Negative Numbers

What Are Negative Numbers?

Negative numbers are numbers less than zero. They are written with a minus sign (−) in front of them, such as $-3$, $-7$, or $-15$.

You encounter negative numbers in everyday life:

• Temperature: $-5°F$ means 5 degrees below zero
• Elevation: $-200$ feet means 200 feet below sea level
• Money: -\50$means you owe$50

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The Number Line

A number line helps us visualize negative numbers. Zero sits in the middle, positive numbers go to the right, and negative numbers go to the left.

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

Key ideas:

• Numbers get smaller as you move left
• Numbers get larger as you move right
• $-4$ is to the left of $-1$, so $-4 < -1$

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Comparing Negative Numbers

When comparing two negative numbers, the one closer to zero is greater.

| Comparison | Result | Why? | |---|---|---| | $-2$ vs $-7$ | $-2 > -7$ | $-2$ is closer to zero | | $-10$ vs $-3$ | $-10 < -3$ | $-3$ is closer to zero | | $0$ vs $-5$ | $0 > -5$ | Zero is always greater than any negative |

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Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction.

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

Absolute value is always non-negative (zero or positive).

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Adding Negative Numbers

Same signs → add the absolute values, keep the sign: $(-3) + (-5) = -(3+5) = -8$

Different signs → subtract the smaller absolute value from the larger, keep the sign of the larger: $7 + (-4) = 7 - 4 = 3$ $(-9) + 2 = -(9-2) = -7$

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Subtracting Negative Numbers

Subtracting a negative = adding a positive: $5 - (-3) = 5 + 3 = 8$ $(-4) - (-6) = -4 + 6 = 2$

Think of it as "two negatives make a positive" when they're right next to each other.

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Ordering Negative Numbers

To order a set of numbers from least to greatest, place them on a number line:

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

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

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Quick Reference

| Operation | Rule | Example | |---|---|---| | Negative + Negative | Add, keep negative | $(-3)+(-4)=-7$ | | Positive + Negative | Subtract, sign of larger | $8+(-3)=5$ | | Subtract a Negative | Add the positive | $6-(-2)=8$ | | Absolute Value | Distance from zero | $\lvert{-9}\rvert=9$ |