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Operations with Integers

NCERT Class 7 · Mathematics Based on NCERT Class 7 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter builds on the basic understanding of integers from Grade 6 and extends it to multiplication and division of integers. It uses real-life contexts like carrom coin movements, elevator descents, and exam scoring to make the concepts relatable. The one big idea a student must walk away with is that the rules for signs in multiplication and division are consistent: same signs give a positive result, different signs give a negative result, and all properties of whole numbers (commutative, associative, distributive) hold true for integers as well.
Key Concepts (14)
Integer
A whole number that can be positive, negative, or zero. It does not include fractions or decimals.
e.g. Examples: 5, -3, 0, -12, 100.
Number Line
A line on which numbers are marked at equal intervals, with positive numbers to the right of zero and negative numbers to the left of zero.
e.g. On a number line, -5 is 5 units to the left of 0, and +3 is 3 units to the right of 0.
Additive Inverse
For any integer a, the additive inverse is -a. The sum of a number and its additive inverse is always zero.
e.g. The additive inverse of 18 is -18, and the additive inverse of -12 is 12.
Magnitude
The absolute value or distance of an integer from zero on the number line, ignoring its sign.
e.g. The magnitude of -4 is 4, and the magnitude of +7 is 7.
Multiplier
The number that tells how many times to add or remove the multiplicand in a multiplication operation.
e.g. In 4 × (-2), 4 is the multiplier.
Multiplicand
The number that is being multiplied in a multiplication operation.
e.g. In 4 × (-2), -2 is the multiplicand.
Product
The result obtained after multiplying two or more numbers.
e.g. The product of 4 and -2 is -8.
Quotient
The result obtained after dividing one number by another.
e.g. The quotient of (-100) ÷ 25 is -4.
Commutative Property (Multiplication)
For any two integers a and b, a × b = b × a. The order of multiplication does not affect the product.
e.g. 3 × (-4) = -12 and (-4) × 3 = -12.
Associative Property (Multiplication)
For any three integers a, b, and c, a × (b × c) = (a × b) × c. The grouping of numbers does not affect the product.
e.g. (5 × -3) × 4 = -60 and 5 × (-3 × 4) = -60.
Distributive Property (Multiplication over Addition)
For any three integers a, b, and c, a × (b + c) = (a × b) + (a × c).
e.g. 5 × (4 + (-2)) = 5 × 2 = 10, and (5 × 4) + (5 × -2) = 20 + (-10) = 10.
Zero Pair
A pair of one positive token (green) and one negative token (red) that together represent zero because they cancel each other out.
e.g. One green token (+1) and one red token (-1) together make a zero pair.
Token Model
A visual model using green tokens for positive integers and red tokens for negative integers to perform addition, subtraction, and multiplication.
e.g. To show 4 × (-2), place 2 red tokens into an empty bag 4 times, resulting in 8 red tokens or -8.
Brahmagupta's Rules
Historical rules from Brahmagupta's Brāhmasphuṭasiddhānta (628 CE) stating: product or quotient of two fortunes (positives) is a fortune; of two debts (negatives) is a fortune; of a debt and a fortune is a debt.
e.g. Positive × Positive = Positive; Negative × Negative = Positive; Positive × Negative = Negative.
Formulas & Equations (9)
P = a + b
Reads as: P equals a plus b
Variables: P = final position of the coin on the number line; a = first movement (positive for right, negative for left); b = second movement (positive for right, negative for left). No SI unit.
Use when: When finding the final position after two successive movements on a number line, such as in the carrom coin or elevator problems.
⚠ Students forget to assign the correct sign (positive for right/up, negative for left/down) to the movements, leading to an incorrect final position.
a × b = b × a
Reads as: a times b equals b times a
Variables: a and b are any two integers. No SI unit.
Use when: When verifying that multiplication is commutative, or when rearranging factors to simplify calculations.
⚠ Students think this property applies to subtraction or division as well, which it does not.
a × (b × c) = (a × b) × c
Reads as: a times the product of b and c equals the product of a and b times c
Variables: a, b, c are any three integers. No SI unit.
Use when: When multiplying three or more integers, to group them in any convenient order.
⚠ Students incorrectly apply this to expressions involving addition or subtraction, confusing it with the distributive property.
a × (b + c) = (a × b) + (a × c)
Reads as: a times the sum of b and c equals a times b plus a times c
Variables: a, b, c are any three integers. No SI unit.
Use when: When simplifying expressions where a number is multiplied by a sum, or when expanding brackets.
⚠ Students forget to multiply the outside term with every term inside the brackets, especially when there is a negative sign.
a ÷ (-b) = -(a ÷ b)
Reads as: a divided by negative b equals negative of a divided by b
Variables: a and b are positive integers, b ≠ 0. No SI unit.
Use when: When dividing a positive integer by a negative integer.
⚠ Students incorrectly think the quotient is positive because they ignore the negative sign in the divisor.
(-a) ÷ b = -(a ÷ b)
Reads as: negative a divided by b equals negative of a divided by b
Variables: a and b are positive integers, b ≠ 0. No SI unit.
Use when: When dividing a negative integer by a positive integer.
⚠ Students forget to put the negative sign in the quotient.
(-a) ÷ (-b) = a ÷ b
Reads as: negative a divided by negative b equals a divided by b
Variables: a and b are positive integers, b ≠ 0. No SI unit.
Use when: When dividing a negative integer by a negative integer.
⚠ Students think the quotient is negative because both numbers are negative.
1 × a = a
Reads as: one times a equals a
Variables: a is any integer. No SI unit.
Use when: When multiplying any integer by 1, the result is the integer itself.
⚠ Students sometimes think 1 × a = 1, confusing it with addition.
-1 × a = -a
Reads as: negative one times a equals negative a
Variables: a is any integer. No SI unit.
Use when: When finding the additive inverse of a number by multiplying by -1.
⚠ Students forget that -1 × (-5) = 5, not -5.
How It Works
Finding two numbers from their sum and difference
  1. Step 1: Guess a pair of numbers (first number and second number).
  2. Step 2: Check if their sum equals the given sum.
  3. Step 3: Check if their difference (first number minus second number) equals the given difference.
  4. Step 4: If both conditions are satisfied, you have found the correct pair. If not, try a different pair.
  5. Step 5: If the difference is negative, swap the numbers from the positive difference case.
💡 The difference is always first number minus second number. If the given difference is negative, the second number is larger than the first.
Adding integers using the carrom coin model (number line)
  1. Step 1: Start at 0 on the number line.
  2. Step 2: For the first movement, move 'a' units: right if 'a' is positive, left if 'a' is negative.
  3. Step 3: From the new position, for the second movement, move 'b' units: right if 'b' is positive, left if 'b' is negative.
  4. Step 4: The final position P is given by P = a + b.
💡 Rightward movement is positive, leftward movement is negative. Always start from 0.
Subtracting integers using the token model
  1. Step 1: Represent the first number using green (positive) and red (negative) tokens.
  2. Step 2: To subtract the second number, you need to remove that many tokens from the bag.
  3. Step 3: If there are not enough tokens to remove, add zero pairs (one green and one red) until you have enough.
  4. Step 4: Remove the required number of tokens.
  5. Step 5: The remaining tokens represent the answer.
💡 Subtracting a number is the same as adding its additive inverse. For example, 7 - 18 = 7 + (-18) = -11.
Multiplying integers using the token model (positive multiplier)
  1. Step 1: Start with an empty bag.
  2. Step 2: If the multiplier is positive, place the multiplicand into the bag that many times.
  3. Step 3: If the multiplicand is positive, place green tokens. If negative, place red tokens.
  4. Step 4: Count the total tokens in the bag. Green tokens give a positive product, red tokens give a negative product.
💡 Positive multiplier means 'put into the bag'. The sign of the product depends on the sign of the multiplicand.
Multiplying integers using the token model (negative multiplier)
  1. Step 1: Start with an empty bag.
  2. Step 2: If the multiplier is negative, you need to remove tokens from the bag that many times.
  3. Step 3: Since the bag is empty, first add zero pairs (one green and one red) to have tokens to remove.
  4. Step 4: Remove the required number of tokens (positive or negative) as indicated by the multiplicand.
  5. Step 5: The remaining tokens in the bag give the product.
💡 Negative multiplier means 'remove from the bag'. Removing a positive leaves negatives, and removing a negative leaves positives.
Using patterns to determine the sign of a product
  1. Step 1: Observe the pattern: 4 × 3 = 12, 3 × 3 = 9, 2 × 3 = 6, 1 × 3 = 3, 0 × 3 = 0.
  2. Step 2: Notice that for every unit decrease in the multiplier, the product decreases by the multiplicand (3).
  3. Step 3: Continue the pattern: (-1) × 3 = -3, (-2) × 3 = -6, (-3) × 3 = -9.
  4. Step 4: Similarly, for a negative multiplicand: 4 × (-3) = -12, 3 × (-3) = -9, ..., 0 × (-3) = 0, (-1) × (-3) = 3, (-2) × (-3) = 6.
  5. Step 5: Conclude: positive × positive = positive; positive × negative = negative; negative × positive = negative; negative × negative = positive.
💡 The pattern continues consistently even when the multiplier becomes negative. The sign rule is: same signs give positive, different signs give negative.
Dividing integers by converting to multiplication
  1. Step 1: Rewrite the division as a multiplication problem: a ÷ b = ? means b × ? = a.
  2. Step 2: Determine the unknown integer that, when multiplied by the divisor, gives the dividend.
  3. Step 3: Use the sign rules for multiplication to determine the sign of the quotient.
  4. Step 4: The quotient is positive if both dividend and divisor have the same sign; negative if they have different signs.
💡 Division is the inverse of multiplication. Always check your answer by multiplying the quotient by the divisor to see if you get the dividend.
Evaluating expressions with multiple integers
  1. Step 1: Follow the order of operations: brackets first, then multiplication and division from left to right, then addition and subtraction from left to right.
  2. Step 2: Use the associative property to group multiplications conveniently.
  3. Step 3: Use the distributive property to simplify expressions like a × (b + c).
  4. Step 4: Count the number of negative signs in a product: an even number of negatives gives a positive product; an odd number gives a negative product.
💡 When multiplying many integers, the sign of the product depends only on the number of negative factors, not on the order of multiplication.
Exam Traps — Don't Make These Mistakes
⚠ Confusing the sign of the difference
✗ For sum = 25 and difference = -11, students write the numbers as 18 and 7 (the same as for difference = 11).
→ The difference is defined as first number minus second number. If the difference is -11, then the first number must be smaller than the second number. Swapping the numbers gives the correct pair: 7 and 18.
✓ For sum = 25 and difference = -11, the numbers are 7 and 18 because 7 - 18 = -11.
⚠ Misapplying the sign rule for multiplication
✗ Students write (-4) × (-2) = -8, thinking that two negatives make a negative.
→ The rule is: product of two numbers with the same sign (both positive or both negative) is positive. Product of two numbers with different signs is negative.
✓ (-4) × (-2) = +8 because both numbers are negative, so the product is positive.
⚠ Forgetting the sign in division
✗ For (-100) ÷ 25, students write the quotient as 4 instead of -4.
→ When dividing a negative number by a positive number, the quotient is negative. The correct quotient is -4 because 25 × (-4) = -100.
✓ (-100) ÷ 25 = -4.
⚠ Incorrectly applying the distributive property with a negative multiplier
✗ For (-2) × (4 + (-3)), students write (-2) × 4 + (-2) × (-3) = -8 + (-6) = -14.
→ The student incorrectly computed (-2) × (-3) as -6 instead of +6. The correct calculation is (-2) × 4 = -8 and (-2) × (-3) = +6, so -8 + 6 = -2.
✓ (-2) × (4 + (-3)) = (-2) × 1 = -2, or using distribution: (-2) × 4 + (-2) × (-3) = -8 + 6 = -2.
⚠ Ignoring the sign when using the carrom coin model
✗ For first movement = -4 and final position = 5, students write the second movement as 9 instead of 9.
→ The student might think -4 + ? = 5, and write ? = 9, which is correct. But the trap is that students often forget that the first movement is negative and add it as positive, leading to wrong answers.
✓ Use the formula P = a + b. Here, 5 = (-4) + b, so b = 5 - (-4) = 5 + 4 = 9. The second movement is +9.
⚠ Confusing the additive inverse with the number itself
✗ For the question 'Find the integer whose product with (-1) is 27', students write 27.
→ The product of (-1) and a number gives the additive inverse of that number. If (-1) × ? = 27, then ? must be -27 because (-1) × (-27) = 27.
✓ The integer is -27 because (-1) × (-27) = 27.
Compare & Contrast
Sign rule for the result
Addition of Integers
Multiplication of Integers
In addition, the sign of the result depends on which number has the larger magnitude (e.g., 5 + (-3) = 2, positive because 5 > 3). In multiplication, the sign depends only on whether the signs are the same (positive result) or different (negative result), regardless of magnitude.
Conversion to another operation
Subtraction of Integers
Division of Integers
Subtraction can be converted to addition of the additive inverse: a - b = a + (-b). Division can be converted to multiplication by the reciprocal (though not explicitly in this chapter), but in practice, division is solved by finding the missing factor: a ÷ b = ? means b × ? = a.
Exam Focus
Marks pattern: Typically, this chapter appears as: 2-3 MCQs (1 mark each), 1-2 very short answer questions (1-2 marks), 1 short answer question (3 marks) on properties or word problems, and 1 long answer question (5 marks) involving multi-step calculations or a complex word problem.
Most asked: Multiplication and division of integers with sign rules · Properties of integers (commutative, associative, distributive) · Word problems involving integers (elevator, temperature, profit/loss, exam scoring) · Finding the value of expressions with multiple integers
Don't skip: Word problems that require setting up an expression with integers and solving it (e.g., profit/loss, elevator, temperature change); Questions that test the distributive property with negative numbers; Finding the product or quotient of multiple integers and determining the sign
One-Line Revision (read aloud before the exam)
  1. Integers are whole numbers that can be positive, negative, or zero.
  2. The additive inverse of a number a is -a, and a + (-a) = 0.
  3. On a number line, rightward movement is positive, leftward movement is negative.
  4. Subtracting a number is the same as adding its additive inverse: a - b = a + (-b).
  5. The product of two integers with the same sign is positive.
  6. The product of two integers with different signs is negative.
  7. The quotient of two integers with the same sign is positive.
  8. The quotient of two integers with different signs is negative.
  9. Multiplication of integers is commutative: a × b = b × a.
  10. Multiplication of integers is associative: a × (b × c) = (a × b) × c.
  11. Multiplication distributes over addition: a × (b + c) = (a × b) + (a × c).
  12. When multiplying many integers, an even number of negative factors gives a positive product; an odd number gives a negative product.
  13. Brahmagupta's rules: fortune × fortune = fortune; debt × debt = fortune; fortune × debt = debt.
  14. In division, always check your answer by multiplying the quotient by the divisor.
  15. For any integer a, 1 × a = a and (-1) × a = -a.

MCQs — 10 Questions with Answers

Q1. Which of the following represents movement to the left on a number line?

  • A. A positive integer
  • B. A negative integer ✓
  • C. Zero
  • D. An additive inverse

Answer: B — Negative integers represent leftward movement or position to the left of zero on a number line.

Q2. What is the additive inverse of 15?

  • A. −15 ✓
  • B. 15
  • C. 0
  • D. 1/15

Answer: A — The additive inverse of 15 is −15 because 15 + (−15) = 0.

Q3. Simplify: 7 − 18 using the additive inverse method.

  • A. 7 + 18 = 25
  • B. 7 + (−18) = −11 ✓
  • C. −7 + 18 = 11
  • D. −18 − 7 = −25

Answer: B — Subtracting 18 is the same as adding its additive inverse, −18, so 7 − 18 = 7 + (−18) = −11.

Q4. In the token model, what do a green token and a red token together form?

  • A. A positive integer
  • B. A negative integer
  • C. A zero pair ✓
  • D. An additive inverse

Answer: C — A green token (+1) and a red token (−1) together form a zero pair because they cancel each other out.

Q5. What is 4 × (−3)?

  • A. 12
  • B. −12 ✓
  • C. 7
  • D. −7

Answer: B — When a positive integer multiplies a negative integer, the result is negative: 4 × (−3) = −12.

Q6. A shopkeeper loses ₹50 per day for 5 days. What is his total loss represented as an integer?

  • A. 50
  • B. −50
  • C. −250 ✓
  • D. 250

Answer: C — A loss of ₹50 per day for 5 days is (−50) × 5 = −250, representing a total loss of ₹250.

Q7. The temperature drops by 3°C every hour for 6 hours. What is the total change in temperature?

  • A. 18°C rise
  • B. −18°C (an 18°C drop) ✓
  • C. 9°C drop
  • D. 3°C drop

Answer: B — A drop of 3°C per hour for 6 hours is (−3) × 6 = −18, representing an 18°C total drop.

Q8. If a carrom coin moves 8 units right, then 12 units left, what is its final position from the starting point?

  • A. 20 units right
  • B. 4 units left
  • C. −4 units (4 units left) ✓
  • D. 0 units

Answer: C — Final position = 8 + (−12) = −4, which means 4 units to the left of the starting point.

Q9. What does (−6) × (−5) equal, and why is the answer positive?

  • A. −30 because both are negative
  • B. 30 because we remove 5 negative tokens 6 times, leaving positive tokens ✓
  • C. 11 because we subtract the magnitudes
  • D. −11 because negatives always give negatives

Answer: B — When multiplying two negative integers, we remove negative tokens and are left with positive tokens, so (−6) × (−5) = 30.

Q10. In Rakesh's first puzzle, two numbers have a sum of 25 and a difference of 11. If the first number is 18, what is the second number, and which operation helps verify this?

  • A. 7; addition only
  • B. 7; both addition and subtraction verify it ✓
  • C. 36; multiplication
  • D. 43; division

Answer: B — The second number is 7 because 18 + 7 = 25 (sum) and 18 − 7 = 11 (difference), so both operations verify the answer.

Flashcards

What does a positive integer represent on the number line?

A positive integer represents movement or position to the right of zero.

What is the additive inverse of −7?

The additive inverse of −7 is +7 because −(−7) = 7.

When you subtract an integer, what operation do you actually perform?

You add the additive inverse of that integer; for example, 5 − 3 = 5 + (−3).

What do green and red tokens represent in the token model?

Green tokens represent +1 and red tokens represent −1; together they form a zero pair.

What is the rule for multiplying two integers with different signs?

The product is always negative when the integers have different signs.

What is (−5) × (−4) and why?

The answer is +20 because the product of two negative integers is always positive.

In the carrom coin puzzle, if the first movement is −4 and the final position is 5, what is the second movement?

The second movement is 9 because −4 + second movement = 5, so second movement = 9.

How do you model 3 × (−2) using tokens?

Place 2 red tokens (negatives) into an empty bag 3 times, giving 6 negatives or −6.

What does (−4) × 2 = −8 tell us about removing tokens?

It means we remove 2 positive tokens from the bag 4 times, leaving 8 negatives.

Why do we need to place zero pairs when subtracting or multiplying with negatives?

Zero pairs are needed because we start with an empty bag and cannot remove tokens that do not exist.

Important Board Questions

What is the additive inverse of −12? [1 mark]

Additive inverse is the number that when added gives zero; use −(−12).

Solve using the token model: (−3) × 4. Show how many zero pairs you need and what remains in the bag. [2 marks]

You need to remove 4 positive tokens 3 times; place zero pairs first, then remove positives to see what remains.

Two integers have a sum of 12 and a difference of 4. Find both integers. Verify your answer by checking both conditions. [3 marks]

Try systematic pairs: if difference is 4, then first number − second number = 4; also first + second = 12. Check both conditions.

A carrom coin starts at position 0. It is struck four times with movements: +6, −3, +5, and −8 units respectively. (a) Find the final position of the coin. (b) Explain what it means if the final position is negative. (c) If the fourth strike had been +8 instead of −8, what would the final position have been? [5 marks]

Add all movements: 6 + (−3) + 5 + (−8); negative position means left of zero; redo with +8 instead of −8 to find new position.

Assertion-Reason Questions for Operations with Integers

Q1. Assertion (A):

The additive inverse of –(–8) is –8.

Reason (R):

The additive inverse of an integer a is –a, so the additive inverse of 8 is –8.

Answer: C — A is true but R is false — The assertion is true: –(–8) = 8, and the additive inverse of 8 is –8. However, the reason is true but doesn't explain why the assertion is true. The reason just defines additive inverse, but doesn't account for the double negative in the assertion. So assertion is correct, but the reason doesn't fully explain it.

Q2. Assertion (A):

Subtracting a negative integer is equivalent to adding its opposite.

Reason (R):

This is because the additive inverse of a negative integer is positive, and subtracting means adding the additive inverse.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains the assertion. For example, 7 – (–3) = 7 + 3 = 10 because the additive inverse of –3 is 3. The reason provides the correct mathematical justification.

Q3. Assertion (A):

In the token model, multiplying by a negative multiplier means removing tokens from the bag.

Reason (R):

Negative numbers always represent subtraction in mathematics.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true: when the multiplier is negative (e.g., –3), we remove tokens 3 times. The reason is also technically true but is NOT the correct explanation. Negative numbers don't always represent subtraction—they represent direction, magnitude, or values. The correct explanation is about the action we perform (remove vs. place) based on multiplier sign, not about what negatives 'always' mean.

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Case-Based Questions (CBQ) for Operations with Integers

Case Study:

Sunita is playing a board game where she moves a token on a number line starting at 0. Each roll of a die with special markings tells her to move right (positive) or left (negative). In Round 1, she rolls +3 and moves 3 units right to position 3. In Round 2, she rolls –5 and moves 5 units left. In Round 3, she rolls +2 and moves 2 units right. In Round 4, she rolls –1 and moves 1 unit left.

Q1. What is Sunita's position after Round 2?

  • A. –2 ✓
  • B. 8
  • C. 2
  • D. –8

Answer: A

Q2. What is Sunita's final position after all 4 rounds?

  • A. –1 ✓
  • B. 11
  • C. –11
  • D. 1

Answer: A

Q3. In which round did Sunita cross from the positive side to the negative side of the number line?

  • A. Round 1
  • B. Round 2 ✓
  • C. Round 3
  • D. Round 4

Answer: B

Q4. If after Round 2 Sunita needed to return to position 0, what single move would she need?

  • A. –2
  • B. +2 ✓
  • C. +5
  • D. –5

Answer: B

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