Integer Operations Examples in Math

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

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

Concept Recap

Adding, subtracting, multiplying, and dividing integers—numbers that include positive values, negative values, and zero.

Think of a number line with zero in the middle. Positive numbers go right, negative numbers go left. Adding a positive moves right; adding a negative moves left. Multiplying two negatives gives a positive because reversing a reversal brings you back to the original direction.

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: Integer operations add, subtract, multiply, and divide positives, negatives, and zero, using a number line for direction and sign rules for products.

Common stuck point: The procedure for integer operations is the easy part; the trap is treating subtracting a negative as subtracting. Asking "Does a negative number enter the operation so I must track sign as well as size?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does a negative number enter the operation so I must track sign as well as size?

Worked Examples

Example 1

easy
Compute (−8)+5(-8) + 5.

Answer

−3-3

First step

1
Notice the addends have different signs, so this becomes a subtraction problem.

Full solution

  1. 2
    Subtract the smaller absolute value from the larger: 8−5=38 - 5 = 3.
  2. 3
    Keep the sign of the number with the larger absolute value (−8-8): the answer is −3-3.
When adding integers with different signs, subtract the absolute values and take the sign of the number with the greater absolute value.

Example 2

medium
Compute (−6)×(−9)(-6) \times (-9).

Example 3

medium
Compute (−3)−(−10)+4(-3) - (-10) + 4.

Example 4

easy
Compute (−6)−(−11)(-6) - (-11).

Example 5

medium
A submarine is at depth −120-120 m. It ascends 4545 m, then descends 8080 m. Find its depth.

Example 6

medium
Compute (−10)+(−7)+15−(−2)(-10) + (-7) + 15 - (-2).

Example 7

hard
Evaluate −18−(−6)−3+(−2)3\dfrac{-18 - (-6)}{-3} + (-2)^3.

Practice Problems

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

Example 1

easy
Compute (−12)+(−5)(-12) + (-5).

Example 2

easy
Compute −3+(−5)-3 + (-5).

Example 3

easy
Compute −3−5-3 - 5.

Example 4

easy
Compute (−4)×(−6)(-4) \times (-6).

Example 5

easy
Compute (−7)×3(-7) \times 3.

Example 6

easy
Compute 5+(−9)5 + (-9).

Example 7

easy
Compute −12÷4-12 \div 4.

Example 8

easy
Compute 0×(−7)0 \times (-7).

Example 9

easy
Compute −8+8-8 + 8.

Example 10

medium
Compute −6−(−9)-6 - (-9).

Example 11

medium
Compute (−2)(3)(−4)(-2)(3)(-4).

Example 12

medium
Evaluate −3+7−10+2-3 + 7 - 10 + 2.

Example 13

medium
The temperature was −5∘-5^\circC and rose 12∘12^\circ. What is the new temperature?

Example 14

medium
Evaluate −24−6+(−3)\frac{-24}{-6} + (-3).

Example 15

medium
A diver descends 18 m, then rises 7 m, then descends 5 m. Find the final depth relative to the surface.

Example 16

medium
Evaluate (−5)2(-5)^2 and −52-5^2 and explain why they differ.

Example 17

medium
Compute −15+6−(−4)-15 + 6 - (-4).

Example 18

medium
A bank account starts at $50\$50, then has withdrawals of $30\$30 and $45\$45 and a deposit of $20\$20. Find the balance.

Example 19

challenge
Evaluate −24-2^4 and explain the sign.

Example 20

challenge
If the product of three integers is negative, what can you conclude about the number of negative factors?

Example 21

challenge
Evaluate −3−(−7)+(−2)×4-3 - (-7) + (-2) \times 4.

Example 22

easy
Compute (−14)+9(-14) + 9.

Example 23

easy
Compute (−7)×(−8)(-7) \times (-8).

Example 24

easy
Compute 15−2015 - 20.

Example 25

easy
Compute 10+(−15)10 + (-15).

Example 26

easy
Compute (−3)×0×7(-3) \times 0 \times 7.

Example 27

medium
Compute (−3)(−4)(−5)(-3)(-4)(-5).

Example 28

medium
Evaluate −4−6+10−7-4 - 6 + 10 - 7.

Example 29

medium
Evaluate −2×(3−8)-2 \times (3 - 8).

Example 30

medium
Evaluate −486−(−2)\frac{-48}{6} - (-2).

Example 31

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

Example 32

medium
Evaluate −(−(−4))−(−2)-(-(-4)) - (-2).

Example 33

medium
Evaluate −2(5−3×4)-2(5 - 3 \times 4).

Example 34

hard
Evaluate (−2)4−(−3)2(−1)3\dfrac{(-2)^4 - (-3)^2}{(-1)^3}.

Example 35

hard
A bank account has these transactions: −$45, +$120, −$30, −$60, +$15-\$45,\ +\$120,\ -\$30,\ -\$60,\ +\$15. Starting balance is $50\$50. Final balance?

Example 36

hard
Evaluate −32−(−4)2+(−2)(−5)-3^2 - (-4)^2 + (-2)(-5).

Example 37

hard
In Death Valley the temperature at noon was 46°46°C; at midnight it dropped to −2°-2°C. How many degrees did it fall?

Example 38

hard
If a<0a < 0, b>0b > 0, and ∣a∣>∣b∣|a| > |b|, determine the sign of a+ba + b and of abab.

Example 39

challenge
Find all integer pairs (a,b)(a, b) with a+b=−5a + b = -5 and ab=−14ab = -14.

Example 40

challenge
Compute the product (−1)(−2)(−3)(−4)(−5)(-1)(-2)(-3)(-4)(-5) and state the sign rule that decides the answer.
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Background Knowledge

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

additionsubtractionmultiplicationdivisionintegers