## 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`

.

Number | Primes | ||
---|---|---|---|

2 | 3 | 5 | |

2/3 | 1 | 1 | 0 |

27/25 | 0 | 3 | 2 |

100/9 | 2 | 2 | 2 |

Multiplying a number by a fraction, is the same as adding the prime numerators and subtracting the prime denominators. For example, multiplying 18, which is made of `2^1 x 3^2`

, by `2/3`

means incrementing prime 2, and decrementing the prime 3, or `2^2 x 3^1`

.

To experiment with primes, have a look at Fractran and CCCC.

### 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.

## Addition/Subtraction

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

**Incoming**: cccc arithmetic fractran