Incommensurability (mathematics)

This article deals with commensurability and incommensurability of real numbers. For commensurability in group theory and topology, see Commensurability (Mathematics) .

In mathematics , two real numbers are called and commensurable (from Latin commensurabilis , `` to be measured equally, evenly '' ) if they are integer multiples of a suitable third real number , i.e. have a common factor . The name comes from the fact that you can then measure it with the common measure . In mathematical notation : ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle c}$

${\displaystyle \exists c\in \mathbb {R} }$so with . ${\displaystyle a=mc\land b=nc}$ ${\displaystyle a,b\in \mathbb {R} ;m,n\in \mathbb {Z} }$

It follows that the ratio of and is a rational number : ${\displaystyle x}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle {\frac {a}{b}}={\frac {m}{n}}=x,\quad x\in \mathbb {Q} }$.

If there is no common measure , no matter how small , then the numerical values ​​are called and incommensurable (from Latin incommensurabilis , ' unmeasurable ' ), i.e. H. their ratio is an irrational number . ${\displaystyle c}$ ${\displaystyle a}$ ${\displaystyle b}$

The term incommensurability, which goes back to Euclid's elements , relates directly to the geometrical measurement of distances with actual measuring rods . It is a good reminder that Greek mathematics was based directly on descriptive geometry, the "descriptiveness" of which was exceeded by incommensurability.

Examples

Five star

• All natural numbers are commensurable because they have the comparative measure c = 1.
• Finally, many arbitrary fractions are commensurable , because they can be reduced to a main denominator , and then a comparison measure is . ${\displaystyle N}$ ${\displaystyle c={\tfrac {1}{N}}}$
• In contrast, all numbers that cannot be written as fractions are incommensurable with the fractions.
• The side a of a square and the length d of its diagonal are incommensurable because, according to the Pythagorean Theorem, is , and the assumption that this is a fraction, can be refuted. ${\displaystyle {\tfrac {d}{a}}={\sqrt {2}}}$
• There are also incommensurable lines in the five-star or pentagram , namely the inner line (BC) and the outer line (AD).

history

The first evidence of the existence of incommensurable stretches since ancient times is ascribed to the Pythagorean Hippasus of Metapontus , who lived in the late 6th and early 5th centuries BC. Lived. This tradition may correspond to the facts. One invention, however, is the related legend, according to which the Pythagoreans treated incommensurability as a secret; Hippasus is said to have revealed this secret, which allegedly resulted in his death. This story arose out of a misunderstanding. In connection with the legend of the betrayal of secrets, the hypothesis was put forward in older research literature that the discovery of incommensurability shocked the Pythagoreans and triggered a fundamental crisis in mathematics and the philosophy of mathematics. However, the assumption of a fundamental crisis, like the alleged betrayal of secrets, is unanimously rejected by recent research. The discovery of incommensurability was seen as an achievement rather than a problem or crisis.

See also

• Coprime numbers for natural numbers
• Golden cut

literature

• H. Vogt: The history of the discovery of the irrational according to Plato and other sources of the 4th century, Bibliotheca Math. (3) 10, 97–155 (1910).
• E. Frank: Plato and the so-called Pythagoreans , Niemeyer, Halle, 1923.
• BL van der Waerden: Zeno and the basic crisis of Greek mathematics. Math. Ann. 117, (1940). 141-161, doi: 10.1007 / BF01450015 .
• K. v. Fritz: The discovery of incommensurability by Hippasus of Metapontum. Ann. of Math. (2) 46, (1945). 242-264. online .
• M. Caveing: The debate between HG Zeuthen and H. Vogt (1909-1915) on the historical source of the knowledge of irrational quantities. Centaurus 38 (1996), no. 2-3, 277-292, doi: 10.1111 / j.1600-0498.1996.tb00611.x .

Individual evidence

1. ^ Karl Ernst Georges : Comprehensive Latin-German concise dictionary . 8th, improved and increased edition. Hahnsche Buchhandlung, Hannover 1918 ( zeno.org [accessed July 30, 2019]).
2. ^ Karl Ernst Georges : Comprehensive Latin-German concise dictionary . 8th, improved and increased edition. Hahnsche Buchhandlung, Hannover 1918 ( zeno.org [accessed July 30, 2019]).
3. ^ David H. Fowler: The Mathematics of Plato's Academy . A new reconstruction . Clarendon Press, Oxford 1987, ISBN 0-19-853912-6 , pp. 302-308; Hans-Joachim Waschkies: Beginnings of arithmetic in the ancient Orient and with the Greeks . Verlag Grüner, Amsterdam 1989, ISBN 90-6032-036-0 , p. 311 and note 23; Walter Burkert: Wisdom and Science. Studies on Pythagoras , Philolaus and Plato . Verlag Carl, Nuremberg 1962, pp. 431-440; Leonid Zhmud: Science, Philosophy and Religion in Early Pythagoreanism . Akademie-Verlag, Berlin 1997, ISBN 3-05-003090-9 , pp. 170–175; Detlef Thiel: The philosophy of Xenocrates in the context of the old academy . Saur, Munich 2006, ISBN 3-598-77843-0 , p. 94, note 65 (plus habilitation, Heidelberg University 2005).