➗ Rational Numbers: Fractions on the Number Line

Where do fractions and decimals sit on the number line? Minus 3 over 4, one half, minus 1.25. Every one of them has a place! Today we explore rational numbers on the number line.

On the number line, the numbers to the right of 0 are positive and the numbers to the left are negative. Going right, numbers get bigger. Going left, they get smaller.

But there are numbers between 0 and 1 too! Split the gap into 2 equal parts and we find 1/2. From 0, split the left side into 4 equal parts and move 3 parts, and we reach −3/4.

We can write the same points as decimals: 1/2 = 0.5 and −3/4 = −0.75. And −1.25 sits between −1 and −2, close to −1: it is −5/4.

A rational number is a number that can be written as a over b. Here a and b are integers and b is not zero. Every integer is rational: 3 = 3/1. So are terminating decimals: 0.2 = 1/5.

Let’s turn a fraction into a decimal by dividing. 3 divided by 4 is 0.75: the division stops. 1 divided by 3 is 0.333… and repeats forever: it is a recurring decimal. We write it with a dot over the repeating digit.

On the number line, the number further to the right is bigger. Minus 1.25 is further left than minus 3/4, and that is further left than 1/2. In order: −1.25 < −3/4 < 1/2.

If we cannot draw the line, we make the denominators equal. Let’s write 2/3 and 3/4 in twelfths: 2/3 = 8/12, 3/4 = 9/12. With a common denominator, the bigger numerator means the bigger number: 8/12 < 9/12. So 2/3 < 3/4.

Careful with negatives! −2/3 = −8/12 and −3/4 = −9/12. Minus 8/12 is closer to 0, so it is further right: −2/3 > −3/4. Among negative numbers, the one closer to 0 is bigger.

Summary: rational numbers have the form a over b, and b cannot be zero. Every rational number has a place on the number line. Its decimal either stops or repeats. To compare, use a common denominator or decimals: the number on the right is bigger. Now try it in Frost & Flame!