Commensurability (mathematics)

In mathematics, two real numbers and commensurable hot (lat. together measurable) if they are integer multiples of a suitable third real number, so have a common divisor. The name comes from the fact that they can then measure with the common measure. In mathematical notation:

It follows that the ratio of a rational number, and is:

Are there no matter how small common measure, then called the number and incommensurable values ​​, their ratio is an irrational number.

The term incommensurability, which goes back to Euclid's Elements, is directly related to the geometric measure distances with actual benchmarks. He is a good reminder that Greek mathematics was based directly on the descriptive geometry whose " clarity " has just exceeded by the incommensurability.

Examples

• All natural numbers are commensurable, because they have the comparative measure c = 1
• Finally, many arbitrary breaks are commensurable, because you can bring them to a common denominator, and a comparative measure is then.
• Incommensurable to the fraction numbers, however, are all the numbers that can not be written as fractions.
• The side a of a square and the length d of its diagonals are incommensurable, because according to the Pythagorean theorem, and the assumption that this is not a whole number, can be refuted.
• Incommensurable distances there are also the five- star or pentagram, namely the inner distance (BC ), and the outer route (AD).

History

The first evidence for the existence of incommensurable distances is attributed since ancient times the Pythagoreans Hippasos of Metapontum, who lived in the late 6th and early 5th century BC. This tradition may correspond to the facts. However, the invention is the fact CONTINUED legend according to the Pythagoreans treated the incommensurability as a mystery; Hippasos should have betrayed this secret, which apparently had resulted in his death. This story is the result of a misunderstanding. In connection with the legend of secret treachery, the hypothesis was in senior research literature represented, the discovery of incommensurability have shocked the Pythagoreans and have triggered a foundational crisis of mathematics and the philosophy of mathematics. The assumption of a fundamental crisis, however, as well as the alleged betrayal of secrets by recent research unanimously rejected. The discovery of incommensurability was regarded as an achievement rather than a problem or crisis.