The Number Line – Visualizing Numbers in Order

A number line is a straight line on which every point represents a number, arranged in order: negative numbers to the left of zero, positive numbers to the right, with each number the same fixed distance from its neighbours. It turns abstract numbers into something you can literally see and point to – and it is one of the most useful pictures in the whole of mathematics.

While ancient mathematicians worked with ratios and geometric lengths, the number line as we know it today – extended to include negative numbers – is usually credited to the English mathematician John Wallis, who included one of the earliest depictions of a full line of positive and negative numbers in his 1685 book A Treatise of Algebra. This simple picture helped mathematicians finally make peace with negative numbers: instead of an abstract oddity, a negative number could now be seen as a perfectly real point, just on the other side of zero.

Reading a Number Line

Moving right on a number line means adding; moving left means subtracting. Numbers further right are always greater than numbers further left.

What number is 6 steps to the right of -4?

Moving right means adding: -4 + 6 = 2.

Distance on a Number Line

What is the distance between -7 and 3 on the number line?

Distance = |a − b| = |-7 − 3| = |-10| = 10.

Real-Life Application

Thermometers: a vertical number line for temperature.
Timelines: years before and after a reference point (like 0 CE).
Rulers: a physical number line for measuring length.

Key Takeaways

A number line arranges numbers in order, with negatives left of zero and positives to the right.
John Wallis's 1685 book gave one of the earliest depictions of a full positive-and-negative number line.
Distance between two points on the number line is the absolute value of their difference, covered next.

Practice: The Number Line

Find the Position