Ratio of two numbers
A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A "common", "vulgar", or "simple" fraction (examples: formula_1 and formula_2) consists of an integer numerator, displayed above a line (or before a slash like ;templatestyles src="Fraction/styles.css" / ;1⁄2), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction , the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates of a cake.
Fractions can be used to represent ratios and division. Thus the fraction can be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division 3 ÷ 4 (three divided by four).
We can also write negative fractions, which represent the opposite of a positive fraction. For example, if represents a half-dollar profit, then − represents a half-dollar loss. Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), −, and all represent the same fraction – negative one-half. And because a negative divided by a negative produces a positive, represents positive one-half.
In mathematics a rational number is a number that can be represented by a fraction of the form , where "a" and "b" are integers and "b" is not zero; the set of all rational numbers is commonly represented by the symbol Q or &amp;NoBreak;}&amp;NoBreak;, which stands for quotient. The term "fraction" and the notation can also be used for mathematical expressions that do not represent a rational number (for example formula_3), and even do not represent any number (for example the rational fraction formula_4).
Vocabulary.
In a fraction, the number of equal parts being described is the numerator (from , "counter" or "numberer"), and the type or variety of the parts is the denominator (from , "thing that names or designates"). As an example, the fraction amounts to eight parts, each of which is of the type named "fifth". In terms of division, the numerator corresponds to the dividend, and the denominator corresponds to the divisor.
Informally, the numerator and denominator may be distinguished by placement alone, but in formal contexts they are usually separated by a fraction bar. The fraction bar may be horizontal (as in ), oblique (as in 2/5), or diagonal (as in ;templatestyles src="Fraction/styles.css" / ;4⁄9). These marks are respectively known as the horizontal bar; the virgule, slash (US), or stroke (UK); and the fraction bar, solidus, or fraction slash. In typography, fractions stacked vertically are also known as "en" or "nut fractions", and diagonal ones as "em" or "mutton fractions", based on whether a fraction with a single-digit numerator and denominator occupies the proportion of a narrow "en" square, or a wider "em" square. In traditional typefounding, a piece of type bearing a complete fraction (e.g. ) was known as a "case fraction", while those representing only part of fraction were called "piece fractions".
The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not 1. (For example, and are both read as a number of "fifths".) Exceptions include the denominator 2, which is always read "half" or "halves", the denominator 4, which may be alternatively expressed as "quarter"/"quarters" or as "fourth"/"fourths", and the denominator 100, which may be alternatively expressed as "hundredth"/"hundredths" or "percent".
When the denominator is 1, it may be expressed in terms of "wholes" but is more commonly ignored, with the numerator read out as a whole number. For example, may be described as "three wholes", or simply as "three". When the numerator is 1, it may be omitted (as in "a tenth" or "each quarter").
The entire fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. (For example, "two-fifths" is the fraction and "two fifths" is the same fraction understood as 2 instances of .) Fractions should always be hyphenated when used as adjectives. Alternatively, a fraction may be described by reading it out as the numerator "over" the denominator, with the denominator expressed as a cardinal number. (For example, may also be expressed as "three over one".) The term "over" is used even in the case of solidus fractions, where the numbers are placed left and right of a slash mark. (For example, 1/2 may be read "one-half", "one half", or "one over two".) Fractions with large denominators that are "not" powers of ten are often rendered in this fashion (e.g., as "one over one hundred seventeen"), while those with denominators divisible by ten are typically read in the normal ordinal fashion (e.g., as "six-millionths", "six millionths", or "six one-millionths").
Forms of fractions.
Simple, common, or vulgar fractions.
A simple fraction (also known as a common fraction or vulgar fraction, where vulgar is Latin for "common") is a rational number written as "a"/"b" or &amp;NoBreak;}&amp;NoBreak;, where "a" and "b" are both integers. As with other fractions, the denominator ("b") cannot be zero. Examples include , −, , and . The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy.
Common fractions can be positive or negative, and they can be proper or improper (see below). Compound fractions, complex fractions, mixed numerals, and decimals (see below) are not "common fractions"; though, unless irrational, they can be evaluated to a common fraction.
In Unicode, precomposed fraction characters are in the Number Forms block.
Proper and improper fractions.
Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise. The concept of an "improper fraction" is a late development, with the terminology deriving from the fact that "fraction" means "a piece", so a proper fraction must be less than 1. This was explained in the 17th century textbook "The Ground of Arts".
In general, a common fraction is said to be a proper fraction, if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1. It is said to be an improper fraction, or sometimes top-heavy fraction, if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, −3/4, and 4/9, whereas examples of improper fractions are 9/4, −4/3, and 3/3.
Reciprocals and the "invisible denominator".
The "reciprocal" of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of , for instance, is . The product of a non-zero fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are not equal) is a proper fraction.
When the numerator and denominator of a fraction are equal (for example, ), its value is 1, and the fraction therefore is improper. Its reciprocal is identical and hence also equal to 1 and improper.
Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as , where 1 is sometimes referred to as the "invisible denominator". Therefore, every fraction or integer, except for zero, has a reciprocal. For example, the reciprocal of 17 is .
Ratios.
A "ratio" is a relationship between two or more numbers that can be sometimes expressed as a fraction. Typically, a number of items are grouped and compared in a ratio, specifying numerically the relationship between each group. Ratios are expressed as "group 1 to group 2 ... to group "n"". For example, if a car lot had 12 vehicles, of which
then the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of yellow cars to white cars is 4 to 2 and may be expressed as 4:2 or 2:1.
A ratio is often converted to a fraction when it is expressed as a ratio to the whole. In the above example, the ratio of yellow cars to all the cars on the lot is 4:12 or 1:3. We can convert these ratios to a fraction, and say that of the cars or of the cars in the lot are yellow. Therefore, if a person randomly chose one car on the lot, then there is a one in three chance or probability that it would be yellow.
Decimal fractions and percentages.
A "decimal fraction" is a fraction whose denominator is not given explicitly, but is understood to be an integer power of ten. Decimal fractions are commonly expressed using decimal notation in which the implied denominator is determined by the number of digits to the right of a decimal separator, the appearance of which (e.g., a period, an interpunct (·), a comma) depends on the locale (for examples, see "Decimal separator"). Thus, for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, namely, 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the fractional part of the number is expressed by the digits to the right of the decimal (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, 375/100, or as a mixed number, .
Decimal fractions can also be expressed using scientific notation with negative exponents, such as , which represents 0.0000006023. The represents a denominator of . Dividing by moves the decimal point 7 places to the left.
Decimal fractions with infinitely many digits to the right of the decimal separator represent an infinite series. For example, = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ...
Another kind of fraction is the percentage (from , meaning "per hundred", represented by the symbol %), in which the implied denominator is always 100. Thus, 51% means 51/100. Percentages greater than 100 or less than zero are treated in the same way, e.g. 311% equals 311/100, and −27% equals −27/100.
The related concept of "permille" or "parts per thousand" (ppt) has an implied denominator of 1000, while the more general parts-per notation, as in 75 "parts per million" (ppm), means that the proportion is 75/1,000,000.
Whether common fractions or decimal fractions are used is often a matter of taste and context. Common fractions are used most often when the denominator is relatively small. By mental calculation, it is easier to multiply 16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more accurate to multiply 15 by 1/3, for example, than it is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions with denominator 100, i.e., with two decimals, for example $3.75. However, as noted above, in pre-decimal British currency, shillings and pence were often given the form (but not the meaning) of a fraction, as, for example, "3/6" (read "three and six") meaning 3 shillings and 6 pence, and having no relationship to the fraction 3/6.
Mixed numbers.
A mixed number (also called a "mixed fraction" or "mixed numeral") is the sum of a non-zero integer and a proper fraction, conventionally written by juxtaposition (or "concatenation") of the two parts, without the use of an intermediate plus (+) or minus (−) sign. When the fraction is written horizontally, a space is added between the integer and fraction to separate them.
As a basic example, two entire cakes and three quarters of another cake might be written as formula_5 cakes or formula_6 cakes, with the numeral formula_7 representing the whole cakes and the fraction formula_8 representing the additional partial cake juxtaposed; this is more concise than the more explicit notation formula_9 cakes. The mixed number is pronounced "two and three quarters", with the integer and fraction portions connected by the