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Griechische Mathematik: Entdeckung der irrationalen Zahlen

If there be an area or field, whose form is a square, and it is required to set out another field whose form is also to be a square, but double in area, as this cannot be accomplished by any numbers or multiplication, it may be found exactly by drawing lines for the purpose, and the demonstration is as follows. A square plot of ground ten feet long by ten feet wide, contains an hundred feet; if we have to double this, that is, to set out a plot also square, which shall contain two hundred, we must find the length of a side of this square, so that its area may be double, that is two hundred feet. By numbers this cannot be done; for if the sides are made fourteen feet, these multiplied into each other give one hundred and ninety-six feet; if fifteen feet, they give a product of two hundred and twenty-five. Marcus Vitruvius Pollio, de Architectura, Book 9

The Greek discovery of geometrical magnitudes that cannot be expressed as rational numbers (numbers expressed as a fraction of integers for example 5/6) – such as the diagonal of a square with sides of size one unit – is remarkable. Vitruvius says that “by numbers this cannot be done” as irrational numbers were unknown. The fact is used twice by Aristotle as a (presumably) well-known example of argument by reductio ad absurdum (Prior Analytics 1.23.41a26 and 1.44.55.a37), and is clearly older than him.

It is always possible to find a rational number between 2 other rational numbers a, b such as the rational number 1/2 *(a + b). For example between 1/3 and 1/2 we have the number 5/12 = 1/2 *(1/3 + 1/2) and by repeating this procedure also for the numbers a and 1/2 *(a + b) and so forth we can find actually infinite rational numbers between a and b. It is therefore not surprising to assume that any number can be expressed as a ratio of integer numbers. The Babylonians already provided an approximation of some irrational numbers such as the square root of 2 with a 4 decimal places accuracy, but Greek mathematicians discovered that whatever this accuracy is it is never exact and it is impossible to expressed these as a ratio of integer numbers.

The number system concerns whole numbers and fractions. In practical computation for building and surveying however, quantities can be more complex. Some arise geometrically as ratios, and turn out to be irrational numbers. A rational number is one which can be expressed as p/q, where p and q are whole numbers and q≠0. An irrational number cannot be so expressed, e.g. √2. Greek geometry involves irrational ratios and measurements that can be manipulated easily by geometrical methods but not represented in their number system and therefore in calculations. These ratios arise naturally, commonly involving √2, √3, … or π. It is commonly held that there was a crisis in Greek mathematics caused by the discovery of irrationals, specifically of quantities inexpressible in the number system. This traditional view sees the crisis as causing a shift in the focus of mathematical activity from number (on which e.g. the Pythagoreans had concentrated) to geometry.

Pythagoras and his students believed the essential unity of things was not in a physical substrate. For them, the one thing that formed the substrate of all things in the universe was number and numerical relations.

According to the theorem of Pythagoras the sum of the squares on the sides of a right-angled triangle is equal to the square of the hypotenuse.

If we have a right-angled triangle whose sides are equal, the square of the hypotenuse is twice the square of one of the sides. The problem is that the square of one whole number cannot be twice the square of another. The hypotenuse cannot be a whole number whatever the length of the two equal sides. If the adjacent and opposite sides contain the same number of atoms then the hypotenuse must contain one incomplete atom. This contradicts the notion of an atom as an indivisible basic unit.

[Theodorus] was proving to us a certain thing about square roots, I mean the side (i.e. root) of a square of three square units and of five square units, that these roots are not commensurable in length with the unit length, and he went on in this way, taking all the separate cases up to the root of seventeen square units, at which point, for some reason, he stopped. Plato

Theodorus of Cyrene (465-398 BC) was a teacher of Plato and Thaetetus. He provided the proof of the irrationality of all integer numbers between 3 and 17 except the square numbers 4, 9 and 16 (the case for n = 2 was well-known before him).

His contribution to Mathematics is part of Euclid'd Element, Book X and XIII.

Hippasus also showed that the ratio (diagonal / side) of a regular Pentagon cannot be expressed as a ratio of integers.

He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side.

The Irrationality of

Prove that is an irrational number.

The number is irrational, i.e. it cannot be expressed as a ratio of integers a, b. Let us assume that is rational so that we may write

= a/b (1)

for a and b = any two integers. To show that is irrational, we must show that no two such integers can be found. We begin by squaring both sides of eq. 1:

2 = a2/b2 or 2b2 = a2 (2)

From Equation 2, we must conclude that a2 (and, therefore, a) is even; b2 (and, therefore, b) may be even or odd. If b is even, the ratio a2/b2 may be immediately reduced by canceling a common factor of 2. If b is odd, it is possible that the ratio a2/b2 is already reduced to smallest possible terms. We assume that b2 (and, therefore, b) is odd.

Now, we set a = 2m, and b = 2n + 1, and require that m and n be integers (to ensure integer values of a and b). Then

a2 = 4m2 (3)


b2 = 4n2 + 4n + 1 (4)

Substituting these expressions into (2), we obtain

2(4n2 + 4n + 1) = 4m2 4n2 + 4n + 1 = 2m2 (5)

The L.H.S. of eq. 5 is an odd integer. The R.H.S., on the other hand, is an even integer. There are no solutions for eq. 5. Therefore, integer values of a, b which satisfy the relationship = a/b cannot be found. We conclude that is irrational.

They say that the man (Hipassus) who first divulged the nature of commensurability and incommensurability to men who were not worthy of being made part of this knowledge, became so much hated by the other Pythagoreans, that not only they cast him out of the community; they built a shrine for him as if he were dead, he who had once been their friend. Others add that even the god became angry with him who had divulged Pythagoras' doctrine; that he who showed how the icosagon (that is the dodecahedron, one of the five solid figures) can be inscribed within a sphere, died at sea like an evil man. Others still say that the same misfortune happened on him who spoke to others of irrational numbers and incommensurability. Hyamblicus (or Iamblichus of Chalkis), De vita pythagorica 246-247

This was not the only crisis for the Pythagoreans, believing that the circle is the most perfect motion the discovery of the retrograde motion of planets was another shocking discovery.

How this discovery changed the history of Mathematics and how it would develop without it is difficult to answer: Was the influence so strong that the Greeks considered mathematical problems and their solution from a geometric and less from an algebraic point and did it require so much time until more algebraic oriented mathematicians like Diophantus emerged?

Ancient Greeks and Prime Numbers

A continuation of Socrates' dialogue with Meno in which the boy proves that the root 2 is irrational (by an anonymous author) (http://www.gutenberg.org/etext/254)

Music of the Ellipses

The Pythagorean Theorem, A version prooved by Babylonians and another proof the irrationality of square root of 2

Theodorus of Cyrene

See also

Irrational number to an irrational power may be rational

What is a number? (From Rational , Irrational to Surreal Numbers)

Ancient Greece

Medieval Greece / Byzantine Empire

Modern Greece

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