Fractions in math can be complex. When you have a command of the basic vocabulary and this pair of important identities, you’ll be better able to understand the more difficult equations that will come later in your math education.

We will follow two rules here, to keep matters simple. First, we will never divide by zero. Second, a “number” is usually a positive integer, unless stated otherwise. Usually mathematicians re-state these conditions every time, but that would just make this article longer than necessary.

## What are Fractions?

In math, a fraction represents part of a whole. Often a fraction has a value between zero and one. Written as “2/3”, this fraction also means “divide two by three”.

## Numerator, Denominator, Proper versus Improper, and Mixed Fractions

In fractions such as “1/3” or “a/b”, ‘1’ and ‘a’ are numerators. The ‘3’ and ‘b’ are denominators.

In a proper fraction, the numerator is less than the denominator, so the value is less than one.

An improper fraction has a numerator greater than its denominator. “7/3” is an example.

Often we convert an improper fraction into the mixed fraction of the same value: “7/3” = “2 + 1/3” = “2 1/3” (in recipe books but not often in math textbooks), or “two and one-third”. It’s very uncommon to have a mixed improper fraction such as “1 + 4/3”.

## Common Denominator, Least Common Multiple, Factors, Prime Factors and Common Factors

Two common phrases used with fractions are common denominator and least common multiple, or LCM.

The fractions “1/3” and “2/3” share the common denominator, ‘3’. Likewise “a/b” and “c/b” have ‘b’ as the common denominator.

In general, a common multiple of two numbers is any third number that can be evenly divided by each of the first two. For example, ‘300’ is a common multiple of ‘2’ and ‘5’.

However, the least common multiple of ‘2’ and ‘5’ is ’10’. The LCM is the lowest number that can be divided by the first two.

Here is another example: the LCM of ’50’ and ’75’ is ‘150’, *not* “50 * 75 = 3750”.

Decoding Science. One article at a time.