 ## A fraction represents a part of a whole.

A fraction consists of a numerator displayed above a line, and a denominator below.

• An proper fraction must be less than 1, like `3/4` and `7/12`.
• An improper fraction is more than 1, like `9/2` and `13/4`.
• A mixed fraction is a whole number and a portion less than 1 together, like `2 3/4`.
• A reciprocal is another fraction with the numerator and denominator exchanged, like `3/7` for `7/3`.

An interesting aspect of fractions and prime factorization is that multiplying fractions is the same as adding the prime numerators and subtracting the prime denominators.

Multiplication
42 5 / 14 15
21 x 31 x 71 / 1 5+1 / 2-1 x 7-1 31 x 51 / 1

For example, multiplying 42 by `5/14` means incrementing the power of prime 5, and decrementing the power of primes 2 and 7 — For a result of 42. A division is simply the inversion of the effects of the numerator and denumerator.

Division
15 5 / 14 42
31 x 51 / 1 5-1 / 2+1 x 7+1 21 x 31 x 71 / 1

To add fractions containing unlike quantities , it is necessary to convert all amounts to like quantities.

```1/4 + 1/3
1*3/4*3 + 1*4/3*4
3/12 + 4/12 = 7/12```

The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator.

## Multiplication

To multiply fractions, multiply the numerators and multiply the denominators.

`2/3 * 3/4 = 6/12`

This particularity serves as the basis for the Fractran programming language, it is also of interest for reversible computation.

### Reducing

Dividing the numerator and denominator of a fraction by the same non-zero number yields an equivalent fraction: if the numerator and the denominator of a fraction are both divisible by a number (called a factor) greater than 1, then the fraction can be reduced to an equivalent fraction with a smaller numerator and a smaller denominator.

#### Recursive Method

```function gcd(a, b)
if b = 0
return a
else
return gcd(b, a mod b)```

## Comparing

Comparing fractions with the same positive denominator yields the same result as comparing the numerators.

Incoming: cccc arithmetic fractran