Operations with Integers
Class 6 introduced negative numbers; now we calculate with them. This Class 7 Ganita Prakash chapter covers all four operations on integers — addition, subtraction, multiplication and division — the rules of signs that govern them, and the key properties that make integers behave predictably.
Learning objectives
- Add and subtract integers.
- Multiply and divide integers.
- Apply the rules of signs.
- Use properties of integer operations.
Key concepts
Adding and subtracting integers
On the number line, adding moves right and subtracting moves left, and this works for negatives too. Adding two negatives gives a more negative result: (−3) + (−4) = −7. Subtracting an integer is the same as adding its opposite: 5 − (−2) = 5 + 2 = 7.
Multiplying integers
When multiplying integers, the rule of signs decides the sign of the answer: a positive times a positive, or a negative times a negative, gives a positive; a positive times a negative gives a negative. So (−4) × (−3) = 12, but (−4) × 3 = −12.
Dividing integers
Division follows the same sign rule as multiplication. Like signs give a positive quotient and unlike signs give a negative one: (−12) ÷ (−3) = 4, while (−12) ÷ 3 = −4. Dividing zero by any non-zero integer gives zero.
Properties of integers
Integer addition and multiplication are commutative (order does not matter) and associative (grouping does not matter). Zero is the additive identity (a + 0 = a) and one is the multiplicative identity (a × 1 = a). These properties let us rearrange calculations conveniently.
Important formulas
Subtracting an integer
a − b = a + (opposite of b)
Sign rule (× and ÷)
like signs → +, unlike signs → −
Key definitions
- Integer
- A positive whole number, zero, or a negative whole number.
- Opposite
- The integer with the same value but the other sign (3 and −3).
- Commutative
- A property where order does not affect the result.
- Additive identity
- Zero, since adding it leaves a number unchanged.
Solved examples
Q1. Find (−3) + (−4).
Solution: Both negative: −7.
Q2. Find (−4) × (−3).
Solution: Like signs give positive: 12.
Q3. Find (−12) ÷ 3.
Solution: Unlike signs give negative: −4.
Common mistakes to avoid
- Thinking two negatives added give a positive.
- Forgetting that subtracting a negative is the same as adding.
- Getting the sign wrong when multiplying or dividing.
- Believing division of integers is commutative (it is not).
Operations with Integers — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
(−5) + (−3) equals:
Practice questions
Short answer
What is the sign of a negative times a negative?
Positive.
What is 6 − (−2)?
6 + 2 = 8.
What is the additive identity for integers?
Zero, since a + 0 = a.
Long answer
Explain the rules of signs for multiplying and dividing integers, with examples.
When multiplying or dividing two integers, the sign of the answer depends only on whether the two signs are alike or unlike. Like signs give a positive result: a positive times a positive is positive (2 × 3 = 6), and a negative times a negative is also positive ((−2) × (−3) = 6). Unlike signs give a negative result: a positive times a negative is negative (2 × (−3) = −6). Division follows the same rule, so (−12) ÷ (−3) = 4 (like signs, positive) while (−12) ÷ 3 = −4 (unlike signs, negative). Getting the size right and then applying the sign rule gives the complete answer.
Explain addition and subtraction of integers using the number line and opposites.
On the number line, adding an integer means moving right and subtracting means moving left, and this idea covers negatives. So (−3) + (−4) starts at −3 and moves 4 more to the left, reaching −7. Subtraction can always be turned into addition of the opposite: subtracting an integer is the same as adding its opposite, so 5 − (−2) becomes 5 + 2 = 7, and 4 − 6 becomes 4 + (−6) = −2. Thinking of subtraction as 'adding the opposite' removes most sign confusion.
HOTS (Higher Order Thinking)
The temperature is −2°C and drops by 5°C, then rises by 3°C. What is the final temperature?
−2 − 5 + 3 = −4°C.
Without computing fully, is (−1) × (−1) × (−1) positive or negative? Why?
Negative, because multiplying three negatives gives a negative (an odd number of negative factors).
Quick revision
Revision notes
- Add → move right; subtract → move left; subtracting a negative adds.
- × and ÷: like signs → positive, unlike signs → negative.
- Addition and multiplication are commutative and associative.
- 0 is additive identity; 1 is multiplicative identity.
Key takeaways
- Subtracting a negative is the same as adding.
- Like signs multiply/divide to positive; unlike to negative.
- Integers obey neat properties that ease calculation.
Frequently asked questions
What is a negative times a negative?
A positive number.
How do I subtract a negative number?
Change it to adding the positive: a − (−b) = a + b.
Is integer division commutative?
No; for example 6 ÷ 2 is not the same as 2 ÷ 6.