Set of four numbers

As a four-set of numbers is called in mathematics a the proportions of numbers in question set out the elements of Euclid .

It is the 19th sentence from Book 7, in which divisibility and prime numbers are dealt with. From this theorem the central theorems of Euclidean number theory are obtained, in particular the fundamental theorem of arithmetic .

Formulation in Euclid

When four numbers are proportioned, the product of the first and fourth is equal to the product of the second and third. And if the product of the first and fourth is equal to the product of the second and third, then such four numbers are proportioned. ${\displaystyle A,B,C,D}$ ${\displaystyle A,B,C,D}$

Explanations

Referring to the definition of "relationship" in Book 5, Euclid defines in Book 7 (Definition 20):

Numbers are proportioned if the first of the second, and the third of the fourth, are either one part or one multiple part.

In today's language, this definition means: For natural numbers , the ratios and in proportion (i.e., it applies ) exactly when there are natural numbers with ${\displaystyle a,b,c,d}$ ${\displaystyle a:b}$ ${\displaystyle c:d}$ ${\displaystyle a:b=c:d}$ ${\displaystyle m,n,x,y}$ ${\displaystyle a=mx,\ b=nx,\ c=my,\ d=ny.}$

The four-digit theorem means in today's language that equality follows from proportionality and vice versa. ${\displaystyle a:b=c:d}$ ${\displaystyle ad=bc}$

Inferences

From the 19th sentence it follows that irreducible elements are prime (Euclid's lemma) as well as the uniqueness of the decomposition into irreducible elements. It is equivalent to that from and always follows. ${\displaystyle a\mid bc}$ ${\displaystyle ggT(a,b)=1}$ ${\displaystyle a\mid c}$

Generalizations

The four-number theorem applies to all GCD rings ( integrity rings in which every two elements have a greatest common factor ). It is especially true in Euclidean rings .

literature

Franz Lemmermeyer : To the number theory of the Greeks. Part 1, Mathematical Semester Reports, Volume 55, 2008, pp. 181–195.

Individual evidence

1. ↑ ^{a b c} Lemmermeyer, op. Cit.
2. ↑ Euclid's Elements, fifteen books , translated by Johann Friedrich Lorenz, Halle 1781 ( online )
3. ↑ Euclid's Elements, fifteen books , translated by Johann Friedrich Lorenz, Halle 1781 ( online )