Stemfractions

This article describes how a fraction is written as the sum of stem fractions.
A stem fraction is a fraction that has 1 as numerator.

A (Windows) program is added that accomplishes the task.
Click on the download (lightning) icon at the top of this page.
There is no installation procedure, simply copy the program to a folder of choice.

In ancient Egypt people did not know fractions like we do.
Instead, stemfractions were used.
However there were some exceptions:
2
3
 and
3
4
 , using special symbols.

So
2
3
 was called “the two parts”.
The addition to 1 is “the third part”.

Each fraction may be written as the sum of different stemfractions.
8
11
 = 
1
2
 + 
1
5
 + 
1
37
 + 
1
4070
But this result is not unique:
2
19
 = 
1
10
 + 
1
190
 , but also
2
19
 = 
1
11
 + 
1
70
 + 
1
14630

However, if we subtract the biggest possible stemfraction, the split up is unique.
This is the procedure for stemfraction splitting.

Take
5
11
 .
Because 2 < 
11
5
 < 3 we have
1
3
 < 
5
11
 < 
1
2
So
1
3
 is the biggest stemfraction that is smaller than
5
11
 .
Decrease
5
11
 by
1
3

This yields
5
11
 − 
1
3
 = 
4
33
 .
Because 8 < 
33
4
 < 9 we get
1
9
 < 
4
33
 < 
1
8
 .
So
1
9
 the biggest stemfraction smaller than
4
33
 .
Decrease
4
33
 by
1
9
 .
This yields
4
33
 − 
1
9
 = 
1
99
 .
And we found
5
11
 = 
1
3
 + 
1
9
 + 
1
99
 .

The same procedure is used over and over.
In this proces, the denominator may grow very large.
In the program, the denominator size may be 19 digits long, otherwise a "denominator overflow" is reported.

Fractions that produce an overflow are: 44/53 , 5/61 , 60/67....
The biggest -non overflow- result is the splitup of 182/193

Interested in the program?
[HERE ] is the complete Delphi-7 project.