What's an irrational number?
Irrational number
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being "incommensurable", meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.
Intermediate value theorem
The theorem depends on, and is equivalent to, the completeness of the real numbers. The intermediate value theorem does not apply to the rational numbers ℚ because gaps exist between rational numbers; irrational numbers fill those gaps. For example, the function formula_16 for formula_17 satisfies formula_18 and formula_19. However, there is no rational number formula_20 such that formula_21, because formula_22 is an irrational number.
Proof that π is irrational
where "n" is a non-negative integer.
Number
The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800 and 500 BC. The first existence proofs of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced Hippasus to death by drowning, to impede spreading of this disconcerting news.
Rational number
Rational numbers together with addition and multiplication form a field which contains the integers and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of are called algebraic number fields, and the algebraic closure of is the field of algebraic numbers.
List of types of numbers
Rational numbers (formula_3): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but the converse is not true.
Ratio
Ratios may also be established between incommensurable quantities (quantities whose ratio, as value of a fraction, amounts to an irrational number). The earliest discovered example, found by the Pythagoreans, is the ratio of the length of the diagonal to the length of a side of a square, which is the square root of 2, formally formula_10 Another example is the ratio of a circle's circumference to its diameter, which is called , and is not just an algebraically irrational number, but a transcendental irrational.
Irrational number
Then
Infinite monkey theorem
Meanwhile, there is an "uncountably" infinite set of strings which do not end in such repetition; these correspond to the irrational numbers. These can be sorted into two uncountably infinite subsets: those which contain "Hamlet" and those which do not. However, the "largest" subset of all the real numbers are those which not only contain "Hamlet", but which contain every other possible string of any length, and with equal distribution of such strings. These irrational numbers are called normal. Because almost all numbers are normal, almost all possible strings contain all possible finite substrings. Hence, the probability of the monkey typing a normal number is 1. The same principles apply regardless of the number of keys from which the monkey can choose; a 90-key keyboard can be seen as a generator of numbers written in base 90.
Georg Cantor's first set theory article
Since the sequence contains all the rational numbers in (0, 1), the construction generates an irrational number, which turns out to be  − 1.
Irrational number
It is suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava (c. 750 – 690 BC) believed that the square roots of numbers such as 2 and 61 could not be exactly determined. However, historian Carl Benjamin Boyer writes that "such claims are not well substantiated and unlikely to be true".
Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective "real" in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as  (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.
Proof that π is irrational
If
Irrational number
As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.
Proof that e is irrational
The number "e" was introduced by Jacob Bernoulli in 1683. More than half a century later, Euler, who had been a student of Jacob's younger brother Johann, proved that "e" is irrational; that is, that it cannot be expressed as the quotient of two integers.
Irrational number
Perhaps the numbers most easy to prove irrational are certain logarithms. Here is a proof by contradiction that log 3 is irrational. Notice that log 3 ≈ 1.58 > 0.
Fraction (mathematics)
In mathematics the set of all numbers that can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. The test for a number being a rational number is that it can be written in that form (i.e., as a common fraction). However, the word "fraction" is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as /2 (see square root of 2) and π/4 (see proof that π is irrational).
Proof that π is irrational
In the 1760's, Johann Heinrich Lambert proved that the number  (pi) is irrational: that is, it cannot be expressed as a fraction "a"/"b", where "a" is an integer and "b" is a non-zero integer. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven, and Nicolas Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich.
Irrational number
The base of the left side is irrational and the right side is rational, so one must prove that the exponent on the left side, formula_17, is irrational. This is so because, by the formula relating logarithms with different bases,
Liouville number
Let formula_39 be an irrational number. If there exists a real number formula_40 with the property that for any formula_41, there is a positive integer formula_42 such that
Quadratic irrational number
It follows that every quadratic irrational number can be written in the form
Euler–Mascheroni constant
The number has not been proved algebraic or transcendental. In fact, it is not even known whether is irrational. Continued fraction analysis reveals that if is rational, its denominator must be greater than 10. The ubiquity of revealed by the large number of equations below makes the irrationality of a major open question in mathematics. Also see Sondow (2003a).
Proof by infinite descent
Therefore, if could be written as a rational number, it could always be written as a rational number with smaller parts, which itself could be written with yet-smaller parts, "ad infinitum". But this is impossible in the set of natural numbers. Since is a real number, which can be either rational or irrational, the only option left is for to be irrational.
Binary number
Binary numerals which neither terminate nor recur represent irrational numbers. For instance,
Irrational number
Irrational numbers may also be dealt with via non-terminating continued fractions.
2
The square root of 2 was the first known irrational number.
Rational number
The term "rational" in reference to the set Q refers to the fact that a rational number represents a "ratio" of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective "rational" sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (that is a point whose coordinates are rational numbers; a "rational matrix" is a matrix of rational numbers; a "rational polynomial" may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, for avoiding confusion with "rational expression" and "rational function" (a polynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a rational curve "is not" a curve defined over the rationals, but a curve which can be parameterized by rational functions.
Fashionable Nonsense
However, with regard to the second sense, which Plotnisky describes by stating that "all imaginary and complex numbers are, by definition, irrational," mathematicians agree with Sokal and Bricmont in not taking complex numbers as irrational. Indeed, the concept of rational numbers has been extended into the complex domain to include Gaussian integers and Gaussian rationals.
Irrational number
Now we multiply this equation by 10 where "r" is the length of the repetend. This has the effect of moving the decimal point to be in front of the "next" repetend. In our example, multiply by 10:
Irrational number
It was not until Eudoxus developed a theory of proportion that took into account irrational as well as rational ratios that a strong mathematical foundation of irrational numbers was created.
Quadratic irrational number
for integers ; with , and non-zero, and with square-free. When is positive, we get real quadratic irrational numbers, while a negative gives complex quadratic irrational numbers which are not real numbers. This implies that the quadratic irrationals have the same cardinality as ordered quadruples of integers, and are therefore countable.
Quadratic irrational number
The definition of quadratic irrationals requires them to satisfy two conditions: they must satisfy a quadratic equation and they must be irrational. The solutions to the quadratic equation "ax" + "bx" + "c" = 0 are
Hermite's problem
Rational numbers are algebraic numbers that satisfy a polynomial of degree 1, while quadratic irrationals are algebraic numbers that satisfy a polynomial of degree 2. For both these sets of numbers we have a way to construct a sequence of natural numbers ("a") with the property that each sequence gives a unique real number and such that this real number belongs to the corresponding set if and only if the sequence is eventually periodic.
Transcendental number
ω("x", 1) is often called the measure of irrationality of a real number "x". For rational numbers, ω("x", 1) = 0 and is at least 1 for irrational real numbers. A Liouville number is defined to have infinite measure of irrationality. Roth's theorem says that irrational real algebraic numbers have measure of irrationality 1.
Periodic continued fraction
A quadratic irrational number is an irrational real root of the quadratic equation
Proof that π is irrational
So
Irrational number
Under the usual (Euclidean) distance function d("x", "y") = |"x" − "y"|, the real numbers are a metric space and hence also a topological space. Restricting the Euclidean distance function gives the irrationals the structure of a metric space. Since the subspace of irrationals is not closed,
the induced metric is not complete. However, being a G-delta set—i.e., a countable intersection of open subsets—in a complete metric space, the space of irrationals is completely metrizable: that is, there is a metric on the irrationals inducing the same topology as the restriction of the Euclidean metric, but with respect to which the irrationals are complete. One can see this without knowing the aforementioned fact about G-delta sets: the continued fraction expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers, which is easily seen to be completely metrizable.
Quadratic irrational number
Assume the square root of "D" is a rational number "p"/"q", assume the "q" here is the smallest for which this is true, hence the smallest number for which "q" is also an integer. Then:
Surreal number
where
Irrational number
A stronger result is the following: Every rational number in the interval formula_23 can be written either as "a" for some irrational number "a" or as "n" for some natural number "n". Similarly, every positive rational number can be written either as formula_24 for some irrational number "a" or as formula_25 for some natural number "n".
Irrational number
Greek mathematicians termed this ratio of incommensurable magnitudes "alogos", or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.” Another legend states that Hippasus was merely exiled for this revelation. Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable–a foundation of their theory.
Numeral system
Note that a number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus ). An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, can be written as the aperiodic 11.001001000011111...
Multiplicative inverse
Every number excluding zero has a reciprocal, and reciprocals of certain irrational numbers can have important special properties. Examples include the reciprocal of e (≈ 0.367879) and the golden ratio's reciprocal (≈ 0.618034). The first reciprocal is special because no other positive number can produce a lower number when put to the power of itself; formula_17 is the global minimum of formula_18. The second number is the only positive number that is equal to its reciprocal plus one:formula_19. Its additive inverse is the only negative number that is equal to its reciprocal minus one:formula_20.
Irrational number
Consider ; if this is rational, then take "a" = "b" = . Otherwise, take "a" to be the irrational number and "b" = . Then "a" = () = = = 2, which is rational.
Proof that e is irrational
Since there is no integer strictly between 0 and 1, we have reached a contradiction, and so "e" must be irrational. Q.E.D.
Proof that π is irrational
then this becomes
Fibonacci number
is known, but the number has been proved irrational by Richard André-Jeannin.
Irrational number
However, the number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made was that log 3 is rational (and so expressible as a quotient of integers "m"/"n" with "n" ≠ 0). The contradiction means that this assumption must be false, i.e. log 3 is irrational, and can never be expressed as a quotient of integers "m"/"n" with "n" ≠ 0.
Rational number
The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set.
Prime constant
The number formula_1 is easily shown to be irrational. To see why, suppose it were rational.
Quadratic irrational number
is also an integer. But 0
Thomae's function
Let formula_26 be any irrational number and define for all formula_27 formula_28
Apéry's theorem
Euler proved that if "n" is a positive integer then
for some rational number "p"/"q". Specifically, writing the infinite series on the left as ζ(2"n") he showed
where the "B" are the rational Bernoulli numbers. Once it was proved that π is always irrational this showed that ζ(2"n") is irrational for all positive integers "n".
Rational number
A nonzero rational number has a multiplicative inverse, also called its "reciprocal",
If is in canonical form, then the canonical form of its reciprocal is either or , depending on the sign of .
Fashionable Nonsense
This latter point has been disputed by Arkady Plotnitsky (one of the authors mentioned by Sokal in his original hoax). Plotnitsky says that "some of their claims concerning mathematical objects in question and specifically complex numbers are incorrect," specifically attacking their statement that complex numbers and irrational numbers "have nothing to do with one another". Plotnisky here defends Lacan's view "of imaginary numbers as an extension of the idea of rational numbers—both in the general conceptual sense, extending to its ancient mathematical and philosophical origins ... and in the sense of modern algebra." The first of these two senses refers to the fact that the extension of real numbers to complex numbers mirrors the extension of rationals to reals, as Plotnitsky points out with a quote from Leibniz: "From the irrationals are born the impossible or imaginary quantities whose nature is very strange but whose usefulness is not to be despised."
Proof that e is irrational
Initially "e" is assumed to be a rational number of the form ⁄. Note that "b" could not be equal to 1 as "e" is not an integer. It can be shown using the above equality that "e" is strictly between 2 and 3:
Quadratic irrational number
so in particular
Commensurability (mathematics)
In fact, it can be proven that if "a" and "b" are any two non-zero rational numbers, then "a" and "b" are commensurable, while if "a" is any irrational number and "b" is any non-zero rational number, then "a" and "b" are incommensurable. On the other hand, if "a" and "b" are both irrational numbers, then "a" and "b" may or may not be commensurable.
Dyadic rational
In mathematics, a dyadic fraction or dyadic rational is a rational number whose denominator, when the ratio is in minimal (coprime) terms, is a power of two, i.e., a number of the form formula_1 where "a" is an integer and "b" is a natural number; for example, 1/2 or 3/8, but not 1/3. These are precisely the numbers possessing a finite binary expansion.
Harold Jeffreys
It is only through an appendix to the third edition of Jeffreys' book "Scientific Inference" that we know about Mary Cartwright's method of proving that the number is irrational.
Irrational number
In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational magnitudes:
Irrational number
Because the algebraic numbers form a subfield of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3 + 2,  +  and "e" are irrational (and even transcendental).
Multiplicative inverse
The function formula_21 gives an infinite number of irrational numbers that differ with their reciprocal by an integer. For example, formula_22 is the irrational formula_23. Its reciprocal formula_24 is formula_25, exactly formula_26 less. Such irrational numbers share a curious property: they have the same fractional part as their reciprocal.
Complete quotient
Clearly this relation is symmetric, reflexive, and transitive, so it is an equivalence relation and it can be used to separate the real numbers into equivalence classes. All the rational numbers are equivalent, because each rational number is equivalent to zero. What can be said about the irrational numbers? Do they also fall into a single equivalence class?
Proof that π is irrational
Proof: Otherwise, there would be a number "y" ≠ 0 and integers "a" and "b" such that "f"("x") = "ay" and "f"("x") = "by". In order to see why, take "y" = "f"("x"), "a" = 0 and "b" = 1 if "f"("x") = 0; otherwise, choose integers "a" and "b" such that "f"("x")/"f"("x") = "b"/"a" and define "y" = "f"("x")/"a" = "f"("x")/"b". In each case, "y" cannot be 0, because otherwise it would follow from claim 1 that each "f"("x") ("n" ∈ N) would be 0, which would contradict claim 2. Now, take a natural number "c" such that all three numbers "bc"/"k", "ck"/"x" and "c"/"x" are integers and consider the sequence
Proof that π is irrational
Then Lambert proved that if "x" is non-zero and rational then this expression must be irrational. Since tan(/4) = 1, it follows that /4 is irrational and therefore that is irrational. A simplification of Lambert's proof is given below.
This proof uses the characterization of as the smallest positive number whose half is a zero of the cosine function and it actually proves that is irrational. As in many proofs of irrationality, the argument proceeds by reductio ad absurdum.
Irrational number
Johann Heinrich Lambert proved (1761) that π cannot be rational, and that "e" is irrational if "n" is rational (unless "n" = 0). While Lambert's proof is often called incomplete, modern assessments support it as satisfactory, and in fact for its time it is unusually rigorous. Adrien-Marie Legendre (1794), after introducing the Bessel–Clifford function, provided a proof to show that π is irrational, whence it follows immediately that π is irrational also. The existence of transcendental numbers was first established by Liouville (1844, 1851). Later, Georg Cantor (1873) proved their existence by a different method, which showed that every interval in the reals contains transcendental numbers. Charles Hermite (1873) first proved "e" transcendental, and Ferdinand von Lindemann (1882), starting from Hermite's conclusions, showed the same for π. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and was finally made elementary by Adolf Hurwitz and Paul Gordan.
Quadratic irrational number
If formula_15 is in formula_10, then formula_20 is too.
0
The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is an integer, and hence a rational number and a real number (as well as an algebraic number and a complex number).
Liouville number
then formula_44 is called the irrationality base of formula_45 and is represented as formula_46
List of numbers
A rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . Since may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface (or blackboard bold formula_1, Unicode ℚ); it was thus denoted in 1895 by Giuseppe Peano after "quoziente", Italian for "quotient".
Normal number
The set of non-normal numbers, though "small" in the sense of being a null set, is "large" in the sense of being uncountable. Also, the non-normal numbers (as well as the normal numbers) are dense in the reals: the set of non-normal numbers between two distinct real numbers is non-empty since it contains every rational number (in fact, it is uncountable and even comeagre). For instance, there are uncountably many numbers whose decimal expansion does not contain the digit 5, and none of these are normal.
Liouville number
Almost all numbers have an irrationality measure equal to 2.
Irrational number
It is not known if formula_32 Catalan's constant, or the Euler–Mascheroni constant formula_33 are irrational. It is not known if either of the tetrations formula_34 or formula_35 is rational for some integer formula_36
Law of excluded middle
In the above argument, the assertion "this number is either rational or irrational" invokes the law of excluded middle. An intuitionist, for example, would not accept this argument without further support for that statement. This might come in the form of a proof that the number in question is in fact irrational (or rational, as the case may be); or a finite algorithm that could determine whether the number is rational.
Proof that π is irrational
Then
Square root of 2
This is a table of recent records in calculating digits of 
A short proof of the irrationality of can be obtained from the rational root theorem, that is, if is a monic polynomial with integer coefficients, then any rational root of is necessarily an integer. Applying this to the polynomial , it follows that is either an integer or irrational. Because is not an integer (2 is not a perfect square), must therefore be irrational. This proof can be generalized to show that any root of any natural number which is not the square of a natural number is irrational.
Quadratic irrational number
Thus quadratic irrationals are precisely those real numbers in this form that are not rational. Since "b" and 2"a" are both integers, asking when the above quantity is irrational is the same as asking when the square root of an integer is irrational. The answer to this is that the square root of any natural number that is not a square number is irrational.
Lochs's theorem
A prominent example of a number not exhibiting this behavior is the golden ratio—sometimes known as the "most irrational" number—whose continued fraction terms are all ones, the smallest possible in canonical form. On average it requires approximately 2.39 continued fraction terms per decimal digit.
Quadratic irrational number
Many proofs of the irrationality of the square roots of non-square natural numbers implicitly assume the fundamental theorem of arithmetic, which was first proven by Carl Friedrich Gauss in his Disquisitiones Arithmeticae. This asserts that every integer has a unique factorization into primes. For any rational non-integer in lowest terms there must be a prime in the denominator which does not divide into the numerator. When the numerator is squared that prime will still not divide into it because of the unique factorization. Therefore, the square of a rational non-integer is always a non-integer; by contrapositive, the square root of an integer is always either another integer, or irrational.
Quotient
The existence of irrational numbers – numbers that are not a quotient of two integers – was first discovered in geometry in such things as the ratio of the diagonal of a square to the side.
Rational number
The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal).
Irrational number
To show this, suppose we divide integers "n" by "m" (where "m" is nonzero). When long division is applied to the division of "n" by "m", only "m" remainders are possible. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most "m" − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats.
Power of two
The numbers formula_1 form an irrationality sequence: for every sequence formula_2 of positive integers, the series
converges to an irrational number. Despite the rapid growth of this sequence, it is the slowest-growing irrationality sequence known.The sum of all -choose binomial coefficients is equal to . Consider the set of all -digit binary integers. Its cardinality is . It is also the sums of the cardinalities of certain subsets: the subset of integers with no 1s (consisting of a single number, written as 0s), the subset with a single 1, the subset with two 1s, and so on up to the subset with 1s (consisting of the number written as 1s). Each of these is in turn equal to the binomial coefficient indexed by and the number of 1s being considered (e.g., there are 10-choose-3 binary numbers with ten digits that include exactly three 1s).
Surreal number
and
Liouville number
is satisfied by an infinite number of integer pairs ("p", "q") with "q" > 0. This least upper bound is defined to be the irrationality measure of "x". For any value "μ" less than this upper bound, the infinite set of all rationals "p"/"q" satisfying the above inequality yield an approximation of "x". Conversely, if "μ" is greater than the upper bound, then there are at most finitely many ("p", "q") with "q" > 0 that satisfy the inequality; thus, the opposite inequality holds for all larger values of "q". In other words, given the irrationality measure "μ" of a real number "x", whenever a rational approximation "x" ≅ "p"/"q", "p","q" ∈ N yields "n" + 1 exact decimal digits, we have
E (mathematical constant)
The real number is irrational. Euler proved this by showing that its simple continued fraction expansion is infinite. (See also Fourier's proof that is irrational.)
Pi
Being an irrational number, cannot be expressed as a common fraction (equivalently, its decimal representation never ends and never settles into a permanently repeating pattern). Still, fractions such as 22/7 and other rational numbers are commonly used to approximate . The digits appear to be randomly distributed. In particular, the digit sequence of is conjectured to satisfy a specific kind of statistical randomness, but to date, no proof of this has been discovered. Also, is a transcendental number; that is, it is not the root of any polynomial having rational coefficients. This transcendence of implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.
Irrational number
is a ratio of integers and therefore a rational number.
Proof that e is irrational
To see that if "e" is rational, then "x" is an integer, substitute "e" = ⁄ into this definition to obtain
Predicative expression
"Every real number is rational OR irrational."
Irrational number
The next step was taken by Eudoxus of Cnidus, who formalized a new theory of proportion that took into account commensurable as well as incommensurable quantities. Central to his idea was the distinction between magnitude and number. A magnitude “...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5.” Numbers are composed of some smallest, indivisible unit, whereas magnitudes are infinitely reducible. Because no quantitative values were assigned to magnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios. By taking quantitative values (numbers) out of the equation, he avoided the trap of having to express an irrational number as a number. “Eudoxus’ theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying the necessary logical foundation for incommensurable ratios.” This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9.
Irrational number
The square root of 2 was the first number proved irrational, and that article contains a number of proofs. The golden ratio is another famous quadratic irrational and there is a simple proof of its irrationality in its article. The square roots of all natural numbers which are not perfect squares are irrational and a proof may be found in quadratic irrationals.
Brjuno number
An irrational number formula_1 is called a Brjuno number when the infinite sum 
converges to a finite number. Here formula_3 is the denominator of the th convergent formula_4 of the continued fraction expansion of formula_1.
Number theory
Euclid IX 21–34 is very probably Pythagorean; it is very simple material ("odd times even is even", "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it"), but it is all that is needed to prove that formula_5
is irrational. Pythagorean mystics gave great importance to the odd and the even.
The discovery that formula_5 is irrational is credited to the early Pythagoreans (pre-Theodorus). By revealing (in modern terms) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; its proof or its divulgation are sometimes credited to Hippasus, who was expelled or split from the Pythagorean sect. This forced a distinction between "numbers" (integers and the rationals—the subjects of arithmetic), on the one hand, and "lengths" and "proportions" (which we would identify with real numbers, whether rational or not), on the other hand.
Proof that e is irrational
Suppose that "e" is a rational number. Then there exist positive integers "a" and "b" such that "e" = ⁄. Define the number
List of works designed with the golden ratio
The golden ratio, an irrational number, is approximately 1.618; it is often denoted by the Greek letter φ (phi).
Square root of 2
Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational.
Irrational number
Therefore, when we subtract the 10"A" equation from the 10,000"A" equation, the tail end of 10"A" cancels out the tail end of 10,000"A" leaving us with:
Hippasus
Hippasus is sometimes credited with the discovery of the existence of irrational numbers, following which he was drowned at sea. Pythagoreans preached that all numbers could be expressed as the ratio of integers, and the discovery of irrational numbers is said to have shocked them. However, the evidence linking the discovery to Hippasus is confused.
Irrational number
It is not known if formula_26 (or formula_27) is irrational. In fact, there is no pair of non-zero integers formula_28 for which it is known whether formula_29 is irrational. Moreover, it is not known whether or not the set formula_30 is algebraically independent over formula_31.