The Number System

Negative Numbers & the Number Line

Representing positive and negative numbers on a number line.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Numbers That Go the Other Direction

A thermometer reads 5 degrees below zero. A submarine is 200 feet below sea level. A bank account is $40 overdrawn. Before reading on, think about what these three situations have in common, and how you might represent each one with a number.

Definition — Negative Number

A number less than zero, representing a quantity in the opposite direction from a positive quantity of the same size — written with a minus sign, like −5. Every positive number has a matching negative number, called its opposite, the same distance from zero in the other direction.

Each of the opening situations has a natural zero point — freezing temperature, sea level, an empty balance — and a quantity that can go above or below it. The number line captures this perfectly: positive numbers extend to the right of 0, and negative numbers extend to the left, mirroring each other exactly.

Worked Example — Representing Real Quantities with Signed Numbers

A hiking trail's elevation markers show a peak 1,200 feet above a base camp and a valley 300 feet below it. Using base camp as 0, the peak is +1,200 and the valley is −300.

Worked Example — Ordering Negative Numbers

Order −8, 3, −2, and 5 from least to greatest. On the number line, numbers further left are always less, no matter how large they look without their sign. Ordered: −8, −2, 3, 5 — notice −8 is less than −2, since −8 sits further left, even though 8 is bigger than 2.

Tip

For two negative numbers, the one with the larger number after the minus sign is actually the smaller value — −8 is less than −2, the opposite of how 8 and 2 compare without their signs.

Common Mistakes

Ordering negative numbers as if the sign didn't matter, such as saying −8 is greater than −2 because 8 is greater than 2.
Always think in terms of position on the number line — further left is always less, so among negative numbers, the one that looks 'bigger' without its sign is actually the smaller value.
Treating a negative number as simply 'a smaller version' of the positive quantity rather than a value in the opposite direction from zero.
A negative number represents the opposite direction from a reference point, not just a lesser amount — −300 feet means 300 feet below the reference, a direction, not a smaller measurement of the same kind.

Key Takeaways

A negative number represents a quantity in the opposite direction from a positive quantity, relative to a zero point.
The number line extends in both directions from 0, with negative numbers to the left and positive numbers to the right.
Among negative numbers, further left (a more negative value) is always less, even when its digits look larger.

Summary

Negative numbers extend the number line to represent real quantities that go the opposite direction from a reference point. The next lesson looks at a related idea — how far a number sits from zero, regardless of which direction.

Track your progress

← Dividing Fractions by Fractions Back to The Number System Absolute Value→