What Does 'Between Two Numbers' Mean?

The number line is one of the most powerful tools in mathematics. It lets us see exactly where any number sits relative to every other number. Between any two points on the number line, no matter how close together they are, there are always more numbers hiding in the gaps.

This is especially true for rational numbers — numbers that can be written as fractions or ratios of whole numbers.

Rational Number

A number that can be written in the form $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq = 0$ . Examples include $\frac{1}{2}$ , $- \frac{3}{4}$ , $2 \frac{1}{5}$ , and even whole numbers like $3 = \frac{3}{1}$ .

The key idea of this subtopic is: given any two rational numbers, you can always find more rational numbers between them. In fact, you can always find infinitely many!

Analogy

Imagine zooming in on a map. No matter how close together two towns look, when you zoom in further you can always see more roads, villages, and landmarks between them. The number line works the same way — zoom in between any two fractions and you'll always find more fractions.

Note

You may have heard of irrational numbers like $2$ or $π$ — these also live on the number line but cannot be written as a fraction of two integers. In this subtopic, we focus specifically on rational numbers and how to find them between any two given values.

Fractions on the Number Line

To place fractions on a number line, we divide each gap between consecutive whole numbers into equal parts. The number of equal parts matches the denominator of the fraction.

If the denominator is 3, divide each unit gap into 3 equal parts — this gives us thirds.
If the denominator is 4, divide into 4 equal parts — quarters.
If the denominator is 6, divide into 6 equal parts — sixths.

The numerator then tells you how many of those parts to count along from zero.

Example

Placing thirds and sixths on a number line

Place $\frac{1}{3}$ , $\frac{2}{3}$ , $\frac{4}{3}$ , and $\frac{5}{6}$ on a number line.

Step 1: Divide each unit gap into 6 equal parts (using 6 as it works for both thirds and sixths).

Step 2: Convert to sixths:

$\frac{1}{3} = \frac{2}{6}$ → 2 parts along
$\frac{2}{3} = \frac{4}{6}$ → 4 parts along
$\frac{4}{3} = \frac{8}{6}$ → 8 parts along (past 1, in the next gap)
$\frac{5}{6}$ → 5 parts along

Step 3: Mark each point on the number line.

Reading left to right: $\frac{1}{3} < \frac{2}{3} < \frac{5}{6} < \frac{4}{3}$

Note

For fractions with the same denominator, you only need to compare the numerators to put them in order. The larger the numerator, the further right on the number line.

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