Negative Numbers
Negative numbers are numbers less than zero. They are written with a minus sign, like −3 or −10. On a number line, negative numbers sit to the left of 0. The farther left a number is, the smaller it is, so -8 \lt -2.
Before this lesson: You only need to know whole numbers (0, 1, 2, 3 and so on). This is the first lesson in the Pre-Algebra course.
Extra practice: The Pre-Algebra Unit 1 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.
What Are Negative Numbers?
A negative number is less than zero. You already see them in everyday life:
- A temperature of −5°F is 5 degrees below zero.
- A diver at −20 feet is 20 feet below sea level.
- A bank balance of −$15 means you owe $15.
A positive number is greater than zero. You can write it with a plus sign, like +4, but most of the time we leave the sign off. Zero is neither positive nor negative.
The integers are the whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …. Each integer has its own spot on the number line.
Opposites
Two numbers are opposites when they are the same distance from 0 but on different sides. The opposite of 4 is −4, and the opposite of −9 is 9. The opposite of 0 is 0. A number’s distance from 0 is called its absolute value.
A minus sign can also mean “the opposite of.” So -(-6) means the opposite of −6, which is 6.
-(-6) = 6
How to Compare Negative Numbers
On a number line, numbers get greater as you move right. They get smaller as you move left.
We use two symbols to compare. The symbol \lt means “is less than,” and \gt means “is greater than.” The open side of the symbol always faces the greater number.
| Numbers you compare | Rule | Example |
|---|---|---|
| A positive and a negative | The positive number is always greater. | 2 \gt -10 |
| Two negatives | The one closer to 0 is greater. | -3 \gt -8 |
| Zero and a negative | Zero is greater. | 0 \gt -1 |
Worked Examples
Example 1: A negative and a positive
Compare −6 and 1.
Step 1. Find both numbers on the number line. −6 is left of 0, and 1 is right of 0.
Step 2. The number farther right is greater, so 1 is greater.
Answer: -6 \lt 1.
Example 2: Two negative numbers
Compare −7 and −2.
Step 1. Both numbers are negative, so check which one is closer to 0.
Step 2. −2 is 2 units from 0. −7 is 7 units from 0. So −2 is closer to 0, which puts it farther right.
Answer: -7 \lt -2. Even though 7 is bigger than 2, −7 is smaller than −2.
Example 3: Order from least to greatest
Put these numbers in order from least to greatest: 4, −9, 0, −1, 6, −5.
Step 1. Start with the negative numbers. The one farthest from 0 is the least: −9, then −5, then −1.
Step 2. Next comes 0.
Step 3. Finish with the positive numbers from smallest to largest: 4, then 6.
Answer: −9, −5, −1, 0, 4, 6.
Example 4: Opposites
Find the opposite of 12 and the opposite of −7. Then simplify -(-15).
Step 1. The opposite of 12 is on the other side of 0, the same distance away: −12.
Step 2. The opposite of −7 is 7.
Step 3. -(-15) means the opposite of −15, so it equals 15.
Answer: −12, 7 and 15.
Example 5: Word problem
Here are the 6 a.m. temperatures in four towns. Pine Lake: −11°F. Elk River: −6°F. Clearwater: 3°F. Red Hill: −14°F. Which town was the coldest? List the towns from coldest to warmest.
Step 1. The coldest temperature is the least number. Of the negatives, −14 is farthest from 0, so it is the least.
Step 2. Order the rest: −11 is less than −6, and every negative is less than 3.
Answer: Red Hill was the coldest. From coldest to warmest: Red Hill (−14°F), Pine Lake (−11°F), Elk River (−6°F), Clearwater (3°F).
Common Mistakes
| Mistake | What it looks like | How to fix it |
|---|---|---|
| Thinking a bigger digit means a bigger negative | -8 \gt -3 | −8 is farther left on the number line, so -8 \lt -3. |
| Forgetting that 0 beats every negative | -1 \gt 0 | 0 is to the right of every negative number: 0 \gt -1. |
| Calling 0 positive | “0 is a positive number” | 0 is neither positive nor negative. |
| Flipping the symbol | -5 \gt -2 | Point the open side at the greater number: -5 \lt -2. |
| Mixing up opposite and negative | The opposite of −4 is −4. | An opposite is on the other side of 0: the opposite of −4 is 4. |
Negative Numbers in Real Life
A negative number always means “less than the zero point.” What zero stands for depends on the situation.
| Situation | Zero means | A negative number means | Example |
|---|---|---|---|
| Temperature | 0 degrees | Below zero | −12°F is 12 degrees below zero |
| Elevation | Sea level | Below sea level | −50 feet is 50 feet below sea level |
| Money | A balance of $0 | Money you owe | −$20 means you owe $20 |
| Football | No gain | Yards lost | −7 yards is a loss of 7 yards |
Adding, subtracting, multiplying and dividing negative numbers each have their own rules. You’ll learn them next in Adding and Subtracting Integers and Multiplying and Dividing Integers. Later in this unit, you’ll use negative numbers to plot points on the coordinate plane.
Practice Problems
For problems 1–4, write \lt or \gt between the numbers.
- −4 and 3
- −10 and −12
- 0 and −7
- −1 and −15
- Put in order from least to greatest: 5, −3, −8, 0, 2.
- Put in order from greatest to least: −6, −2, −11, 1.
- What is the opposite of −25?
- Simplify -(-9).
- A submarine is 300 feet below sea level. Write its position as a number.
- At 7 a.m. the temperature was −4°F. At noon it was −1°F. Was it warmer at 7 a.m. or at noon?
Show answers
- -4 \lt 3. A negative number is always less than a positive number.
- -10 \gt -12. −10 is closer to 0, so it is farther right.
- 0 \gt -7. Zero is greater than every negative number.
- -1 \gt -15. −1 is only 1 unit from 0, so it is greater.
- −8, −3, 0, 2, 5.
- 1, −2, −6, −11. The negative closest to 0 comes first after the positive number.
- 25. It is the same distance from 0, on the other side.
- -(-9) = 9. The opposite of −9 is 9.
- −300 feet. Below sea level is negative.
- Noon was warmer. -1 \gt -4, because −1 is closer to 0.
Want more practice? The Pre-Algebra Unit 1 Practice Pack has 36 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.
Negative Numbers FAQ
Is 0 a negative number?
No. Zero is neither positive nor negative. It sits between the negative and positive numbers on the number line.
Is −5 greater than −2?
No. −5 is farther left on the number line, so -5 \lt -2. With two negative numbers, the one closer to 0 is greater.
Are negative numbers integers?
Some are. Numbers like −3 and −40 are integers. Numbers like −2.5 and −½ are negative, but they are not integers.
What is the opposite of a negative number?
It is a positive number. The opposite of −8 is 8, because both are 8 units from 0.
Summary
- Negative numbers are less than 0 and sit to the left of 0 on the number line.
- Zero is neither positive nor negative.
- Numbers get greater as you move right on the number line.
- Any positive number is greater than any negative number.
- With two negatives, the one closer to 0 is greater: -2 \gt -7.
- Opposites are the same distance from 0 on different sides, like 9 and −9.
Next lesson: Absolute Value
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