds by asking what number when increased by 1, and 5 is added to it, makes 7. The answer is 1, and is written down in the result's hundred's place.

There is an additional subtlety in that the student always employs a mental subtraction table in the American method. The Austrian method often encourages the student to mentally use the addition table in reverse. In the example above, rather than adding 1 to 5, getting 6, and subtracting that from 7, the student is asked to consider what number, when increased by 1, and 5 is added to it, makes 7.

Subtraction by hand 

Austrian method 

Example:

File:Vertical Subtraction Method B Step 1.JPG|1 + … = 3
File:Vertical Subtraction Method B Step 2.JPG|The difference is written under the line.
File:Vertical Subtraction Method B Step 3.JPG|9 + … = 5The required sum (5) is too small!
File:Vertical Subtraction Method B Step 4.JPG|So, we add 10 to it and put a 1 under the next higher place in the subtrahend.
File:Vertical Subtraction Method B Step 5.JPG|9 + … = 15Now we can find the difference like before.
File:Vertical Subtraction Method B Step 6.JPG|(4 + 1) + … = 7
File:Vertical Subtraction Method B Step 7.JPG|The difference is written under the line.
File:Vertical Subtraction Method B Step 8.JPG|The total difference.

Subtraction from left to right 

Example:

File:LeftToRight Subtraction Step 1.JPG|7 − 4 = 3This result is only penciled in.
File:LeftToRight Subtraction Step 2.JPG|Because the next digit of the minuend is smaller than the subtrahend, we subtract one from our penciled-in-number and mentally add ten to the next.
File:LeftToRight Subtraction Step 3.JPG|15 − 9 = 6
File:LeftToRight Subtraction Step 4.JPG|Because the next digit in the minuend is not smaller than the subtrahend, We keep this number.
File:LeftToRight Subtraction Step 5.JPG|3 − 1 = 2

American method 

In this method, each digit of the subtrahend is subtracted from the digit above it starting from right to left. If the top number is too small to subtract the bottom number from it, we add 10 to it; this 10 is 'borrowed' from the top digit to the left, which we subtract 1 from. Then we move on to subtracting the next digit and borrowing as needed, until every digit has been subtracted.
Example:

File:Vertical Subtraction Method A Step 1.JPG|3 − 1 = …
File:Vertical Subtraction Method A Step 2.JPG|We write the difference under the line.
File:Vertical Subtraction Method A Step 3.JPG|5 − 9 = …The minuend (5) is too small!
File:Vertical Subtraction Method A Step 4.JPG|So, we add 10 to it. The 10 is 'borrowed' from the digit on the left, which goes down by 1. 
File:Vertical Subtraction Method A Step 5.JPG|15 − 9 = …Now the subtraction works, and we write the difference under the line.
File:Vertical Subtraction Method A Step 6.JPG|6 − 4 = …
File:Vertical Subtraction Method A Step 7.JPG|We write the difference under the line.
File:Vertical Subtraction Method A Step 8.JPG|The total difference.

Trade first 

A variant of the American method where all borrowing is done before all subtraction. 

Example:

File:Trade First Subtraction Step 1.JPG|1 − 3 =  not possible.We add a 10 to the 1. Because the 10 is 'borrowed' from the nearby 5, the 5 is lowered by 1.
File:Trade First Subtraction Step 2.JPG|4 − 9 = not possible.So we proceed as in step 1.
File:Trade First Subtraction Step 3.JPG|Working from right to left:11 − 3 = 8
File:Trade First Subtraction Step 4.JPG|14 − 9 = 5
File:Trade First Subtraction Step 5.JPG|6 − 4 = 2

Partial differences 

The partial differences method is different from other vertical subtraction methods because no borrowing or carrying takes place. In their place, one places plus or minus signs depending on whether the minuend is greater or smaller than the subtrahend. The sum of the partial differences is the total difference. 

Example:

File:Partial-Differences Subtraction Step 1.JPG|The smaller number is subtracted from the greater:700 − 400 = 300Because the minuend is greater than the subtrahend, this difference has a plus sign.
File:Partial-Differences Subtraction Step 2.JPG|The smaller number is subtracted from the greater:90 − 50 = 40Because the minuend is smaller than the subtrahend, this difference has a minus sign.
File:Partial-Differences Subtraction Step 3.JPG|The smaller number is subtracted from the greater:3 − 1 = 2Because the minuend is greater than the subtrahend, this difference has a plus sign.
File:Partial-Differences Subtraction Step 4.JPG|+ 300 − 40 + 2 = 262

Nonvertical methods 

Counting up 

Instead of finding the difference digit by digit, one can count up the numbers between the subtrahend and the minuend. 

Example:

1234 − 567 = can be found by the following steps:
* 567 + 3 = 570
* 570 + 30 = 600
* 600 + 400 = 1000
* 1000 + 234 = 1234
Add up the value from each step to get the total difference: 3 + 30 + 400 + 234 = 667.

Breaking up the subtraction 

Another method that is useful for mental arithmetic is to split up the subtraction into small steps. 

Example:

1234 − 567 = can be solved in the following way:
* 1234 − 500 = 734
* 734 − 60 = 674
* 674 − 7 = 667

Same change 

The same change method uses the fact that adding or subtracting the same number from the minuend and subtrahend does not change the answer. One adds the amount needed to get zeros in the subtrahend. 

Example:

"1234 − 567 =" can be solved as follows:
*1234 − 567 1237 − 570 
 1267 − 600 = 667 Multiplication (often denoted by the cross symbol "×", by a point "·", or by the absence of symbol) is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division.

The multiplication of whole numbers may be thought as a repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the value of the other one, the multiplier. Normally, the multiplier is written first and multiplicand second,  though this can vary, as the distinction is not very meaningful:
a\times b \underbrace{b + \cdots + b}_a 
 \underbrace{a + \cdots + a}_b
For example, 4 multiplied by 3 (often written as  3 \times 4  and said as "3 times 4") can be calculated by adding 3 copies of 4 together:
3 \times 4 4 + 4 + 4 
 12
Here 3 and 4 are the "factors" and 12 is the "product".

One of the main properties of multiplication is the commutative property, adding 3 copies of 4 gives the same result as adding 4 copies of 3:
4 \times 3 3 + 3 + 3 + 3 
 12

The multiplication of integers (including negative numbers), rational numbers (fractions) and real numbers is defined by a systematic generalization of this basic definition.

Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers) or as finding the area of a rectangle whose sides have given lengths. The area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for instance multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis.

The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number (since the division of a number other than 0 by itself equals 1).

Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter.  A listing of the many different kinds of products that are used in mathematics is given in the product (mathematics) page.

Notation and terminology

In arithmetic, multiplication is often written using the sign "×" between the terms; that is, in infix notation.  For example,
2\times 3 = 6 (verbally, "two times three equals six")
3\times 4 = 12
2\times 3\times 5 6\times  5 
 30
2\times 2\times 2\times 2\times 2 = 32

The sign is encoded in Unicode at .

There are other mathematical notations for multiplication:

*Multiplication is also denoted by dot signs, usually a middle-position dot (rarely period):
5 \cdot 2 \quad\text{or}\quad 5\,.\,2
The middle dot notation, encoded in Unicode as , is standard in the United States, the United Kingdom, and other countries where the period is used as a decimal point. When the dot operator character is not accessible, the interpunct (·) is used. In other countries that use a comma as a decimal mark, either the period or a middle dot is used for multiplication.

*In algebra, multiplication involving variables is often written as a juxtaposition (e.g., xy for x times y or 5x for five times x). The notation can also be used for quantities that are surrounded by parentheses (e.g., 5(2) or (5)(2) for five times two). This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the order of operations.
*In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a cross product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar.

In computer programming, the asterisk (as in 5*2) is still the most common notation. This is due to the fact that most computers historically were limited to small character sets (such as ASCII and EBCDIC) that lacked a multiplication sign (such as • or ×), while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language.

The numbers to be multiplied are generally called the "factors".  The number to be multiplied is called the "multiplicand", while the number of times the multiplicand is to be multiplied comes from the "multiplier". Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand and the second the multiplier.  Additionally, there are some sources in which the term "multiplicand" is regarded as a synonym for "factor".  In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in 3xy2) is called a coefficient.

The result of a multiplication is called a product. A product of integers is a multiple of each factor. For example, 15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5.

Computation

The common methods for multiplying numbers using pencil and paper require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9), however one method, the peasant multiplication algorithm, does not. 

Multiplying numbers to more than a couple of decimal places by hand is tedious and error prone. Common logarithms were invented to simplify such calculations. The slide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early twentieth century, mechanical calculators, such as the Marchant, automated multiplication of up to 10 digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.

Historical algorithms

Methods of multiplication were documented in the Egyptian, Greek, Indian and Chinese civilizations.

The Ishango bone, dated to about 18,000 to 20,000 BC, hints at a knowledge of multiplication in the Upper Paleolithic era in Central Africa.

Egyptians

The Egyptian method of multiplication of integers and fractions, documented in the Ahmes Papyrus, was by successive additions and doubling.  For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining , , .  The full product could then be found by adding the appropriate terms found in the doubling sequence:
13 × 21 (1 + 4 + 8) × 21 
 (1 × 21) + (4 × 21) + (8 × 21) 21 + 84 + 168 
 273.

Babylonians

The Babylonians used a sexagesimal positional number system, analogous to the modern day decimal system.  Thus, Babylonian multiplication was very similar to modern decimal multiplication.  Because of the relative difficulty of remembering 60 × 60 different products, Babylonian mathematicians employed multiplication tables.  These tables consisted of a list of the first twenty multiples of a certain principal number n: n, 2n, ..., 20n; followed by the multiples of 10n: 30n 40n, and 50n.  Then to compute any sexagesimal product, say 53n, one only needed to add 50n and 3n computed from the table.

Chinese

In the mathematical text Zhou Bi Suan Jing, dated prior to 300 BC, and the Nine Chapters on the Mathematical Art, multiplication calculations were written out in words, although the early Chinese mathematicians employed  Rod calculus  involving place value addition, subtraction, multiplication and division. These place value decimal arithmetic algorithms were introduced by Al Khwarizmi to  Arab countries in the early 9th century.

Modern methods

The modern method of multiplication based on the Hindu–Arabic numeral system was first described by Brahmagupta.  Brahmagupta gave rules for addition, subtraction, multiplication and division. Henry Burchard Fine, then professor of Mathematics at Princeton University, wrote the following:
The Indians are the inventors not only of the positional decimal system itself, but of most of the processes involved in elementary reckoning with the system. Addition and subtraction they performed quite as they are performed nowadays; multiplication they effected in many ways, ours among them, but division they did cumbrously. 

Grid Method

Grid method multiplication or the box method, is used in primary schools in England and Wales to help teach and understanding of how multiple digit multiplication works. An example of multiplying 34 by 13 would be to lay the numbers out in a grid like:

and then add the entries. 

Computer algorithms

The classical method of multiplying two n-digit numbers requires n2 simple multiplications. Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers. In particular for very large numbers methods based on the Discrete Fourier Transform can reduce the number of simple multiplications to the order of n log2(n) log2 log2(n).

Products of measurements

One can only meaningfully add or subtract quantities of the same type but can multiply or divide quantities of different types. Four bags with three marbles each can be though of as:

[4 bags] × [3 marbles per bag] = 12 marbles.

When two measurements are multiplied together the product is of a type depending on the types of the measurements. The general theory is given by dimensional analysis. This analysis is routinely applied in physics but has also found applications in finance. 

A common example is multiplying speed by time gives distance, so

50 kilometers per hour × 3 hours = 150 kilometers.

Other examples:
2.5 \mbox{ meters} \times 4.5 \mbox{ meters} = 11.25 \mbox{ square meters}
11 \mbox{ meters/second} \times 9 \mbox{ seconds} = 99 \mbox{ meters}

Products of sequences

Capital Pi notation

The product of a sequence of terms can be written with the product symbol, which derives from the capital letter Π (Pi) in the Greek alphabet.  Unicode position U+220F (∏) contains a glyph for denoting such a product, distinct from U+03A0 (Π), the letter.  The meaning of this notation is given by:

 \prod_{i1}^4 i 
 1\cdot 2\cdot 3\cdot 4,  
that is
 \prod_{i1}^4 i 
 24. 
The subscript gives the symbol for a dummy variable (i in this case), called the "index of multiplication" together with its lower bound (1), whereas the superscript (here 4) gives its upper bound. The lower and upper bound are expressions denoting integers. The factors of the product are obtained by taking the expression following the product operator, with successive integer values substituted for the index of multiplication, starting from the lower bound and incremented by 1 up to and including the upper bound. So, for example:

 \prod_{i1}^6 i 
 1\cdot 2\cdot 3\cdot 4\cdot 5 \cdot 6 = 720 

More generally, the notation is defined as
 \prod_{im}^n x_i 
 x_m \cdot x_{m+1} \cdot x_{m+2} \cdot \,\,\cdots\,\, \cdot x_{n-1} \cdot x_n, 
where m and n are integers or expressions that evaluate to integers. In case m = n, the value of the product is the same as that of the single factor xm. If m > n, the product is the empty product, with the value 1.

Infinite products

One may also consider products of infinitely many terms; these are called infinite products. Notationally, we would replace n above by the lemniscate ∞. The product of such a series is defined as the limit of the product of the first n terms, as n grows without bound. That is, by definition,

 \prod_{im}^{\infty} x_{i} 
 \lim_{n\to\infty} \prod_{i=m}^{n} x_{i}. 

One can similarly replace m with negative infinity, and define:

\prod_{i-\infty}^\infty x_i 
 \left(\lim_{m\to-\infty}\prod_{im}^0 x_i\right) \cdot \left(\lim_{n\to\infty}\prod_{i
1}^n x_i\right),

provided both limits exist.

Properties

For the real and complex numbers, which includes for example natural numbers, integers and fractions, multiplication has certain properties:

;Commutative property
The order in which two numbers are multiplied does not matter:
:x\cdot y = y\cdot x.

;Associative property
Expressions solely involving multiplication or addition are invariant with respect to order of operations:
:(x\cdot y)\cdot z = x\cdot(y\cdot z)

;Distributive property
Holds with respect to multiplication over addition.  This identity is of prime importance in simplifying algebraic expressions:
:x\cdot(y + z) = x\cdot y + x\cdot z 

;Identity element
The multiplicative identity is 1; anything multiplied by one is itself. This is known as the identity property:
:x\cdot 1 = x

;Property of Zero
Any number multiplied by zero is zero. This is known as the zero property of multiplication:
:x\cdot  0 = 0

;Negation
Negative one times any number is equal to the additive inverse of that number.
:(-1)\cdot x = (-x)

Negative one times negative one is positive one.
:(-1)\cdot (-1) = 1

The natural numbers do not include negative numbers.

;Inverse element
Every number x, except zero, has a multiplicative inverse, \frac{1}{x}, such that x\cdot\left(\frac{1}{x}\right) = 1.

;Order preservation
Multiplication by a positive number preserves order:
:if a > 0, then if b > c then ab > ac.
Multiplication by a negative number reverses order:
:if a  c then ab x \times 0 = 0
x \times S(y) = (x \times y) + x

Here S(y) represents the successor of y, or the natural number that follows y. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic including induction. For instance S(0). denoted by 1, is a multiplicative identity because

x \times 1 x \times S(0) 
 (x \times 0) + x 0 + x 
 x 

The axioms for integers typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (x,y) as equivalent to x−y when x and y are treated as integers. Thus both (0,1) and (1,2) are equivalent to −1. The multiplication axiom for integers defined this way is

(x_p,\, x_m) \times (y_p,\, y_m) = (x_p \times y_p + x_m \times y_m,\; x_p \times y_m + x_m \times y_p)

The rule that −1 × −1 = 1 can then be deduced from

(0, 1) \times (0, 1) (0 \times 0 + 1 \times 1,\, 0 \times 1 + 1 \times 0) 
 (1,0)

Multiplication is extended in a similar way to rational numbers and then to real numbers.

Multiplication with set theory

The product of non-negative integers can be defined with set theory using cardinal numbers or the Peano axioms. See below how to extend this to multiplying arbitrary integers, and then arbitrary rational numbers. The product of real numbers is defined in terms of products of rational numbers, see construction of the real numbers.

Multiplication in group theory

There are many sets that, under the operation of multiplication, satisfy the axioms that define group structure.  These axioms are closure, associativity, and the inclusion of an identity element and inverses.

A simple example is the set of non-zero rational numbers.  Here we have identity 1, as opposed to groups under addition where the identity is typically 0.  Note that with the rationals, we must exclude zero because, under multiplication, it does not have an inverse: there is no rational number that can be multiplied by zero to result in 1.  In this example we have an abelian group, but that is not always the case.

To see this, look at the set of invertible square matrices of a given dimension, over a given field.  Now it is straightforward to verify closure, associativity, and inclusion of identity (the identity matrix) and inverses.  However, matrix multiplication is not commutative, therefore this group is nonabelian.

Another fact of note is that the integers under multiplication is not a group, even if we exclude zero.  This is easily seen by the nonexistence of an inverse for all elements other than 1 and -1.

Multiplication in group theory is typically notated either by a dot, or by juxtaposition (the omission of an operation symbol between elements).  So multiplying element a by element b could be notated a \cdot b or ab.  When referring to a group via the indication of the set and operation, the dot is used, e.g., our first example could be indicated by \left( \mathbb{Q}\smallsetminus \{ 0 \} ,\cdot \right) 

Multiplication of different kinds of numbers 

Numbers can count (3 apples), order (the 3rd apple), or measure (3.5 feet high); as the history of mathematics has progressed from counting on our fingers to modelling quantum mechanics, multiplication has been generalized to more complicated and abstract types of numbers, and to things that are not numbers (such as matrices) or do not look much like numbers (such as quaternions).

;Integers
N\times M is the sum of M copies of N when N and M are positive whole numbers. This gives the number of things in an array N wide and M high. Generalization to negative numbers can be done by
N\times (-M) (-N)\times M 
 - (N\times M) and
(-N)\times (-M) = N\times M
The same sign rules apply to rational and real numbers.

;Rational numbers
Generalization to fractions \frac{A}{B}\times \frac{C}{D} is by multiplying the numerators and denominators respectively:  \frac{A}{B}\times \frac{C}{D} = \frac{(A\times C)}{(B\times D)}. This gives the area of a rectangle \frac{A}{B} high and \frac{C}{D} wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.

;Real numbers
Real numbers and their products can be defined in terms of sequences of rational numbers.

;Complex numbers
Considering complex numbers z_1 and z_2 as ordered pairs of real numbers (a_1, b_1) and (a_2, b_2), the product z_1\times z_2 is (a_1\times a_2 - b_1\times b_2, a_1\times b_2 + a_2\times b_1). This is the same as for reals, a_1\times a_2, when the imaginary parts b_1 and b_2 are zero.

Equivalently, denoting \sqrt{-1} as i, we have z_1 \times z_2 (a_1+b_1i)(a_2+b_2i)
(a_1 \times a_2)+(a_1\times b_2i)+(b_1\times a_2i)+(b_1\times b_2i^2)=(a_1a_2-b_1b_2)+(a_1b_2+b_1a_2)i.

;Further generalizations
See Multiplication in group theory, above, and Multiplicative Group, which for example includes matrix multiplication. A very general, and abstract, concept of multiplication is as the "multiplicatively denoted" (second) binary operation in a ring. An example of a ring that is not any of the above number systems is a polynomial ring (you can add and multiply polynomials, but polynomials are not numbers in any usual sense.)

;Division
Often division, \frac{x}{y}, is the same as multiplication by an inverse, x\left(\frac{1}{y}\right). Multiplication for some types of "numbers" may have corresponding division, without inverses; in an integral domain x may have no inverse "\frac{1}{x}" but \frac{x}{y} may be defined. In a division ring there are inverses, but \frac{x}{y} may be ambiguous in non-commutative rings since x\left(\frac{1}{y}\right) need not the same as \left(\frac{1}{y}\right)x.

Exponentiation

When multiplication is repeated, the resulting operation is known as exponentiation.  For instance, the product of three factors of two (2×2×2) is "two raised to the third power", and is denoted by 23, a two with a superscript three.  In this example, the number two is the base, and three is the exponent.  In general, the exponent (or superscript) indicates how many times the base appears in the expression, so that the expression

a^n = \underbrace{a\times a \times \cdots  \times a}_n

indicates that n copies of the base a are to be multiplied together. This notation can be used whenever multiplication is known to be power associative. Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. The division of two natural numbers is the process of calculating the number of times one number is contained within one another.  For example, in the picture on the right, the 20 apples are divided into groups of five apples, and there exist four groups, meaning that five can be contained within 20 four times, or . Division can also be thought of as the process of evaluating a fraction, and fractional notation ( and ) is commonly used to represent division. 

Division is the inverse of multiplication; if , then , as long as  is not zero. Division by zero is undefined for the real numbers and most other contexts,  because if , then  cannot be deduced from  and , as then  will always equal zero regardless of . In some contexts, division by zero can be defined although to a limited extent, and limits involving division of a real number as it approaches zero are defined. 

In division, the dividend is divided by the divisor to get a quotient. In the above example, 20 is the dividend, five is the divisor, and the quotient is four. In some cases, the divisor may not be contained fully by the dividend; for example,  leaves a remainder of  as 10 is not a multiple of three. Normally, this remainder is added to the quotient so  would equal  or , but in the context of integer division, where numbers have no fractional part, the remainder is discarded. 

Besides dividing apples, division can be applied to other physical and abstract objects. Division has been defined in several contexts, such as for the real and complex numbers and for more abstract contexts such as for vector spaces and fields.

Teaching division usually leads to the concept of fractions being introduced to school pupils. Unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder. To complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called.

Notation 

Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a fraction bar, between them. For example, a divided by b is written
\frac ab
This can be read out loud as "a divided by b", "a by b" or "a over b". A way to express division all on one line is to write the dividend (or numerator), then a slash, then the divisor (or denominator), like this:

a/b\,

This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of ASCII characters. Some mathematical software, such as GNU Octave, allows the operands to be written in the reverse order by using the backslash as the division operator:

b\backslash a

A typographical variation halfway between these two forms uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:

Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (typically called the numerator and denominator), and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:

a \div b

This form is infrequent except in elementary arithmetic. ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.

In some non-English-speaking cultures, "a divided by b" is written a : b. This notation was introduced in 1631 by William Oughtred in his Clavis Mathematicae and later popularized by Gottfried Wilhelm Leibniz.  However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b").

In elementary classes of some countries, the notation b)~a or b \overline{)a} is used to denote a divided by b, especially when discussing long division; similarly, but less commonly, b \underline{)a} for short division (as shown in an example on that page). This notation was first introduced by Michael Stifel in Arithmetica integra, published in 1544.

Computing 

Manual methods

Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of sweets, into a number of equal portions. Distributing the objects several at a time in each round of sharing to each portion leads to the idea of "chunking", i.e., division by repeated subtraction.

More systematic and more efficient (but also more formalised and more rule-based, and more removed from an overall holistic picture of what division is achieving), a person who knows the multiplication tables can divide two integers using pencil and paper using the method of short division, if the divisor is simple. Long division is used for larger integer divisors. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, we can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.

A person can calculate division with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset.

A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.

A person can calculate division with a slide rule by aligning the divisor on the C scale with the dividend on the D scale. The quotient can be found on the D scale where it is aligned with the left index on the C scale. The user is responsible, however, for mentally keeping track of the decimal point.

By computer or with computer assistance

Modern computers compute division by methods that are faster than long division: see Division algorithm.

In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. We can calculate division by multiplication in such a case. This approach is useful in computers that do not have a fast division instruction.

Properties 

Division is right-distributive over addition and subtraction. That means:

\frac{a + b}{c} (a + b) \div c 
 \frac{a}{c} + \frac{b}{c}

in the same way as in multiplication (a + b) \times c = a \times c + b \times c. But division is not left-distributive, i.e. we have

\frac{a}{b + c} = a \div (b + c) \ne \frac{a}{b} + \frac{a}{c}

unlike multiplication.

Euclidean division 

The Euclidean division is the mathematical formulation of the outcome of the usual process of division of integers. It asserts that, given two integers, a, the dividend, and b, the divisor, such that b ≠ 0, there are unique integers q, the quotient, and r, the remainder,  such that a = bq + r and 0 ≤ r \tfrac{26}{11} \simeq 2.36 or \tfrac{26}{11} \simeq 2 \tfrac {36}{100}. This is the approach usually taken in numerical computation.
# Give the answer as a fraction representing a rational number, so the result of the division of 26 by 11 is \tfrac{26}{11}. But, usually, the resulting fraction should be simplified: the result of the division of 52 by 22 is also \tfrac{26}{11}. This simplification may be done by factoring out the greatest common divisor.
# Give the answer as an integer quotient and a remainder, so \tfrac{26}{11} = 2 \mbox{ remainder } 4. To make the distinction with the previous case, this division, with two integers as result, is sometimes called Euclidean division, because it is the basis of the Euclidean algorithm.
# Give the integer quotient as the answer, so \tfrac{26}{11} = 2. This is sometimes called integer division.
Dividing integers in a computer program requires special care. Some programming languages, such as C, treat integer division as in case 5 above, so the answer is an integer. Other languages, such as MATLAB and every computer algebra system return a rational number as the answer, as in case 3 above. These languages also provide functions to get the results of the other cases, either directly or from the result of case 3.

Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative: rounding may be toward zero (so called T-division) or toward −∞ (F-division); rarer styles can occur – see Modulo operation for the details.

Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another.

Of rational numbers 

The result of dividing two rational numbers is another rational number when the divisor is not 0. The division of two rational numbers p/q and r/s is defined as

{p/q \over r/s} {p \over q} \times {s \over r} 
 {ps \over qr}.

All four quantities are integers, and only p may be 0. This definition ensures that division is the inverse operation of multiplication.

Of real numbers 

Division of two real numbers results in another real number when the divisor is not 0. It is defined such a/b c if and only if a 
 cb and b ≠ 0.

By zero 

Division of any number by zero (where the divisor is zero) is undefined. This is because zero multiplied by any finite number always results in a product of zero. Entry of such an expression into most calculators produces an error message.

Of complex numbers 

Dividing two complex numbers results in another complex number when the divisor is not 0, which is defined as:

{p+iq \over r+is} {(p+iq)(r-is) \over (r+is)(r-is)} 
 {pr+qs + i(qr-ps) \over r^2+s^2} = {pr+qs \over r^2+s^2} + i{qr-ps \over r^2+s^2}.

This process of multiplying and dividing by r-is is called rationalisation. All four quantities p, q, r, s are real numbers, and r and s may not both be 0.

Division for complex numbers expressed in polar form is simpler than the definition above:

{p e^{iq} \over r e^{is}} {p e^{iq} e^{-is} \over r e^{is} e^{-is}} 
 {p \over r}e^{i(q - s)}. 

Again all four quantities p, q, r, s are real numbers, and r may not be 0.

Of polynomials 

One can define the division operation for polynomials in one variable over a field. Then, as in the case of integers, one has a remainder. See Euclidean division of polynomials, and, for hand-written computation, polynomial long division or synthetic division.

Of matrices 

One can define a division operation for matrices. The usual way to do this is to define , where  denotes the inverse of B, but it is far more common to write out  explicitly to avoid confusion.

Left and right division 

Because matrix multiplication is not commutative, one can also define a left division or so-called backslash-division as . For this to be well defined,  need not exist, however  does need to exist. To avoid confusion, division as defined by  is sometimes called right division or slash-division in this context.

Note that with left and right division defined this way,  is in general not the same as  and nor is  the same as , but  and .

Pseudoinverse 

To avoid problems when  and/or  do not exist, division can also be defined as multiplication with the pseudoinverse, i.e.,  and , where  and  denote the pseudoinverse of A and B.

In abstract algebra 

In abstract algebras such as matrix algebras and quaternion algebras, fractions such as {a \over b} are typically defined as a \cdot {1 \over b} or a \cdot b^{-1} where b is presumed an invertible element (i.e., there exists a multiplicative inverse b^{-1} such that bb^{-1} b^{-1}b 
 1 where 1 is the multiplicative identity). In an integral domain where such elements may not exist, division can still be performed on equations of the form ab ac or ba 
 ca by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any ring with the aforementioned cancellation properties. If such a ring is finite, then by an application of the pigeonhole principle, every nonzero element of the ring is invertible, so division by any nonzero element is possible in such a ring. To learn about when algebras (in the technical sense) have a division operation, refer to the page on division algebras. In particular Bott periodicity can be used to show that any real normed division algebra must be isomorphic to either the real numbers R, the complex numbers C, the quaternions H, or the octonions O.

Calculus 

The derivative of the quotient of two functions is given by the quotient rule:

{\left(\frac fg\right)}' = \frac{f'g - fg'}{g^2}. Arithmetic or arithmetics (from the Greek ἀριθμός arithmos, "number") is the oldest  and most elementary branch of mathematics. It consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are sometimes still used to refer to a wider part of number theory. 

History

The prehistory of arithmetic is limited to a small number of artifacts which may indicate the conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC, although its interpretation is disputed. 

The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC. These artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, descended from tally marks used for counting. In both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a counting board or the Roman abacus to obtain the results.

Early number systems that included positional notation were not decimal, including the sexagesimal (base 60) system for Babylonian numerals and the vigesimal (base 20) system that defined Maya numerals. Because of this place-value concept, the ability to reuse the same digits for different values contributed to simpler and more efficient methods of calculation.

The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, although it originated much later than the Babylonian and Egyptian examples. Prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, and their relationships to each other, in his Introduction to Arithmetic.

Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero (until the Hellenistic period), they used three separate sets of symbols. One set for the unit's place, one for the ten's place, and one for the hundred's. Then for the thousand's place they would reuse the symbols for the unit's place, and so on. Their addition algorithm was identical to ours, and their multiplication algorithm was only very slightly different. Their long division algorithm was the same, and the square root algorithm that was once taught in school was known to Archimedes, who may have invented it. He preferred it to Hero's method of successive approximation because, once computed, a digit doesn't change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a fractional part, such as 546.934, they used negative powers of 60 instead of negative powers of 10 for the fractional part 0.934.  The ancient Chinese used a similar positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the unit's place, and a second set for the ten's place. For the hundred's place they then reused the symbols for the unit's place, and so on. Their symbols were based on the ancient counting rods. It is a complicated question to determine exactly when the Chinese started calculating with positional representation, but it was definitely before 400 BC.  The Bishop of Syria, Severus Sebokht (650 AD), "Indians possess a method of calculation that no word can praise enough. Their rational system of mathematics, or of their method of calculation. I mean the system using nine symbols." 

Leonardo of Pisa (Fibonacci) in 1200 AD wrote in Liber Abaci "The method of the Indians (Modus Indoram) surpasses any known method to compute. It's a marvelous method. They do their computations using nine figures and symbol zero". 

The gradual development of Hindu–Arabic numerals independently devised the place-value concept and positional notation, which combined the simpler methods for computations with a decimal base and the use of a digit representing 0. This allowed the system to consistently represent both large and small integers. This approach eventually replaced all other systems. In the early  the Indian mathematician Aryabhata incorporated an existing version of this system in his work, and experimented with different notations. In the 7th century, Brahmagupta established the use of 0 as a separate number and determined the results for multiplication, division, addition and subtraction of zero and all other numbers, except for the result of division by 0. His contemporary, the Syriac bishop Severus Sebokht described the excellence of this system as "... valuable methods of calculation which surpass description". The Arabs also learned this new method and called it hesab.

Although the Codex Vigilanus described an early form of Arabic numerals (omitting 0) by 976 AD, Fibonacci was primarily responsible for spreading their use throughout Europe after the publication of his book Liber Abaci in 1202. He considered the significance of this "new" representation of numbers, which he styled the "Method of the Indians" (Latin Modus Indorum), so fundamental that all related mathematical foundations, including the results of Pythagoras and the algorism describing the methods for performing actual calculations, were "almost a mistake" in comparison.

In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities.

The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimal notation.

Various types of tools have been invented and widely used to assist in numeric calculations. Before Renaissance, they were various types of abaci. More recent examples include slide rules, nomograms and mechanical calculators, such as Pascal's calculator. Presently, they have been supplanted by electronic calculators and computers.

Arithmetic operations

The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots, exponentiation, and logarithmic functions. Arithmetic is performed according to an order of operations. Any set of objects upon which all four arithmetic operations (except division by 0) can be performed, and where these four operations obey the usual laws, is called a field.

Addition (+)

Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum of the numbers (Such as  or ).

Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series; repeated addition of the number 1 is the most basic form of counting.

Addition is commutative and associative so the order the terms are added in does not matter. The identity element of addition (the additive identity) is 0, that is, adding 0 to any number yields that same number. Also, the inverse element of addition (the additive inverse) is the opposite of any number, that is, adding the opposite of any number to the number itself yields the additive identity, 0. For example, the opposite of 7 is −7, so .

Addition can be given geometrically as in the following example:
If we have two sticks of lengths 2 and 5, then if we place the sticks one after the other, the length of the stick thus formed is .

Subtraction (−)

Subtraction is the inverse of addition. Subtraction finds the difference between two numbers, the minuend  minus the subtrahend. If the minuend is larger than the subtrahend, the difference is positive; if the minuend is smaller than the subtrahend, the difference is negative; if they are equal, the difference is 0.

Subtraction is neither commutative nor associative. For that reason, it is often helpful to look at subtraction as addition of the minuend and the opposite of the subtrahend, that is . When written as a sum, all the properties of addition hold.

There are several methods for calculating results, some of which are particularly advantageous to machine calculation. For example, digital computers employ the method of two's complement. Of great importance is the counting up method by which change is made. Suppose an amount P is given to pay the required amount Q, with P greater than Q. Rather than performing the subtraction  and counting out that amount in change, money is counted out starting at Q and continuing until reaching P. Although the amount counted out must equal the result of the subtraction , the subtraction was never really done and the value of  might still be unknown to the change-maker.

Multiplication (× or · or *)

Multiplication is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product. The two original numbers are called the multiplier and the multiplicand, sometimes both simply called factors.

Multiplication may be viewed as a scaling operation. If the numbers are imagined as lying in a line, multiplication by a number, say x, greater than 1 is the same as stretching everything away from 0 uniformly, in such a way that the number 1 itself is stretched to where x was. Similarly, multiplying by a number less than 1 can be imagined as squeezing towards 0. (Again, in such a way that 1 goes to the multiplicand.)

Multiplication is commutative and associative; further it is distributive over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 yields that same number. Also, the multiplicative inverse is the reciprocal of any number (except 0; 0 is the only number without a multiplicative inverse), that is, multiplying the reciprocal of any number by the number itself yields the multiplicative identity.

The product of a and b is written as  or . When a or b are expressions not written simply with digits, it is also written by simple juxtaposition: ab. In computer programming languages and software packages in which one can only use characters normally found on a keyboard, it is often written with an asterisk: 

Division (÷ or /)

Division is essentially the inverse of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by 0 is undefined. For distinct positive numbers, if the dividend is larger than the divisor, the quotient is greater than 1, otherwise it is less than 1 (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.

Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is   When written as a product, it obeys all the properties of multiplication.

Decimal arithmetic

Decimal representation refers exclusively, in common use, to the written numeral system employing arabic numerals as the digits for a radix 10 ("decimal") positional notation; however, any numeral system based on powers of 10, e.g., Greek, Cyrillic, Roman, or Chinese numerals may conceptually be described as "decimal notation" or "decimal representation".

Modern methods for four fundamental operations (addition, subtraction, multiplication and division) were first devised by Brahmagupta of India. This was known during medieval Europe as "Modus Indoram" or Method of the Indians. Positional notation (also known as "place-value notation") refers to the representation or encoding of numbers using the same symbol for the different orders of magnitude (e.g., the "ones place", "tens place", "hundreds place") and, with a radix point, using those same symbols to represent fractions (e.g., the "tenths place", "hundredths place"). For example, 507.36 denotes 5 hundreds (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10−1) plus 6 hundredths (10−2).

The concept of 0 as a number comparable to the other basic digits is essential to this notation, as is the concept of 0's use as a placeholder, and as is the definition of multiplication and addition with 0. The use of 0 as a placeholder and, therefore, the use of a positional notation is first attested to in the Jain text from India entitled the Lokavibhâga, dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the scholarship of the Arabic world, were introduced into Europe by Fibonacci  using the Hindu–Arabic numeral system.

Algorism comprises all of the rules for performing arithmetic computations using this type of written numeral. For example, addition produces the sum of two arbitrary numbers. The result is calculated by the repeated addition of single digits from each number that occupies the same position, proceeding from right to left. An addition table with ten rows and ten columns displays all possible values for each sum. If an individual sum exceeds the value 9, the result is represented with two digits. The rightmost digit is the value for the current position, and the result for the subsequent addition of the digits to the left increases by the value of the second (leftmost) digit, which is always one. This adjustment is termed a carry of the value 1.

The process for multiplying two arbitrary numbers is similar to the process for addition. A multiplication table with ten rows and ten columns lists the results for each pair of digits. If an individual product of a pair of digits exceeds 9, the carry adjustment increases the result of any subsequent multiplication from digits to the left by a value equal to the second (leftmost) digit, which is any value from  (). Additional steps define the final result.

Similar techniques exist for subtraction and division.

The creation of a correct process for multiplication relies on the relationship between values of adjacent digits. The value for any single digit in a numeral depends on its position. Also, each position to the left represents a value ten times larger than the position to the right. In mathematical terms, the exponent for the radix (base) of 10 increases by 1 (to the left) or decreases by 1 (to the right). Therefore, the value for any arbitrary digit is multiplied by a value of the form 10n with integer n. The list of values corresponding to all possible positions for a single digit is written 

Repeated multiplication of any value in this list by 10 produces another value in the list. In mathematical terminology, this characteristic is defined as closure, and the previous list is described as closed under multiplication. It is the basis for correctly finding the results of multiplication using the previous technique. This outcome is one example of the uses of number theory.

Compound unit arithmetic

Compound  unit arithmetic is the application of arithmetic operations to mixed radix quantities such as feet and inches, gallons and pints, pounds shillings and pence, and so on. Prior to the use of decimal-based systems of money and units of measure, the use of compound unit arithmetic formed a significant part of commerce and industry.

Basic arithmetic operations

The techniques used for compound unit arithmetic were developed over many centuries and are well-documented in many textbooks in many different languages.  In addition to the basic arithmetic functions encountered in decimal arithmetic, compound unit arithmetic employs three more functions:
*Reduction where a compound quantity is reduced to a single quantity, for example conversion of a distance expressed in yards, feet and inches to one expressed in inches.  
*Expansion, the inverse function to reduction, is the conversion of a quantity that is expressed as a single unit of measure to a compound unit, such as expanding 24 oz to .
*Normalization is the conversion of a set of compound units to a standard form – for example rewriting "" as "".

Knowledge of the relationship between the various units of measure, their multiples and their submultiples forms an essential part of compound unit arithmetic.

Principles of compound unit arithmetic

There are two basic approaches to compound unit arithmetic:
*Reduction–expansion method where all the compound unit variables are reduced to single unit variables, the calculation performed and the result expanded back to compound units. This approach is suited for automated calculations. A typical example is the handling of time by Microsoft Excel where all time intervals are processed internally as days and decimal fractions of a day.
*On-going normalization method in which each unit is treated separately and the problem is continuously normalized as the solution develops. This approach, which is widely described in classical texts, is best suited for manual calculations. An example of the ongoing normalization method as applied to addition is shown below.

Operations in practice

During the 19th and 20th centuries various aids were developed to aid the manipulation of compound units, particularly in commercial applications. The most common aids were mechanical tills which were adapted in countries such as the United Kingdom to accommodate pounds, shillings, pennies and farthings and "Ready Reckoners" – books aimed at traders that catalogued the results of various routine calculations such as the percentages or multiples of various sums of money. One typical booklet  that ran to 150 pages tabulated multiples "from one to ten thousand at the various prices from one farthing to one pound".

The cumbersome nature of compound unit arithmetic has been recognized for many years – in 1586, the Flemish mathematician Simon Stevin published a small pamphlet called De Thiende ("the tenth")  in which he declared that the universal introduction of decimal coinage, measures, and weights to be merely a question of time while in the modern era, many conversion programs, such as that embedded in the calculator supplied as a standard part of the Microsoft Windows 7 operating system display compound units in a reduced decimal format rather than using an expanded format (i.e. "2.5 ft" is displayed rather than ).

Number theory

Until the 19th century, number theory was a synonym of "arithmetic". The addressed problems were directly related to the basic operations and concerned primality, divisibility, and the solution of equations in integers, such as Fermat's last theorem. It appeared that most of these problems, although very elementary to state, are very difficult and may not be solved without very deep mathematics involving concepts and methods from many other branches of mathematics. This led to new branches of number theory such as analytic number theory, algebraic number theory, Diophantine geometry and arithmetic algebraic geometry. Wiles' proof of Fermat's Last Theorem is a typical example of the necessity of sophistical methods, which go far beyond the classical methods of arithmetic, for solving problems that can be stated in elementary arithmetic.

Arithmetic in education

Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, fractions, and decimals (using the decimal place-value system). This study is sometimes known as algorism.

The difficulty and unmotivated appearance of these algorithms has long led educators to question this curriculum, advocating the early teaching of more central and intuitive mathematical ideas. One notable movement in this direction was the New Math of the 1960s and 1970s, which attempted to teach arithmetic in the spirit of axiomatic development from set theory, an echo of the prevailing trend in higher mathematics.[http://www.mathematicallycorrect.com/glossary.htm Mathematically Correct: Glossary of Terms]

Also, arithmetic was used by Islamic Scholars in order to teach application of the rulings related to Zakat and Irth. This was done in a book entitled The Best of Arithmetic by Abd-al-Fattah-al-Dumyati. 

The book begins with the foundations of mathematics and proceeds to its application in the later chapters.
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Arithmetic
 Which branch of science has four fundamental principles,  Adishin, subtraction, multiplication and division. 

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