Rule of the middle numbers

The rule of the middle numbers , French regle des nombres moyens , is a mathematical theorem from the field of analysis , which is attributed to the French mathematician Nicolas Chuquet . The theorem contains two elementary inequalities of fractions on the relationship of the so-called mediants to their original fractions .

Formulation of the rule

If you have two fractions with positive denominators on the number line and you form a third fraction whose numerator is equal to the sum of the numerators and whose denominator is equal to the sum of the denominators of the two given fractions, this third fraction always lies between the two given fractions . Expressed in a formula:

For four real numbers with , the inequalities always follow from the inequality ${\displaystyle a,b,x,y}$ ${\displaystyle b,y>0}$ ${\displaystyle {\frac {a}{b}}<{\frac {x}{y}}}$ ${\displaystyle {\frac {a}{b}}<{\frac {a+x}{b+y}}<{\frac {x}{y}}}$ .

The same applies if instead of the small character the less equal to sign exists.

example

For true and therefore ${\displaystyle a=-2,b=3,x=-1,y=2}$ ${\displaystyle {\frac {-2}{3}}=-0{,}{\bar {6}}<-0{,}5={\frac {-1}{2}}}$

${\displaystyle {\frac {-2}{3}}<{\frac {(-2)+(-1)}{3+2}}=-0{,}6<{\frac {-1}{2}}}$.

Explanations and Notes

• One calls occurring above average breaking the mediant of the two output breaks and . ${\displaystyle {\frac {a+x}{b+y}}}$ ${\displaystyle {\frac {a}{b}}}$ ${\displaystyle {\frac {x}{y}}}$
• Starting with the two fractions and gradually forming mediants you arrive at a typical Farey sequence . ${\displaystyle {\frac {0}{1}}}$ ${\displaystyle {\frac {1}{1}}}$
• As the lexicon of eminent mathematicians expressly emphasizes, Nicolas Chuquet claimed the rule of the mean numbers as a discovery of his own.

swell

• Howard Eves : An Introduction to the History of Mathematics (=  The Saunders Series ). 5th edition. Saunders College Publishing , Philadelphia et al. 1983, ISBN 0-03-062064-3 , pp. 214 ( MR0684360 ).
• Siegfried Gottwald , Hans-Joachim Ilgauds and Karl-Heinz Schlote (ed.): Lexicon of important mathematicians . Verlag Harri Deutsch , Thun 1990, ISBN 3-8171-1164-9 , p. 103-104 ( MR1089881 ).
• Guido Walz [Red.]: Lexicon of Mathematics in six volumes . Third volume. Inp to Mon. Spektrum Akademischer Verlag , Heidelberg, Berlin 2001, ISBN 3-8274-0436-3 .

Individual evidence

1. ↑ ^{a b} Siegfried Gottwald et al. (Ed.): Lexicon of important mathematicians. 1990, p. 104
2. ^ Howard Eves: An Introduction to the History of Mathematics. 1983, p. 214
3. ↑ ^{a b c} Guido Walz [Red.]: Lexicon of Mathematics. Third volume. 2001, p. 397
4. ↑ In Eve's Introduction to the History of Mathematics , the positivity of the four numbers is assumed, while the lexicon of important mathematicians does not mention any requirements. In any case, the possible case must be ruled out. The situation is then unproblematic if it is accepted as a convention that the sign of a fraction is fundamentally part of the numerator, i.e. that there is always a positive denominator. ${\displaystyle {\text{'}}}$ ${\displaystyle b+y=0}$