Irrational number

An irrational number is a real number that is not a rational number. An irrational number is characterized in that it is not a ratio of integers. The term "ratio" Here we are talking money and not, as in everyday parlance reason. Known irrational numbers, for example, Euler's number, the county number, the square root of two and the division ratio of the golden section.


A real number is called irrational if it can not be expressed as a fraction of two integers, that is not as with and.

In contrast to rational numbers can be represented as a finite or periodic decimal irrational numbers are those whose decimal not break and is not periodic.

There are two types of irrational numbers:

  • Algebraic numbers (such as roots, for example, or ) and
  • Transcendental numbers ( the circle number π = 3.14159 ..., the Euler number e = 2.71828 ... ).

The set of irrational numbers can also be short than write with a set of real numbers and as a set of rational numbers. In addition, it is often used as an indication of the amount of the irrational numbers.

Discovery of the irrationality

The first evidence for irrational in size, it was in Ancient Greece in the 5th century BC with the Pythagoreans. Definitions for irrational numbers that satisfy the current requirements of accuracy, can already be found in the elements of Euclid. Translations into today's language of mathematics first gave to Karl Weierstrass and Richard Dedekind.

If you have a square with sides of length 1 and calculates its diagonal, it follows from the Pythagorean theorem. The positive solution of this equation is called today. Leaves for Greek mathematician, the question arose as to whether the length of the diagonal exactly by a ratio of two integers p and q, ie, a fraction p / q represent. A proof by contradiction, which is still taught in school is handed down to us from Euclid. Whether the discovery of the irrationality by applying the Pythagorean theorem was carried on square or, as Kurt von Fritz said, through constant pitch on the pentagram, is unknown.

The older history of science research assumed that the discovery of the irrationality of a " foundational crisis " of the then Greek mathematics and the Pythagorean theory of numbers led. It was previously assumed that is the basic assumption that everything is expressed by integer ratios, and the refutation of this view have shaken the world view of the Pythagoreans. Thus an ancient legend has been associated, according to which the Pythagoreans Hippasos of Metaponto in the 5th century BC by the written notice of this discovery made ​​a secret of betrayal and later drowned in the sea, which was interpreted as divine punishment. Some of the sources narrated Hippasos myself have discovered the irrationality. Science historians now believe that there has not been such a crisis and the irrationality was not considered a secret. One possible explanation for the betrayal legend is that it was created by a misunderstanding, because the Greek adjective that was used for " irrational" (in the mathematical sense), also the meanings " unspeakable " and " secret" was. But the fact is that Greek mathematics changed fundamentally in the period after Hippasos.

Numbers whose irrationality is proved

  • Even the Pythagorean Archytas proved the irrationality of natural numbers. The proof for the case in Euclid's Elements, delivered ( Euclid's proof of the irrationality of the square root of 2). The set of Archytas generalized Euclid himself in his music theory in which he proved the irrationality of any roots.
  • Another important quadratic irrational number is the golden ratio.
  • The Euler number is irrational. This Leonhard Euler proved in 1737. Your transcendence was proved in 1873 by Charles Hermite.
  • 1761 Johann Heinrich Lambert proved the irrationality of the circle number, their transcendence was proved in 1882 by Ferdinand von Lindemann.
  • The non-integer zeros of a polynomial with integer coefficients are normalized irrational. In particular, the square roots of non- square numbers are irrational.
  • In 1979, Roger apery proved the irrationality of apery constant
  • Is transcendent.
  • Is transcendent, this Carl Ludwig Siegel proved.
  • The lemniscate constant = 2.622057 ... is transcendent. ( Theodor Schneider, 1937)

Numbers whose irrationality is suspected

The irrationality of the numbers is suspected, but is still unproven. One can easily see that at least one of any pair formation must be irrational. This is generally true for any two transcendental numbers a and b.

It is even for a single pair of whole, non-zero integers m and n known whether π n m is irrational. However, it is known that in the case of the existence of linear combinations of the rational value assumes a constant value. Furthermore, it is unknown whether 2e, πe, π √ 2, ππ, ee or the Euler constant γ = 0.57721 ... are irrational. However, it seems reasonable to suspect this.

The uncountability of irrational numbers

Like the first diagonal argument of Cantor shows that the set of rational numbers is countable; so there is a sequence of rational numbers, which contains every rational number. Cantor's second diagonal argument proves that there are uncountably many real numbers. This means at the same time that there must be uncountably many irrational numbers; because otherwise the real numbers would be as a union of two countable sets is countable itself.

Cantor has further shown that the set of algebraic numbers, with all root expressions belong, is still countable. In addition, is that the algebraic hull of each countable subset of the real or complex numbers ( such amounts can especially be made of transcendental numbers ) is also countable, so be sure not contain all real numbers.