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.

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.

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

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.

The line that separates the numerator and the denominator is called a vinculum, which is also the word used to describe ‘a connecting band of tissue, such as that attaching a flexor tendon to the bone of a finger or toe’.

incoming cccc arithmetic fractran 2021 malleable computing