Transcendental number

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https://en.wikipedia.org/wiki/Transcendental_number 

Pi (π) is a well-known transcendental number.

In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are π and e.

Though only a few classes of transcendental numbers are known, in part as it can be extremely difficult to show that a given number is transcendental, transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers compose a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping rational, algebraic non-rational and transcendental real numbers. For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x² − 2 = 0. The golden ratio (denoted ${\displaystyle \varphi }$ or ${\displaystyle \phi }$) is another irrational number that is not transcendental, as it is a root of the polynomial equation x² − x − 1 = 0. The quality of a number being transcendental is called transcendence.

History

The name "transcendental" comes from the Latin transcendĕre 'to climb over or beyond, surmount', and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that sin x is not an algebraic function of x. Euler, in the 18th century, was probably the first person to define transcendental numbers in the modern sense.

Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch of a proof of π's transcendence.

Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant

${\displaystyle {\begin{aligned}L_{b}&=\sum _{n=1}^{\infty }10^{-n!}\\&=10^{-1}+10^{-2}+10^{-6}+10^{-24}+10^{-120}+10^{-720}+10^{-5040}+10^{-40320}+\ldots \\&=0.{\textbf {1}}{\textbf {1}}000{\textbf {1}}00000000000000000{\textbf {1}}00000000000000000000000000000000000000000000000000000\ldots \\\end{aligned}}}$

in which the nth digit after the decimal point is 1 if n is equal to k! (k factorial) for some k and 0 otherwise. In other words, the nth digit of this number is 1 only if n is one of the numbers 1! = 1, 2! = 2, 3! = 6, 4! = 24, etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by rational numbers than can any irrational algebraic number, and this class of numbers are called Liouville numbers, named in his honour. Liouville showed that all Liouville numbers are transcendental.

The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was e, by Charles Hermite in 1873.

In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a new method for constructing transcendental numbers. Although this was already implied by his proof of the countability of the algebraic numbers, Cantor also published a construction that proves there are as many transcendental numbers as there are real numbers. Cantor's work established the ubiquity of transcendental numbers.

In 1882, Ferdinand von Lindemann published the first complete proof of the transcendence of π. He first proved that e^a is transcendental if a is a non-zero algebraic number. Then, since e^iπ = −1 is algebraic (see Euler's identity), iπ must be transcendental. But since i is algebraic, π therefore must be transcendental. This approach was generalized by Karl Weierstrass to what is now known as the Lindemann–Weierstrass theorem. The transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle.

In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number that is not zero or one, and b is an irrational algebraic number, is a^b necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).

Properties

A transcendental number is a (possibly complex) number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree. Every real transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one. The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, it is impossible for both subsets to be countable. This makes the transcendental numbers uncountable.

No rational number is transcendental and all real transcendental numbers are irrational. The irrational numbers contain all the real transcendental numbers and a subset of the algebraic numbers, including the quadratic irrationals and other forms of algebraic irrationals.

Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument. For example, from knowing that π is transcendental, it can be immediately deduced that numbers such as 5π, π-3/√2, (√π-√3)⁸, and ⁴√π⁵+7 are transcendental as well.

However, an algebraic function of several variables may yield an algebraic number when applied to transcendental numbers if these numbers are not algebraically independent. For example, π and (1 − π) are both transcendental, but π + (1 − π) = 1 is obviously not. It is unknown whether e + π, for example, is transcendental, though at least one of e + π and eπ must be transcendental. More generally, for any two transcendental numbers a and b, at least one of a + b and ab must be transcendental. To see this, consider the polynomial (x − a)(x − b) = x² − (a + b)x + ab. If (a + b) and ab were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, a and b, must be algebraic. But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental.

The non-computable numbers are a strict subset of the transcendental numbers.

All Liouville numbers are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its continued fraction expansion. Using a counting argument one can show that there exist transcendental numbers which have bounded partial quotients and hence are not Liouville numbers.

Using the explicit continued fraction expansion of e, one can show that e is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). Kurt Mahler showed in 1953 that π is also not a Liouville number. It is conjectured that all infinite continued fractions with bounded terms that are not eventually periodic are transcendental (eventually periodic continued fractions correspond to quadratic irrationals).

Numbers proven to be transcendental

Numbers proven to be transcendental:

• e^a if a is algebraic and nonzero (by the Lindemann–Weierstrass theorem).
• π (by the Lindemann–Weierstrass theorem).
• e^π, Gelfond's constant, as well as e^−π/2 = iⁱ (by the Gelfond–Schneider theorem).
• a^b where a is algebraic but not 0 or 1, and b is irrational algebraic (by the Gelfond–Schneider theorem), in particular:

2^√2, the Gelfond–Schneider constant (or Hilbert number)

• sin a, cos a, tan a, csc a, sec a, and cot a, and their hyperbolic counterparts, for any nonzero algebraic number a, expressed in radians (by the Lindemann–Weierstrass theorem).
• The fixed point of the cosine function (also referred to as the Dottie number d) – the unique real solution to the equation cos x = x, where x is in radians (by the Lindemann–Weierstrass theorem).
• ln a if a is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
• log_b a if a and b are positive integers not both powers of the same integer (by the Gelfond–Schneider theorem).
• W(a) if a is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann–Weierstrass theorem), in particular: Ω the omega constant
• √x_s, the square super-root of any natural number is either an integer or transcendental (by the Gelfond–Schneider theorem)
• Γ(1/3), Γ(1/4), and Γ(1/6).
• 0.64341054629..., Cahen's constant.
• The Champernowne constants, the irrational numbers formed by concatenating representations of all positive integers.
• Ω, Chaitin's constant (since it is a non-computable number).
• The so-called Fredholm constants, such as

  ${\displaystyle \sum _{n=0}^{\infty }10^{-2^{n}}=0.{\textbf {1}}{\textbf {1}}0{\textbf {1}}000{\textbf {1}}0000000{\textbf {1}}\ldots }$

which also holds by replacing 10 with any algebraic b > 1.

• The Gauss constant.
• The two lemniscate constants L₁ (sometimes denoted as ϖ) and L₂.
• The aforementioned Liouville constant for any algebraic b ∈ (0, 1).
• The Prouhet–Thue–Morse constant.
• The Komornik–Loreti constant.
• Any number for which the digits with respect to some fixed base form a Sturmian word.
• For β > 1

${\displaystyle \sum _{k=0}^{\infty }10^{-\left\lfloor \beta ^{k}\right\rfloor };}$

where ${\displaystyle \beta \mapsto \lfloor \beta \rfloor }$ is the floor function.

• 3.300330000000000330033... and its reciprocal 0.30300000303..., two numbers with only two different decimal digits whose nonzero digit positions are given by the Moser–de Bruijn sequence and its double.
• The number π/2Y₀(2)/J₀(2)-γ, where Y_α(x) and J_α(x) are Bessel functions and γ is the Euler–Mascheroni constant.

Possible transcendental numbers

Numbers which have yet to be proven to be either transcendental or algebraic:

• Most sums, products, powers, etc. of the number π and the number e, e.g. eπ, e + π, π − e, π/e, π^π, e^e, π^e, π^√2, e^π2 are not known to be rational, algebraic, irrational or transcendental. A notable exception is e^π√n (for any positive integer n) which has been proven transcendental.
• The Euler–Mascheroni constant γ: In 2010 M. Ram Murty and N. Saradha considered an infinite list of numbers also containing γ/4 and showed that all but at most one of them have to be transcendental. In 2012 it was shown that at least one of γ and the Euler–Gompertz constant δ is transcendental.
• Catalan's constant, not even proven to be irrational.
• Khinchin's constant, also not proven to be irrational.
• Apéry's constant ζ(3) (which Apéry proved is irrational).
• The Riemann zeta function at other odd integers, ζ(5), ζ(7), ... (not proven to be irrational).
• The Feigenbaum constants δ and α, also not proven to be irrational.
• Mills' constant, also not proven to be irrational.
• The Copeland–Erdős constant, formed by concatenating the decimal representations of the prime numbers.

Conjectures:

• Schanuel's conjecture,
• Four exponentials conjecture.

Sketch of a proof that e is transcendental

The first proof that the base of the natural logarithms, e, is transcendental dates from 1873. We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. The idea is the following:

Assume, for purpose of finding a contradiction, that e is algebraic. Then there exists a finite set of integer coefficients c₀, c₁, ..., c_n satisfying the equation:

${\displaystyle c_{0}+c_{1}e+c_{2}e^{2}+\cdots +c_{n}e^{n}=0,\qquad c_{0},c_{n}\neq 0.}$

Now for a positive integer k, we define the following polynomial:

${\displaystyle f_{k}(x)=x^{k}\left[(x-1)\cdots (x-n)\right]^{k+1},}$

and multiply both sides of the above equation by

${\displaystyle \int _{0}^{\infty }f_{k}e^{-x}\,dx,}$

to arrive at the equation:

${\displaystyle c_{0}\left(\int _{0}^{\infty }f_{k}e^{-x}\,dx\right)+c_{1}e\left(\int _{0}^{\infty }f_{k}e^{-x}\,dx\right)+\cdots +c_{n}e^{n}\left(\int _{0}^{\infty }f_{k}e^{-x}\,dx\right)=0.}$

By splitting respective domains of integration, this equation can be written in the form

${\displaystyle P+Q=0}$

where

${\displaystyle {\begin{aligned}P&=c_{0}\left(\int _{0}^{\infty }f_{k}e^{-x}\,dx\right)+c_{1}e\left(\int _{1}^{\infty }f_{k}e^{-x}\,dx\right)+c_{2}e^{2}\left(\int _{2}^{\infty }f_{k}e^{-x}\,dx\right)+\cdots +c_{n}e^{n}\left(\int _{n}^{\infty }f_{k}e^{-x}\,dx\right)\\Q&=c_{1}e\left(\int _{0}^{1}f_{k}e^{-x}\,dx\right)+c_{2}e^{2}\left(\int _{0}^{2}f_{k}e^{-x}\,dx\right)+\cdots +c_{n}e^{n}\left(\int _{0}^{n}f_{k}e^{-x}\,dx\right)\end{aligned}}}$

Lemma 1. For an appropriate choice of k, ${\displaystyle {\tfrac {P}{k!}}}$ is a non-zero integer.

Proof. Each term in P is an integer times a sum of factorials, which results from the relation

${\displaystyle \int _{0}^{\infty }x^{j}e^{-x}\,dx=j!}$

which is valid for any positive integer j (consider the Gamma function).

It is non-zero because for every a satisfying 0< a ≤ n, the integrand in

${\displaystyle c_{a}e^{a}\int _{a}^{\infty }f_{k}e^{-x}\,dx}$

is e^−x times a sum of terms whose lowest power of x is k+1 after substituting x for x+a in the integral. Then this becomes a sum of integrals of the form

${\displaystyle A_{j-k}\int _{0}^{\infty }x^{j}e^{-x}\,dx}$ Where A_j-k is integer.

with k+1 ≤ j, and it is therefore an integer divisible by (k+1)!. After dividing by k!, we get zero modulo (k+1). However, we can write:

${\displaystyle \int _{0}^{\infty }f_{k}e^{-x}\,dx=\int _{0}^{\infty }\left(\left[(-1)^{n}(n!)\right]^{k+1}e^{-x}x^{k}+\cdots \right)dx}$

and thus

${\displaystyle {\frac {1}{k!}}c_{0}\int _{0}^{\infty }f_{k}e^{-x}\,dx\equiv c_{0}[(-1)^{n}(n!)]^{k+1}\not \equiv 0{\pmod {k+1}}.}$

So when dividing each integral in P by k!, the initial one is not divisible by k+1, but all the others are, as long as k+1 is prime and larger than n and |c₀|. It follows that ${\displaystyle {\tfrac {P}{k!}}}$ itself is not divisible by the prime k+1 and therefore cannot be zero.

Lemma 2. ${\displaystyle \left|{\tfrac {Q}{k!}}\right|<1}$ for sufficiently large ${\displaystyle k}$.

Proof. Note that

${\displaystyle {\begin{aligned}f_{k}e^{-x}&=x^{k}[(x-1)(x-2)\cdots (x-n)]^{k+1}e^{-x}\\&=\left(x(x-1)\cdots (x-n)\right)^{k}\cdot \left((x-1)\cdots (x-n)e^{-x}\right)\\&=u(x)^{k}\cdot v(x)\end{aligned}}}$

where ${\displaystyle u(x)}$ and ${\displaystyle v(x)}$ are continuous functions of ${\displaystyle x}$ for all ${\displaystyle x}$, so are bounded on the interval ${\displaystyle [0,n]}$. That is, there are constants ${\displaystyle G,H>0}$ such that

${\displaystyle \left|f_{k}e^{-x}\right|\leq |u(x)|^{k}\cdot |v(x)|<G^{k}H\quad {\text{ for }}0\leq x\leq n.}$

So each of those integrals composing ${\displaystyle Q}$ is bounded, the worst case being

${\displaystyle \left|\int _{0}^{n}f_{k}e^{-x}\,dx\right|\leq \int _{0}^{n}\left|f_{k}e^{-x}\right|\,dx\leq \int _{0}^{n}G^{k}H\,dx=nG^{k}H.}$

It is now possible to bound the sum ${\displaystyle Q}$ as well:

${\displaystyle |Q|<G^{k}\cdot nH\left(|c_{1}|e+|c_{2}|e^{2}+\cdots +|c_{n}|e^{n}\right)=G^{k}\cdot M,}$

where ${\displaystyle M}$ is a constant not depending on ${\displaystyle k}$. It follows that

${\displaystyle \left|{\frac {Q}{k!}}\right|<M\cdot {\frac {G^{k}}{k!}}\to 0\quad {\text{ as }}k\to \infty ,}$

finishing the proof of this lemma.

Choosing a value of ${\displaystyle k}$ satisfying both lemmas leads to a non-zero integer (${\displaystyle P/k!}$) added to a vanishingly small quantity (${\displaystyle Q/k!}$) being equal to zero, is an impossibility. It follows that the original assumption, that e can satisfy a polynomial equation with integer coefficients, is also impossible; that is, e is transcendental.

The transcendence of π

A similar strategy, different from Lindemann's original approach, can be used to show that the number π is transcendental. Besides the gamma-function and some estimates as in the proof for e, facts about symmetric polynomials play a vital role in the proof.

 

Irrational number

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Irrational_number 

 

The number √2 is irrational.

In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two 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.

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 all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor 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 and 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.

History

Set of real numbers (R), which include the rationals (Q), which include the integers (Z), which include the natural numbers (N). The real numbers also include the irrationals (R\Q).

Ancient Greece

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the pentagram. The then-current 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. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact 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. 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: c² = a²+b² = b²+b² = 2b². (Since the triangle is isosceles, a = b).
• Since c² = 2b², c² is divisible by 2, and therefore even.
• Since c² 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 c² = (2y)², or c² = 4y².
• Substituting 4y² for c² in the first equation (c² = 2b²) gives us 4y²= 2b².
• Dividing by 2 yields 2y² = b².
• Since y is an integer, and 2y² = b², b² is divisible by 2, and therefore even.
• Since b² 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 number and geometry were inseparable–a foundation of their theory.

The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of the discrete to the continuous. This was brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a finite number of units of a given size. Past Greek conceptions dictated that they necessarily must be, for “whole numbers represent discrete objects, and a commensurable ratio represents a relation between two collections of discrete objects,” but Zeno found that in fact “[quantities] in general are not discrete collections of units; this is why ratios of incommensurable [quantities] appear….[Q]uantities are, in other words, continuous.” What this means is that, contrary to the popular conception of the time, there cannot be an indivisible, smallest unit of measure for any quantity. That in fact, these divisions of quantity must necessarily be infinite. For example, consider a line segment: this segment can be split in half, that half split in half, the half of the half in half, and so on. This process can continue infinitely, for there is always another half to be split. The more times the segment is halved, the closer the unit of measure comes to zero, but it never reaches exactly zero. This is just what Zeno sought to prove. He sought to prove this by formulating four paradoxes, which demonstrated the contradictions inherent in the mathematical thought of the time. While Zeno's paradoxes accurately demonstrated the deficiencies of current mathematical conceptions, they were not regarded as proof of the alternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore further investigation had to occur.

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. 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.

As a result of the distinction between number and magnitude, geometry became the only method that could take into account incommensurable ratios. Because previous numerical foundations were still incompatible with the concept of incommensurability, Greek focus shifted away from those numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases algebraic conceptions were reformulated into geometric terms. This may account for why we still conceive of x² and x³ as x squared and x cubed instead of x to the second power and x to the third power. Also crucial to Zeno’s work with incommensurable magnitudes was the fundamental focus on deductive reasoning that resulted from the foundational shattering of earlier Greek mathematics. The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that underlie that theory. Out of this necessity, Eudoxus developed his method of exhaustion, a kind of reductio ad absurdum that “...established the deductive organization on the basis of explicit axioms...” as well as “...reinforced the earlier decision to rely on deductive reasoning for proof.” This method of exhaustion is the first step in the creation of calculus.

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.

India

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). (See Bag, Indian Journal of History of Science, 25(1-4), 1990).

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".

It is also suggested that Aryabhata (5th century AD), in calculating a value of pi to 5 significant figures, used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational).

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.

Middle Ages

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:

"It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes etc."

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:

"their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it."

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, often in the form of square roots, cube 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. Iranian mathematician, Abū Ja'far al-Khāzin (900–971) provides a definition of rational and irrational magnitudes, stating that if a definite quantity is:

"contained in a certain given magnitude once or many times, then this (given) magnitude corresponds to a rational number. . . . Each time when this (latter) magnitude comprises a half, or a third, or a quarter of the given magnitude (of the unit), or, compared with (the unit), comprises three, five, or three fifths, it is a rational magnitude. And, in general, each magnitude that corresponds to this magnitude (i.e. to the unit), as one number to another, is rational. If, however, a magnitude cannot be represented as a multiple, a part (1/n), or parts (m/n) of a given magnitude, it is irrational, i.e. it cannot be expressed other than by means of roots."

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, ${\displaystyle {\frac {3\quad 1}{5\quad 3}}}$." This same fractional notation appears soon after in the work of Leonardo Fibonacci in the 13th century.

Modern period

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 eⁿ 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 π² 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.

Examples

Square roots

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 number. The square roots of all natural numbers which are not perfect squares are irrational and a proof may be found in quadratic irrationals.

General roots

The proof above for 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 kth power of another integer, then that first integer's kth root is irrational.

Logarithms

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

${\displaystyle \log _{2}3={\frac {m}{n}}.}$

It follows that

${\displaystyle 2^{m/n}=3}$

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

${\displaystyle 2^{m}=3^{n}.}$

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 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.

Types

• number theoretic distinction : transcendental/algebraic
• normal/ abnormal (non-normal)

Transcendental/algebraic

Almost all irrational numbers are transcendental and all real transcendental numbers are irrational (there are also complex transcendental numbers): the article on transcendental numbers lists several examples. So e ^r and π ^r are irrational for all nonzero rational r, and, e.g., e^π is irrational, too.

Irrational numbers can also be found within the countable set of real algebraic numbers (essentially defined as the real roots of polynomials with integer coefficients), i.e., as real solutions of polynomial equations

${\displaystyle p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}=0\;,}$

where the coefficients ${\displaystyle a_{i}}$ are integers and ${\displaystyle a_{n}\neq 0}$. Any rational root of this polynomial equation must be of the form r /s, where r is a divisor of a₀ and s is a divisor of a_n. If a real root ${\displaystyle x_{0}}$ of a polynomial ${\displaystyle p}$ is not among these finitely many possibilities, it must be an irrational algebraic number. An exemplary proof for the existence of such algebraic irrationals is by showing that x₀  = (2^1/2 + 1)^1/3 is an irrational root of a polynomial with integer coefficients: it satisfies (x³ − 1)² = 2 and hence x⁶ − 2x³ − 1 = 0, and this latter polynomial has no rational roots (the only candidates to check are ±1, and x₀, being greater than 1, is neither of these), so x₀ is an irrational algebraic 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, 3π + 2, π + √2 and e√3 are irrational (and even transcendental).

Decimal expansions

The decimal expansion of an irrational number never repeats or terminates (the latter being equivalent to repeating zeroes), 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, 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.

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

${\displaystyle A=0.7\,162\,162\,162\,\ldots }$

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:

${\displaystyle 10A=7.162\,162\,162\,\ldots }$

Now we multiply this equation by 10^r 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 10³:

${\displaystyle 10,000A=7\,162.162\,162\,\ldots }$

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:

${\displaystyle 9990A=7155.}$

Then

${\displaystyle A={\frac {7155}{9990}}={\frac {53}{74}}}$

is a ratio of integers and therefore a rational number.

Irrational powers

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

Consider √2^√2; if this is rational, then take a = b = √2. Otherwise, take a to be the irrational number √2^√2 and b = √2. Then a^b = (√2^√2)^√2 = √2^√2·√2 = √2² = 2, which is rational.

Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem shows that √2^√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^b 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

${\displaystyle \left({\sqrt {2}}\right)^{\log _{\sqrt {2}}3}=3.}$

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, ${\displaystyle \log _{\sqrt {2}}3}$, is irrational. This is so because, by the formula relating logarithms with different bases,

${\displaystyle \log _{\sqrt {2}}3={\frac {\log _{2}3}{\log _{2}{\sqrt {2}}}}={\frac {\log _{2}3}{1/2}}=2\log _{2}3}$

which we can assume, for the sake of establishing a contradiction, equals a ratio m/n of positive integers. Then ${\displaystyle \log _{2}3=m/2n}$ hence ${\displaystyle 2^{\log _{2}3}=2^{m/2n}}$ hence ${\displaystyle 3=2^{m/2n}}$ hence ${\displaystyle 3^{2n}=2^{m}}$, 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 ${\displaystyle ((1/e)^{1/e},\infty )}$ can be written either as a^a for some irrational number a or as nⁿ for some natural number n. Similarly, every positive rational number can be written either as ${\displaystyle a^{a^{a}}}$ for some irrational number a or as ${\displaystyle n^{n^{n}}}$ for some natural number n.

Open questions

It is not known if ${\displaystyle \pi +e}$ (or ${\displaystyle \pi -e}$) is irrational. In fact, there is no pair of non-zero integers ${\displaystyle m,n}$ for which it is known whether ${\displaystyle m\pi +ne}$ is irrational. Moreover, it is not known if the set ${\displaystyle \{\pi ,e\}}$ is algebraically independent over ${\displaystyle \mathbb {Q} }$.

It is not known if ${\displaystyle \pi e,\ \pi /e,\ 2^{e},\ \pi ^{e},\ \pi ^{\sqrt {2}},\ \ln \pi ,}$ Catalan's constant, or the Euler–Mascheroni constant ${\displaystyle \gamma }$ are irrational. It is not known if either of the tetrations ${\displaystyle ^{n}\pi }$ or ${\displaystyle ^{n}e}$ is rational for some integer ${\displaystyle n>1.}$

Set of all irrationals

Since the reals form an uncountable set, of which the rationals are a countable subset, the complementary set of irrationals is uncountable.

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.

Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals equipped with the subspace topology have a basis of clopen sets so the space is zero-dimensional.