Rational Numbers on the Number Line

Key points

• Denominator q = how many equal parts in each unit.
• Numerator p = how many parts to count from 0.
• Positive → count right; negative → count left.
• Between any two rationals there is another: their average (a + b)/2.

Cut, then count

• Draw the line and mark 0, 1, 2, … (and −1, −2, … if needed).
• Cut each unit into q equal parts (q = denominator).
• From 0, count p parts: right if positive, left if negative.
• Where you stop is p/q.

Worked examples

Density of rational numbers

Between 1/2 and 3/4 lies their average (1/2 + 3/4)/2 = 5/8; between 1/2 and 5/8 lies 9/16; and so on forever. There are infinitely many rationals between any two.

Common mistakes: cutting the whole visible line instead of each unit; counting tick marks instead of the gaps; putting −1/2 to the right of 0.

Try it yourself

Goal: Place any fraction on the number line by cutting each unit into q parts and counting p of them.

• Set p and q with the + and − buttons (or pick an example).
• Press Cut and count: each unit is cut into q parts, then p parts are counted from 0.
• Read where it lands and between which integers it lies.
• Try a negative and an improper fraction.

Open the simulation full screen

Quick quiz

5 questions. Choose an answer for each, then submit to see your score and the explanations.