Negative Numbers

What Are Negative Numbers?

Negative numbers are numbers less than zero. They are written with a minus sign in front, like $-3$, $-7$, or $-15$.

You see negative numbers in real life all the time:

• Temperature: $-5°F$ means 5 degrees below zero
• Elevation: $-200$ ft means 200 feet below sea level
• Money: Owing $20 can be represented as $-20$

Zero is the dividing line — it is neither positive nor negative.

The Number Line

Picture a horizontal line with zero in the middle.

$\cdots\;\; -5 \;\; -4 \;\; -3 \;\; -2 \;\; -1 \;\; 0 \;\; 1 \;\; 2 \;\; 3 \;\; 4 \;\; 5 \;\;\cdots$

• Right of zero → positive numbers (get larger)
• Left of zero → negative numbers (get smaller)

The farther left you go, the smaller the number.

A city records a temperature of $-11°F$. What does the negative sign tell us?

Comparing Negative Numbers

Here is the key rule:

The negative number closer to zero is the greater one.

Think of it on the number line — the number further to the right is always greater.

Which number is greater: $-4$ or $-9$?

Absolute Value

The absolute value of a number is its distance from zero on the number line — no matter which direction.

$|{-6}| = 6 \qquad |{6}| = 6 \qquad |{0}| = 0$

We write absolute value with vertical bars: $|x|$.

Absolute value is never negative — it measures distance, and distance is always zero or positive.

What is $|{-12}|$?

Adding Negative Numbers — Same Signs

When both numbers have the same sign, add their absolute values and keep the sign.

$(-3) + (-5) = -(3 + 5) = -8$

Think: if you owe $3 and then owe $5 more, you owe $8 total → $-8$.

What is $(-4) + (-7)$?

Adding Negative Numbers — Different Signs

When the signs are different, subtract the smaller absolute value from the larger and take the sign of the number with the larger absolute value.

$7 + (-4) = 7 - 4 = 3 \quad\text{(positive wins, } |7| > |{-4}|\text{)}$

$(-9) + 2 = -(9 - 2) = -7 \quad\text{(negative wins, } |{-9}| > |2|\text{)}$

What is $10 + (-15)$?

Subtracting Negative Numbers

The golden rule:

Subtracting a negative = adding a positive.

$5 - (-3) = 5 + 3 = 8$

$(-4) - (-6) = -4 + 6 = 2$

Think of the two minus signs canceling each other out.

What is $(-2) - (-9)$?

A diver is at $-30$ feet. She ascends $18$ feet and then descends $7$ feet. What is her depth now?

Ordering Negative Numbers

To put numbers in order, picture them on a number line.

Example: Order $-3, 5, -8, 0, 2$ from least to greatest.

1. Find the most negative (farthest left): $-8$
2. Next: $-3$
3. Then zero
4. Then the positives: $2, 5$

$-8 < -3 < 0 < 2 < 5$

Which lists $-6, 1, -2, 4, 0$ from least to greatest?

Quick Review

Great job — you now have a solid understanding of negative numbers! 🎉

The temperature started at $-5°F$ and rose $12$ degrees. What is the new temperature?

What is $(-3) + (-8) - (-5)$?