Negative Numbers Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Negative Numbers.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Negative numbers are numbers less than zero, used to represent direction, deficit, or values below a reference point.

If zero is sea level, negative numbers are depths below the surface — temperature −5°-5° is 5 degrees below freezing.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Negative numbers extend counting past zero to track deficits, depths, and directions opposite a chosen reference.

Common stuck point: The procedure for negative numbers is the easy part; the trap is thinking −5>−2-5>-2 because 5 looks bigger. Asking "Is there a meaningful zero point, and does this value sit on the opposite side of it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is there a meaningful zero point, and does this value sit on the opposite side of it?

Worked Examples

Example 1

easy
The temperature was −8°C-8°\text{C} in the morning and rose by 15°C15°\text{C} by noon. What was the noon temperature?

Answer

7°C7°\text{C}

First step

1
Start at −8-8 and add 1515: −8+15-8 + 15.

Full solution

  1. 2
    Since 15>815 > 8, the result is positive: 15−8=715 - 8 = 7.
  2. 3
    The noon temperature was 7°C7°\text{C}.
Adding a positive number to a negative number is equivalent to finding the difference and taking the sign of the number with the greater absolute value. Real-world contexts like temperature make negative numbers concrete.

Example 2

medium
Evaluate (−3)2−(−2)3(-3)^2 - (-2)^3.

Example 3

medium
Evaluate −(−5)+2×(−3)-(-5) + 2 \times (-3).

Example 4

medium
Simplify −2−3(−4+1)-2 - 3(-4 + 1).

Example 5

hard
Three temperatures average −4°-4°C. Two of them are −9°-9° and 2°2°. Find the third.

Example 6

challenge
Show that for any negative number aa, a+∣a∣=0a + |a| = 0.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Evaluate (−4)×(−7)×(−1)(-4) \times (-7) \times (-1).

Example 2

easy
A submarine is at −35-35 meters. It rises 1212 meters and then sinks 77 meters. What is its final position?

Example 3

easy
Which is greater: −3-3 or −7-7?

Example 4

easy
What is the opposite of −5-5?

Example 5

easy
Compute −4+9-4 + 9.

Example 6

easy
Compute −6−4-6 - 4.

Example 7

easy
Compute 5−85 - 8.

Example 8

easy
Compute −3×5-3 \times 5.

Example 9

easy
Compute −2×−6-2 \times -6.

Example 10

easy
Order from least to greatest: 0,−4,2,−10, -4, 2, -1.

Example 11

medium
Compute −7−(−3)-7 - (-3).

Example 12

medium
Compute −12÷3-12 \div 3.

Example 13

medium
The temperature drops from 5°5° to −8°-8°. By how many degrees?

Example 14

medium
Compute (−2)4(-2)^4.

Example 15

medium
Compute −3×4×(−2)-3 \times 4 \times (-2).

Example 16

medium
A submarine at −120-120 m descends another 4545 m. What is its new depth?

Example 17

medium
Compute −24−6\frac{-24}{-6}.

Example 18

medium
Evaluate 3−5×(−2)3 - 5 \times (-2).

Example 19

medium
If x=−3x = -3, evaluate x2−2xx^2 - 2x.

Example 20

challenge
For which integers nn is (−1)n+(−1)n+1=0(-1)^n + (-1)^{n+1} = 0?

Example 21

challenge
Find all integers xx with −2≤x<3-2 \leq x < 3 such that x2<4x^2 < 4.

Example 22

challenge
Why is a negative times a negative positive? Give an argument using the distributive property.

Example 23

easy
Compute −9+4-9 + 4.

Example 24

easy
Compute −8−5-8 - 5.

Example 25

easy
Compute −7×2-7 \times 2.

Example 26

easy
Compute −18÷−3-18 \div -3.

Example 27

easy
Order from least to greatest: −2,−10,0,−1,5-2, -10, 0, -1, 5.

Example 28

easy
A balance of \$15 has \$22 withdrawn. What is the new balance?

Example 29

medium
Evaluate (−2)3×(−1)2(-2)^3 \times (-1)^2.

Example 30

medium
Compute 10−(−4)−710 - (-4) - 7.

Example 31

medium
If a=−4a = -4 and b=3b = 3, find a2−2aba^2 - 2ab.

Example 32

medium
Plane altitude is changing at −200-200 ft/min. After 7 minutes, how much has its altitude changed?

Example 33

medium
Find −364−−15−3\frac{-36}{4} - \frac{-15}{-3}.

Example 34

medium
A scuba diver descends 14 m, ascends 6 m, then descends 9 m. Starting from sea level, what is the diver's final position?

Example 35

medium
Compute (−1)100+(−1)99(-1)^{100}+(-1)^{99}.

Example 36

hard
Evaluate −24−(−2)4-2^4 - (-2)^4.

Example 37

hard
Find all integers xx with −3<x≤2-3 < x \leq 2 satisfying ∣x∣<2|x| < 2.

Example 38

hard
Solve for xx: −2(x−3)=−10-2(x - 3) = -10.

Example 39

hard
Solve the inequality −3x+5>11-3x + 5 > 11 for xx.

Example 40

challenge
Find all integer pairs (a,b)(a, b) with a⋅b=−12a \cdot b = -12 and a+b=−1a + b = -1.
← Back to Negative Numbers Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

integersnumber linesubtraction