Broken trunk

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The stem fraction is a term from mathematics and describes a fraction with a 1 in the numerator and any natural number in the denominator. This results in original fractions as the reciprocal of natural numbers. Examples are the stem fractions and , while there is no stem fraction. ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle {\frac {1}{6}}}$ ${\displaystyle {\frac {2}{3}}}$

Trunk break development

Any fraction of the form with natural numbers can be represented as the sum of stem fractions (and a natural number, if any ). For example ${\displaystyle {\frac {a}{b}}}$ ${\displaystyle a,b}$ ${\displaystyle a>b}$

${\displaystyle {\frac {2}{3}}={\frac {1}{2}}+{\frac {1}{6}}}$

One method for developing the stem fraction is to first subtract the integer part and then subtract the largest stem fraction that is less than or equal to the rest (this is called a greedy algorithm ).

Procedure

With this method, a real abbreviated fraction is broken down into a sum of original fractions, where all original fractions have different denominators:

A real, already shortened fraction is given: with . ${\displaystyle {\frac {a}{b}}}$ ${\displaystyle a<b}$

Step 1

Form the new fraction , where: and and minimal, i.e. H., ${\displaystyle {\frac {c}{d}}}$ ${\displaystyle c=a}$ ${\displaystyle d=n\cdot a\;>b\;,n\in \mathbb {N} }$ ${\displaystyle n}$

the new numerator is equal to the old numerator, and the new denominator is equal to the least multiple of the old numerator that is greater than the old denominator.

The new fraction can always be shortened to the trunk fraction due to the formation regulation . ${\displaystyle {\frac {1}{n}}}$

2nd step

So it applies with . ${\displaystyle {\frac {a}{b}}={\frac {c}{d}}\;+\;{\frac {a}{b}}\;-\;{\frac {c}{d}}}$ ${\displaystyle {\frac {c}{d}}={\frac {1}{n}}}$

3rd step

Calculate the difference . ${\displaystyle {\frac {a}{b}}={\frac {c}{d}}+\left({\frac {a}{b}}-{\frac {c}{d}}\right)={\frac {1}{n}}+{\frac {na-b}{nb}}}$

4th step

If possible, reduce the difference . ${\displaystyle {\frac {na-b}{nb}}}$

5th step

Abort the procedure if the difference is a fraction of the stem, otherwise repeat steps 1 to 4 for the difference . ${\displaystyle {\frac {na-b}{nb}}}$ ${\displaystyle {\frac {na-b}{nb}}}$

Example: ${\displaystyle {\frac {2}{3}}}$

1. Step: New break: ${\displaystyle {\frac {2}{4}}}$
2. Step: ${\displaystyle {\frac {2}{3}}={\frac {2}{4}}+{\frac {2}{3}}-{\frac {2}{4}}}$
3. Step: ${\displaystyle {\frac {2}{3}}={\frac {2}{4}}+\left({\frac {2}{3}}-{\frac {2}{4}}\right)={\frac {1}{2}}+{\frac {2}{12}}}$
4. Step: ${\displaystyle {\frac {2}{3}}={\frac {1}{2}}+{\frac {1}{6}}}$
5. Step: The process is canceled because the difference is already a trunk break. ${\displaystyle {\frac {1}{6}}}$

This process always ends after a finite number of steps. However, it does not always provide the shortest possible representation as the sum of fractions. For example, this method provides the representation

${\displaystyle {\frac {59}{120}}={\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{65}}+{\frac {1}{10920}}}$,

but there is the shorter version

${\displaystyle {\frac {59}{120}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{8}}}$

history

The ancient Egyptians only recorded real fractions. Since they only had hieroglyphs for stem fractions apart from 2/3, they had to split all other fractions into sums of stem fractions ( see also Egyptian numerals ).

Leonardo Fibonacci published the above algorithm in Liber abaci ( 1202 ). The British mathematician James Joseph Sylvester was only able to prove the general validity of the algorithm in 1880 .

Further occurrences

An unsolved mathematical problem in connection with the development of trunk fractures is the Erdős-Straus conjecture .

Some statistically recorded quantities are distributed proportionally to stem fractions; this represents a simple Zipf distribution .