A **fraction** is most easily thought of as representing some *portion* of a unit.

The simplest fractions to think about are numbers like "one half" or "one tenth." For example, *one third* is the number you get when you divide *one* into three equal parts and take one of them.

We write *one third* as .

Every fraction looks like that, having an integer on top, which we call the **numerator**, and an integer on the bottom, which we call the **denominator**. This *fractional notation* represents the number that we get when we divide the numerator by the denominator. In our simple examples we had a numerator of 1.

So, what do we mean by the expression ?

When we read this in English, we say "two thirds." That sounds like we have "two" of something, namely two "thirds" or two of: , which we naturally think of as or . But in our more technical definition of a fraction we said it was a number we get by dividing a pair of integers, so that is *defined* as .

Well fortunately for our intuitions, it will turn out that .

Thinking about a fraction as some portion of a unit works nicely so long as the numerator is less than the denominator. But what do we mean by ? Well, intuitively, we think of having four of , which, thinking about huge slices of cake, must surely be equal to . And, sure enough, if you do the division of 4 by 3 you'll end up with 1.33333..., which is 1 + 0.3333... or .

Fractions can represent whole numbers, too. Nothing but our boring desire for simplicity and clarity keeps us from writing 12 as whenever we like.

Further topics in fractions: