Negative Numbers on a Number Line

The idea in one sentence

Negative numbers live to the left of zero; farther left always means smaller.

Use a real-world picture

Imagine temperature. A temperature of $-2°$ is colder than $3°$, so $-2 < 3$. And $-8°$ is colder than $-2°$, so $-8 < -2$.

That last comparison can feel backward. Remember: on the number line, the number farther right is greater.

Add by walking

Start on the first number. A positive number means walk right; a negative number means walk left.

For $-3 + 5$:

1. Start at $-3$.
2. Walk 5 spaces right.
3. Land at $2$.

So $-3 + 5 = 2$.

Subtraction means add the opposite

Turn subtraction into addition by changing the second number to its opposite:

\[4 - (-3) = 4 + 3 = 7\]

You can picture this as removing a debt of 3. Losing a debt leaves you with 3 more.

Common mix-ups

• Thinking $-10$ is greater than $-2$ because 10 is greater than 2. On the number line, $-10$ is farther left.
• Automatically making every answer negative. The direction and distance decide the sign.
• Treating two negative signs as a magic rule. Say “subtracting a negative means adding its opposite.”

Check yourself

Which is greater: $-6$ or $-9$? Then solve $-6 + 4$.

$-6$ is greater because it is farther right. $-6 + 4 = -2$.