Irrational number

In mathematics, the irrational numbers are all the real numbers that are not rational numbers; that is, irrational numbers are those that cannot be expressed as the ratio of two integers. Geometrically, 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.

Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational.

Like any real number, an irrational number can be expressed in positional notation; however, it does not terminate or end with a repeating sequence. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics.

Irrational numbers can also be expressed as non-terminating continued fractions (which in some cases are periodic), and in many other ways.

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.

The first proof of the existence of irrational numbers is attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the pentagram. The Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. Hippasus in the 5th century BC, however, was able to deduce that there was no common unit of measure, and that the assertion of such an existence was a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with a leg, then one of those lengths measured in that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:

• Start with an isosceles right triangle with side lengths of integers a, b, and c (a = b since it is isosceles). The ratio of the hypotenuse to a leg is represented by c:b.
• Assume a, b, and c are in the smallest possible terms (i.e. they have no common factors).
• By the Pythagorean theorem: c = a+b = b+b = 2b. (Since the triangle is isosceles, a = b).
• Since c = 2b, c is divisible by 2, and therefore even.
• Since c is even, c must be even.
• Since c is even, dividing c by 2 yields an integer. Let y be this integer (c = 2y).
• Squaring both sides of c = 2y yields c = (2y), or c = 4y.
• Substituting 4y for c in the first equation (c = 2b) gives us 4y= 2b.
• Dividing by 2 yields 2y = b.
• Since y is an integer, and 2y = b, b is divisible by 2, and therefore even.
• Since b is even, b must be even.
• We have just shown that both b and c must be even. Hence they have a common factor of 2. However, this contradicts the assumption that they have no common factors. This contradiction proves that c and b cannot both be integers and thus the existence of a number that cannot be expressed as a ratio of two integers.

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 numbers and geometry were inseparable; a foundation of their theory.

Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there probably because the algebra he used could not be applied to the square root of 17.

Condensed: the full section is in Wikipedia.

Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the Vedic period in India. There are references to such calculations in the Samhitas, Brahmanas, and the Shulba Sutras (800 BC or earlier).

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. Historian Carl Benjamin Boyer, however, writes that "such claims are not well substantiated and unlikely to be true".

Later, in their treatises, Indian mathematicians wrote on the arithmetic of surds including addition, subtraction, multiplication, rationalization, as well as separation and extraction of square roots.

Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed. In the 12th century Bhāskara II evaluated some of these formulas and critiqued them, identifying their limitations.

During the 14th to 16th centuries, Madhava of Sangamagrama and the Kerala school of astronomy and mathematics discovered the infinite series for several irrational numbers such as π and certain irrational values of trigonometric functions. Jyeṣṭhadeva provided proofs for these infinite series in the Yuktibhāṣā.

In the Middle Ages, the development of algebra by Muslim mathematicians allowed irrational numbers to be treated as algebraic objects. Middle Eastern mathematicians also merged the concepts of "number" and "magnitude" into a more general idea of real numbers, criticized Euclid's idea of ratios, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude. In his commentary on Book 10 of the Elements, the Persian mathematician Al-Mahani (d. 874/884) examined and classified quadratic irrationals and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows:

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:

The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850, 930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation in the form of square roots and fourth roots. In the 10th century, the Iraqi mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions.

Many of these concepts were eventually accepted by European mathematicians sometime after the Latin translations of the 12th century. Al-Hassār, a Moroccan mathematician from Fez specializing in Islamic inheritance jurisprudence during the 12th century, first mentions the use of a fractional bar, where numerators and denominators are separated by a horizontal bar. In his discussion he writes, "..., for example, if you are told to write three-fifths and a third of a fifth, write thus, \(\frac{3 \quad 1}{5 \quad 3}\)." This same fractional notation appears soon after in the work of Leonardo Fibonacci in the 13th century.

The 17th century saw imaginary numbers become a powerful tool in the hands of Abraham de Moivre, and especially of Leonhard Euler. The completion of the theory of complex numbers in the 19th century entailed the differentiation of irrationals into algebraic and transcendental numbers, the proof of the existence of transcendental numbers, and the resurgence of the scientific study of the theory of irrationals, largely ignored since Euclid. The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Ernst Kossak), Eduard Heine (Crelle's Journal, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by Salvatore Pincherle in 1880, and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of all rational numbers, separating them into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Leopold Kronecker (Crelle, 101), and Charles Méray.

Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the 19th century were brought into prominence through the writings of Joseph-Louis Lagrange. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.

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.

The square root of 2 was likely the first number proved irrational. The golden ratio (\(\varphi\)) is another well-known quadratic irrational number. It can be defined as the positive solution of the quadratic equation \(x^2-x-1=0\) and has the value \(\varphi = (1+\sqrt{5})/2\). The square roots of all natural numbers that are not perfect squares are irrational and a proof may be found in quadratic irrationals.

The proof for the irrationality of the square root of two can be generalized using the fundamental theorem of arithmetic. This asserts that every integer has a unique factorization into primes. Using it we can show that if a rational number is not an integer then no integral power of it can be an integer, as in lowest terms there must be a prime in the denominator that does not divide into the numerator whatever power each is raised to. Therefore, if an integer is not an exact k-th power of another integer, then that first integer's k-th root is irrational.

Perhaps the numbers most easy to prove irrational are certain logarithms. Here is a proof by contradiction that log₂ 3 is irrational (log₂ 3 ≈ 1.58 > 0).

Assume log₂ 3 is rational. For some positive integers m and n, we have

\(\log_2 3 = \frac{m}{n}.\)

It follows that

\(2^{m/n}=3\)

\((2^{m/n})^n = 3^n\)

\(2^m=3^n.\)

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 log₂ 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. log₂ 3 is irrational, and can never be expressed as a quotient of integers m/n with n ≠ 0.

Cases such as log₁₀ 2 can be treated similarly.

An irrational number may be algebraic, that is a real root of a polynomial with integer coefficients. Those that are not algebraic are transcendental.

The real algebraic numbers are the real solutions of polynomial equations

\(p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0\;,\)

where the coefficients \(a_i\) are integers and \(a_n \ne 0\). An example of an irrational algebraic number is x₀ = (2 + 1). It is clearly algebraic since it is the root of an integer polynomial, \((x^3 - 1)^2= 2\), which is equivalent to \((x^6 - 2x^3-1)= 0\). This polynomial has no rational roots, since the rational root theorem shows that the only possibilities are ±1, but x₀ is greater than 1. So x₀ is an irrational algebraic number. There are countably many algebraic numbers, since there are countably many integer polynomials.

Almost all irrational numbers are transcendental. Examples are e and π, which are transcendental for all nonzero rational r.

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, 3π + 2, π + √2 and e√3 are irrational (and even transcendental).

The decimal expansion of an irrational number never repeats (meaning the decimal expansion does not repeat the same number or sequence of numbers) or terminates (this means there is not a finite number of nonzero digits), unlike any rational number. The same is true for binary, octal or hexadecimal expansions, and in general for expansions in every positional notation with natural bases.

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, there can never be a remainder greater than or equal to m. 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.

Conversely, suppose we are faced with a repeating decimal, we can prove that it is a fraction of two integers. For example, consider:

\(A=0.7\,162\,162\,162\,\ldots\)

Here the repetend is 162 and the length of the repetend is 3. First, we multiply by an appropriate power of 10 to move the decimal point to the right so that it is just in front of a repetend. In this example we would multiply by 10 to obtain:

\(10A = 7.162\,162\,162\,\ldots\)

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:

\(10,000A=7\,162.162\,162\,\ldots\)

The result of the two multiplications gives two different expressions with exactly the same "decimal portion", that is, the tail end of 10,000A matches the tail end of 10A exactly. Here, both 10,000A and 10A have .162162162... after the decimal point.

Therefore, when we subtract the 10A equation from the 10,000A equation, the tail end of 10A cancels out the tail end of 10,000A leaving us with:

\(9990A=7155.\)

\(A= \frac{7155}{9990} = \frac{53}{74}\)

Condensed: the full section is in Wikipedia.

Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that a is rational:

Consider \(\sqrt{2}\); if this is rational, then take a = b = \(\sqrt{2}\). Otherwise, take a to be the irrational number \(\sqrt{2}\) and b = \(\sqrt{2}\). Then a = (\(\sqrt{2}\)) \(=\) \(\sqrt{2}\) \(=\) \(\sqrt{2}\) \(=\) \(2\), which is rational.

Although the above argument does not decide between the two cases, the Gelfond-Schneider theorem shows that \(\sqrt{2}\) is transcendental, hence irrational. This theorem states that if a and b are both algebraic numbers, and a is not equal to 0 or 1, and b is not a rational number, then any value of a is a transcendental number (there can be more than one value if complex number exponentiation is used).

An example that provides a simple constructive proof is

\(\left(\sqrt{2}\right)^{\log_{\sqrt{2}}3}=3.\)

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, \(\log_{\sqrt{2}}3\), is irrational. This is so because, by the formula relating logarithms with different bases,

\(\log_{\sqrt{2}}3=\frac{\log_2 3}{\log_2 \sqrt{2}}=\frac{\log_2 3}{1/2} = 2\log_2 3\)

which we can assume, for the sake of establishing a contradiction, equals a ratio m/n of positive integers. Then \(\log_2 3 = m/2n\) hence \(2^{\log_2 3}=2^{m/2n}\) hence \(3=2^{m/2n}\) hence \(3^{2n}=2^m\), which is a contradictory pair of prime factorizations and hence violates the fundamental theorem of arithmetic (unique prime factorization).

A stronger result is the following: Every rational number in the interval \(((1/e)^{1/e}, \infty)\) 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 \(a^{a^a}\) for some irrational number a or as \(n^{n^n}\) for some natural number n.

• Various combinations of e, π and elementary functions (such as e + π, eπ, e, π, π, ln π) are not known to be irrational, in part because e and π are not known to be algebraically independent. Schanuel's conjecture would imply that all of the above numbers are irrational and even transcendental.
• The question about the irrationality of Euler's constant γ is a long standing open problem in number theory.
• Other important numbers which are not known to be irrational include odd zeta constants ζ(5), ζ(7), ζ(9), ... for n > 3 and Catalan's constant β(2).