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Integers — Operations, Number Line, and Word Problems

Grade: 6-7 | Topic: Arithmetic

What You Will Learn​

By the end of this guide you will understand what integers are, how to locate them on a number line, and how to add, subtract, multiply, and divide positive and negative numbers confidently. You will also learn to apply these skills to real-world word problems involving temperature, money, and elevation.

Theory​

What is an integer?​

An integer is any whole number — positive, negative, or zero. The set of integers is written as:

Z={…,−3,−2,−1,0,1,2,3,…}\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}

Integers do not include fractions (12\frac{1}{2}) or decimals (3.73.7). Here are a few everyday examples of integers in action:

  • A temperature of −5°C-5°\text{C} (below zero)
  • A bank balance of +120+120 dollars (credit)
  • Sea level at 00 metres

The number line​

A number line places integers in order from left to right. Numbers to the right are always greater, and numbers to the left are always smaller.

⟵  −5    −4    −3    −2    −1      0      1      2      3      4      5  ⟶\longleftarrow \;-5 \;\; -4 \;\; -3 \;\; -2 \;\; -1 \;\;\; 0 \;\;\; 1 \;\;\; 2 \;\;\; 3 \;\;\; 4 \;\;\; 5\; \longrightarrow

Key ideas:

  • Every positive integer is greater than every negative integer: 1>−1001 > -100.
  • Zero is neither positive nor negative.
  • The absolute value of an integer is its distance from zero: ∣−7∣=7|{-7}| = 7 and ∣7∣=7|7| = 7.

Adding integers​

Same signs — add the absolute values and keep the sign:

(−4)+(−6)=−(4+6)=−10(-4) + (-6) = -(4 + 6) = -10

Different signs — subtract the smaller absolute value from the larger one and take the sign of the number with the larger absolute value:

(−9)+5=−(9−5)=−4(-9) + 5 = -(9 - 5) = -4

Subtracting integers​

Subtraction is the same as adding the opposite (also called the additive inverse). Change the subtraction sign to addition and flip the sign of the second number:

a−b=a+(−b)a - b = a + (-b)

This means:

7−10=7+(−10)=−37 - 10 = 7 + (-10) = -3

3−(−8)=3+8=113 - (-8) = 3 + 8 = 11

The rule "subtracting a negative is adding" is one of the most important ideas in integer arithmetic.

Multiplying and dividing integers​

The sign rules for multiplication and division are identical:

OperationResult
(+)×(+)(+) \times (+)++
(−)×(−)(-) \times (-)++
(+)×(−)(+) \times (-)−-
(−)×(+)(-) \times (+)−-

In short: same signs give a positive result; different signs give a negative result.

(−6)×(−4)=+24(-6) \times (-4) = +24

(−15)÷3=−5(-15) \div 3 = -5

When multiplying more than two integers, count the negative factors. An even number of negatives gives a positive product; an odd number gives a negative product.

Integers in word problems​

Many real-world situations involve integers:

  • Temperature: A drop of 8 degrees from −2°C-2°\text{C} is −2+(−8)=−10°C-2 + (-8) = -10°\text{C}.
  • Money: Owing $30 and then paying off $12 leaves -30 + 12 = -\18$ (still in debt).
  • Elevation: A submarine at −200-200 m rises 7575 m to reach −200+75=−125-200 + 75 = -125 m.

Translate the words into an integer expression, then apply the rules above.

Worked Examples​

Example 1: Adding integers with different signs​

Problem: Calculate (−14)+9(-14) + 9.

Step 1: The signs are different, so subtract the smaller absolute value from the larger one. 14−9=514 - 9 = 5

Step 2: The number with the larger absolute value is −14-14 (negative), so the result is negative.

Answer: (−14)+9=−5(-14) + 9 = \mathbf{-5}

Example 2: Subtracting a negative integer​

Problem: Calculate 6−(−11)6 - (-11).

Step 1: Rewrite subtraction as adding the opposite. 6−(−11)=6+116 - (-11) = 6 + 11

Step 2: Both numbers are now positive, so add normally. 6+11=176 + 11 = 17

Answer: 6−(−11)=176 - (-11) = \mathbf{17}

Example 3: Multiplying three integers​

Problem: Calculate (−3)×4×(−2)(-3) \times 4 \times (-2).

Step 1: Multiply the first two factors. (−3)×4=−12(-3) \times 4 = -12

Step 2: Multiply the result by the third factor. (−12)×(−2)=+24(-12) \times (-2) = +24

Quick check: There are two negative factors (even count), so the product should be positive.

Answer: (−3)×4×(−2)=24(-3) \times 4 \times (-2) = \mathbf{24}

Example 4: Division with negative integers​

Problem: Calculate (−48)÷(−6)(-48) \div (-6).

Step 1: Divide the absolute values. 48÷6=848 \div 6 = 8

Step 2: Both integers are negative (same sign), so the result is positive.

Answer: (−48)÷(−6)=+8(-48) \div (-6) = \mathbf{+8}

Example 5: Word problem — temperature change​

Problem: At 6 a.m. the temperature is −7°C-7°\text{C}. By noon it has risen by 13°C13°\text{C}, and by midnight it drops 19°C19°\text{C} from the noon temperature. What is the temperature at midnight?

Step 1: Find the noon temperature. −7+13=6°C-7 + 13 = 6°\text{C}

Step 2: Find the midnight temperature. 6+(−19)=6−19=−13°C6 + (-19) = 6 - 19 = -13°\text{C}

Answer: The temperature at midnight is −13°C\mathbf{-13°\text{C}}.

Common Mistakes​

Mistake 1: Confusing subtraction of a negative with subtraction of a positive

❌ 5−(−3)=5−3=25 - (-3) = 5 - 3 = 2

✅ 5−(−3)=5+3=85 - (-3) = 5 + 3 = 8

Why this matters: Subtracting a negative number means adding its opposite. Forgetting to flip the sign is one of the most frequent errors in integer arithmetic and leads to answers that are off by double the value.

Mistake 2: Using addition sign rules for multiplication

❌ (−4)×(−5)=−20(-4) \times (-5) = -20 (thinking "two negatives stay negative")

✅ (−4)×(−5)=+20(-4) \times (-5) = +20

Why this matters: In multiplication and division, two negatives always give a positive. The sign rules for addition are different from those for multiplication, and mixing them up causes systematic errors.

Mistake 3: Forgetting that zero is an integer

❌ Listing integers between −2-2 and 33 as: −1,1,2-1, 1, 2

✅ Correct list: −2,−1,0,1,2,3-2, -1, 0, 1, 2, 3 (zero is included, and "between" usually means inclusive at both ends in math problems)

Why this matters: Zero is an integer, and skipping it when counting or listing integers changes the answer, especially in statistics and probability problems.

Practice Problems​

Try these on your own before checking the answers:

  1. Calculate (−8)+(−15)(-8) + (-15).
  2. Calculate 12−(−7)12 - (-7).
  3. Simplify (−5)×6×(−2)(-5) \times 6 \times (-2).
  4. Calculate (−72)÷9(-72) \div 9.
  5. A diver is at −40-40 metres. She ascends 1818 metres, then descends 1010 metres. What is her final depth?
Click to see answers
  1. (−8)+(−15)=−(8+15)=−23(-8) + (-15) = -(8 + 15) = \mathbf{-23} (same signs: add absolute values, keep the negative sign)
  2. 12−(−7)=12+7=1912 - (-7) = 12 + 7 = \mathbf{19} (subtracting a negative becomes addition)
  3. (−5)×6=−30(-5) \times 6 = -30, then (−30)×(−2)=60(-30) \times (-2) = \mathbf{60} (two negatives give a positive)
  4. (−72)÷9=−8(-72) \div 9 = \mathbf{-8} (different signs give a negative result)
  5. −40+18=−22-40 + 18 = -22, then −22+(−10)=−32-22 + (-10) = \mathbf{-32} metres

Summary​

  • Integers are whole numbers: positive, negative, or zero (Z={…,−2,−1,0,1,2,…}\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}).
  • On a number line, numbers increase to the right and decrease to the left.
  • Adding: same signs — add and keep the sign; different signs — subtract and take the sign of the larger absolute value.
  • Subtracting: change to adding the opposite (a−b=a+(−b)a - b = a + (-b)).
  • Multiplying/dividing: same signs give positive; different signs give negative.

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