What are the sign rules for multiplying and dividing integers?

For multiplying or dividing integers: positive times positive equals positive, positive times negative equals negative, negative times positive equals negative, and negative times negative equals positive. The same sign rules apply to division. This is taught in NCERT Class 7 Chapter 2.

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Multiplication and Division of Integers

🎓 Class 7 Mathematics CBSE Theory Ch 2 — Integers ⏱ ~35 min

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2.2 Multiplication of Integers

We used the token model to represent addition and subtraction of integers. Now we explore how to model multiplication^? of integers using tokens.

Positive × Positive (Placing Tokens)

Suppose we put some positive tokens into an empty bag. There are 8 positives in the bag. We can see this as adding 2 positives to the bag 4 times:

\(4 \times 2 = 8\) (two green tokens, 4 times = 8 green tokens)

++++ ++++

Two green tokens, 4 times → \(4 \times 2 = 8\)

Positive × Negative (Placing Red Tokens)

Can we use tokens to give meaning to multiplications like \(4 \times (-2)\)?

\(4 \times (-2)\) can be interpreted as placing 2 negatives into an empty bag 4 times. There are now 8 red tokens, meaning −8.

−−−− −−−−

Two red tokens, 4 times → \(4 \times (-2) = -8\)

Negative × Positive (Removing Green Tokens)

When the multiplier is positive, we place tokens into the bag. When the multiplier is negative, we remove tokens from the bag.

For \((-4) \times 2\), we need to remove 2 positives from the bag 4 times. But the bag is empty! Just as in the case of subtraction, we first place zero pairs inside and then remove the positives.

We need to remove 2 green tokens 4 times = 8 green tokens total. So we place 8 zero pairs, then remove 8 green tokens. What is left? 8 red tokens = −8.

\((-4) \times 2 = -8\)

Negative × Negative (Removing Red Tokens)

For \((-4) \times (-2)\), we need to remove 2 negatives from the bag 4 times. Again, the bag is empty, so we place 8 zero pairs, then remove 8 red tokens. What is left?

8 green tokens = +8!

\((-4) \times (-2) = +8\)

Sign Rules for Multiplication

(+) × (+) = (+)

Positive × Positive = Positive

(+) × (−) = (−)

Positive × Negative = Negative

(−) × (+) = (−)

Negative × Positive = Negative

(−) × (−) = (+)

Negative × Negative = Positive

Patterns in Integer Multiplication

Let us construct a sequence of multiplications and observe the patterns:

\(4 \times 3 = 12\) \(3 \times 3 = 9\) ↓ −3 \(2 \times 3 = 6\) ↓ −3 \(1 \times 3 = 3\) ↓ −3 \(0 \times 3 = 0\) ↓ −3 \((-1) \times 3 = -3\) ↓ −3 \((-2) \times 3 = -6\) ↓ −3 \((-3) \times 3 = -9\) ↓ −3

\(4 \times (-3) = -12\) \(3 \times (-3) = -9\) ↓ +3 \(2 \times (-3) = -6\) ↓ +3 \(1 \times (-3) = -3\) ↓ +3 \(0 \times (-3) = 0\) ↓ +3 \((-1) \times (-3) = 3\) ↓ +3 \((-2) \times (-3) = 6\) ↓ +3 \((-3) \times (-3) = 9\) ↓ +3

Observation

Left column: When the multiplicand is +3, each unit decrease in multiplier decreases the product by 3.
Right column: When the multiplicand is −3, each unit decrease in multiplier increases the product by 3. This confirms that negative × negative = positive!

Key Results

For any integer \(a\):
• \(1 \times a = a\) (identity)
• \((-1) \times a = -a\) (additive inverse)
• \(a \times b = b \times a\) (commutative^? property)

Figure it Out (Multiplication)

Q1. Using tokens, find: (a) \(3 \times (-2)\) (b) \((-5) \times (-2)\) (c) \((-4) \times (-1)\) (d) \((-7) \times 3\)

Answers:
(a) \(3 \times (-2) = -6\) (place 2 reds, 3 times = 6 reds)
(b) \((-5) \times (-2) = +10\) (remove 2 reds, 5 times → place 10 zero pairs, remove 10 reds, 10 greens left)
(c) \((-4) \times (-1) = +4\) (remove 1 red, 4 times → 4 greens left)
(d) \((-7) \times 3 = -21\) (remove 3 greens, 7 times → 21 reds left)

Q2. If \(123 \times 456 = 56088\), without calculating, find: (a) \((-123) \times 456\) (b) \((-123) \times (-456)\) (c) \(123 \times (-456)\)

Answers: The magnitude stays the same; only the sign changes.
(a) \((-123) \times 456 = -56088\) (negative × positive = negative)
(b) \((-123) \times (-456) = +56088\) (negative × negative = positive)
(c) \(123 \times (-456) = -56088\) (positive × negative = negative)

Q3. Try to frame a simple rule to multiply two integers.

Rule: Multiply the magnitudes (absolute values). Then determine the sign:
• Same signs → Positive (both + or both −)
• Different signs → Negative (one + and one −)
In short: \(|a \times b| = |a| \times |b|\), and the sign follows the rule above.

Historical Note

Brahmagupta's Rules (628 CE): The Indian mathematician Brahmagupta^? in his Brahmasphutasiddhanta explicitly stated rules for multiplication and division of positive and negative numbers — using "fortunes" (dhana) for positive and "debts" (rina) for negative values. This was the first time in recorded history that these rules were formally articulated!

Worked Examples

Example 1: An exam has 50 MCQs. 5 marks for each correct answer and −2 for each wrong answer. Mala got 30 correct and 20 wrong. What is her total?

Solution:
Marks for correct: \(30 \times 5 = 150\)
Marks for wrong: \(20 \times (-2) = -40\)
Total: \(150 + (-40) = 150 - 40 = \mathbf{110}\)
Mala got 110 marks.

Example 2: An elevator in a mining shaft descends at 3 metres per minute. (a) Starting from ground level (0), what is its position after 1 hour? (b) Starting from 15 m above ground, what is its position after 45 minutes?

Solution:
(a) Speed = −3 m/min (downward). Time = 60 min.
Position = \(0 + 60 \times (-3) = 0 + (-180) = -180\) metres.
The elevator is 180 metres below ground level.

(b) Starting position = +15. Time = 45 min.
Distance = \(45 \times (-3) = -135\)
Position = \(15 + (-135) = -120\) metres.
The elevator is 120 metres below ground level.

Mining elevator shaft: positions above ground are positive, positions below ground are negative

🔵 What are the maximum possible marks in the exam? What are the minimum possible marks? Maximum = \(50 \times 5 = 250\) (all correct). Minimum = \(50 \times (-2) = -100\) (all wrong).

A Magic Grid of Integers

A grid containing some numbers is given below. Follow the steps as shown until no number is left:

8	−4	12	−6
−28	14	−42	21
12	−6	18	−9
20	−10	30	−15

Circle any number.
Strike out the row and the column containing that number.
Circle any unstruck number.
Repeat until there are no more unstruck numbers.
Multiply the circled numbers together.

🔵 Try this with different numbers. What product did you get? Was it different from the first time? Try a few more times with different numbers! What is so special about these grids — is it the magic in the numbers or the way they are arranged, or both? Can you make more such grids?

Observation

No matter which numbers you circle (following the rules), the product always comes out the same! This is because the grid has a special multiplicative structure — each entry is the product of its row factor and column factor.

Division of Integers

Division can be converted into multiplication. For example, \((-100) \div 25\) can be reframed as: "What should be multiplied to 25 to get −100?"

\(25 \times \,? = (-100)\)

We know that \(25 \times (-4) = (-100)\). Therefore, \((-100) \div 25 = (-4)\).

Similarly, \((-100) \div (-4)\) means: "What should be multiplied to \((-4)\) to get \((-100)\)?"

Since \((-4) \times 25 = (-100)\), we get \((-100) \div (-4) = 25\).

And \(50 \div (-25) = (-2)\), since \((-25) \times (-2) = 50\).

Sign Rules for Division

For any two positive integers \(a\) and \(b\) (where \(b \ne 0\)):
• \(a \div (-b) = -(a \div b)\)
• \((-a) \div b = -(a \div b)\)
• \((-a) \div (-b) = a \div b\)

Same sign → positive quotient. Different signs → negative quotient. (Same rules as multiplication!)

Figure it Out (Division)

Q1. Find: (a) \(14 \times (-15)\) (b) \(-16 \times (-5)\) (c) \((-5) \times (-1)\) (d) \((-8) \times 4\) (e) \((-9) \times 10\) (f) \(10 \times (-17)\)

(a) \(14 \times (-15) = -210\) (b) \(-16 \times (-5) = +80\) (c) \((-5) \times (-1) = +5\) (d) \((-8) \times 4 = -32\) (e) \((-9) \times 10 = -90\) (f) \(10 \times (-17) = -170\)

Q2. Find: (a) \((-27) \div 9\) (b) \(84 \div (-4)\) (c) \((-56) \div (-2)\) (d) \((-46) \div (-23)\)

(a) \(-27 \div 9 = -3\) (b) \(84 \div (-4) = -21\) (c) \((-56) \div (-2) = 28\) (d) \((-46) \div (-23) = 2\)

Q3. A cement company earns a profit of ₹8 per bag of white cement sold and a loss of ₹5 per bag of grey cement sold. (a) The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss? (b) If the number of bags of grey cement sold is 6,400, what is the number of bags of white cement to have neither profit nor loss?

(a) White: \(3000 \times 8 = 24000\). Grey: \(5000 \times (-5) = -25000\). Total: \(24000 + (-25000) = -1000\). Loss of ₹1,000.

(b) Grey loss: \(6400 \times (-5) = -32000\). To break even: \(n \times 8 = 32000\), so \(n = 4000\). 4,000 bags of white cement.

Q4. Replace the blank with an integer to make a true statement:
(a) \((-3) \times \text{___} = 27\)
(b) \(\text{___} \times (-8) = (-56)\)
(c) \(36 \div (-18) = \text{___}\)
(d) \(\text{___} \times (-12) = 132\)
(e) \(\text{___} \times (-8) = 7\)
(f) \(\text{___} \times 12 = -11\)

(a) \((-3) \times (-9) = 27\) → −9
(b) \(7 \times (-8) = -56\) → 7
(c) \(36 \div (-18) = -2\) → −2
(d) \((-11) \times (-12) = 132\) → −11
(e) This requires \(-8x = 7\), so \(x = -7/8\) — no integer solution (not exactly divisible)
(f) This requires \(12x = -11\), so \(x = -11/12\) — no integer solution

Q3. A freezing process requires the room temperature to be lowered from 32°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?

Solution: Temperature drop per hour = −5°C. After 10 hours: \(10 \times (-5) = -50\).
Final temperature = \(32 + (-50) = 32 - 50 = \mathbf{-18°C}\).

Expressions Using Integers — Properties

Let us explore the key properties of integer multiplication and check if they hold for negative integers too.

Commutative Property

For any two integers \(a\) and \(b\): \(a \times b = b \times a\)

Example: \((-3) \times 5 = -15\) and \(5 \times (-3) = -15\) ✓

Associative Property

For any three integers \(a\), \(b\), and \(c\): \(a \times (b \times c) = (a \times b) \times c\)

Example: \(5 \times (-3) \times 4\). Either way: \((5 \times (-3)) \times 4 = (-15) \times 4 = -60\), or \(5 \times ((-3) \times 4) = 5 \times (-12) = -60\) ✓

Distributive Property

For any three integers \(a\), \(b\), and \(c\):

\(a \times (b + c) = (a \times b) + (a \times c)\)

Example: \(5 \times (4 + (-2)) = 5 \times 2 = 10\). Also: \(5 \times 4 + 5 \times (-2) = 20 + (-10) = 10\) ✓

Sign of Product of Many Integers

When \(-1\) is multiplied an even number of times, the product is positive.
When \(-1\) is multiplied an odd number of times, the product is negative.
In general: if a product has an even number of negative factors → result is positive; odd number → negative.

Pick the Pattern — Machine Puzzles

Two pattern machines are given below. Each machine takes 3 numbers, does some operations, and gives the result.

Machine 1
5	8	3	10
10	11	12	9
5	8	3	10
−3	10	2	5
−4	−1	−6	1
10	−11	−1	?

🔵 Find the operations being done by Machine 1.

The operation done by Machine 1 is: (first number) + (second number) − (third number).

Written as an expression: \(a + b - c\). For example, \(5 + 8 - 3 = 10\) ✓, and \((-4) + (-1) - (-6) = -4 -1 + 6 = 1\) ✓.

So the last row: \(10 + (-11) - (-1) = 10 - 11 + 1 = 0\).

Machine 2
4	9	−3	−21
6	−11	12	54
5	3	−2	−22
3	9	−8	35
−1	5	−4	?
−10	−11	−9	?

🔵 Find the operations being done by Machine 2. Make your own machine and challenge your peers in finding its operations!

Activity: Design Your Own Pattern Machine

L6 Create

Challenge: Can you design a machine that takes 3 integers and always produces a positive output, regardless of whether the inputs are positive or negative?

Choose an operation involving multiplication and addition of the three inputs.
Test your machine with: (a) all positive inputs, (b) all negative, (c) mixed positive/negative.
Does it always give a positive result? If not, modify your operation.
Hint: Think about what happens when you square a number or use absolute values.

Example: Machine operation = \(a^2 + b^2 + c^2\). Since squaring any integer gives a non-negative result, the sum of three squares is always ≥ 0. It equals 0 only if all inputs are 0.

Another: \(a \times a + b \times b + 1\) — always positive since we add 1.

Competency-Based Questions

Scenario: A cement company earns a profit of ₹8 per bag of white cement sold and incurs a loss of ₹5 per bag of grey cement sold. In one month, the company sells 3,000 bags of white cement and 5,000 bags of grey cement.

Q1. What is the company's total profit or loss for the month?

L3 Apply

(a) ₹1,000 profit
(b) ₹1,000 loss
(c) ₹49,000 profit
(d) ₹25,000 loss

Answer: (b) ₹1,000 loss.
White cement: \(3000 \times 8 = +24000\)
Grey cement: \(5000 \times (-5) = -25000\)
Total: \(24000 + (-25000) = -1000\). A loss of ₹1,000.

Q2. If the number of grey cement bags sold is 6,400, what minimum number of white cement bags must be sold to avoid a loss?

L4 Analyse

Answer: Grey loss: \(6400 \times (-5) = -32000\). To break even: white profit ≥ 32000. So \(n \times 8 \ge 32000\), giving \(n \ge 4000\). At least 4,000 bags of white cement.

Q3. The company considers changing the grey cement price to earn ₹2 profit per bag instead of ₹5 loss. With 3,000 white and 5,000 grey bags, evaluate the difference in monthly outcome.

L5 Evaluate

Answer: Old: \(3000 \times 8 + 5000 \times (-5) = 24000 - 25000 = -1000\) (loss).
New: \(3000 \times 8 + 5000 \times 2 = 24000 + 10000 = +34000\) (profit).
Difference: \(34000 - (-1000) = 35000\). The price change improves the outcome by ₹35,000.

Q4. Create an expression using all four operations (+, −, ×, ÷) with integers that evaluates to exactly −7. Use at least one negative integer.

L6 Create

One solution: \((-3) \times 4 + 10 \div 2 - 2 = -12 + 5 - 2 = -9 + 2 = -7\) ✓
Another: \((-20) \div 4 + 3 \times 1 - 5 = -5 + 3 - 5 = -7\) ✓
Many answers are possible!

Assertion–Reason Questions

Assertion (A): \((-6) \times (-7) = 42\)
Reason (R): The product of two negative integers is always positive.

(a) Both true, R explains A.

(b) Both true, R doesn't explain A.

(c) A true, R false.

(d) A false, R true.

Answer: (a) — \(6 \times 7 = 42\), and negative × negative = positive, so \((-6) \times (-7) = +42\). R correctly explains A.

Assertion (A): \((-5) \times 3 \times (-2) = -30\)
Reason (R): A product with an odd number of negative factors is negative.

(a) Both true, R explains A.

(b) Both true, R doesn't explain A.

(c) A true, R false.

(d) A false, R true.

Answer: (d) — A is false: \((-5) \times 3 \times (-2) = (-15) \times (-2) = +30\) (not −30). There are 2 negative factors (even), so the result is positive. R is true as a general rule, but doesn't apply here since there are 2 (even) negatives.

Assertion (A): Integer multiplication is distributive over addition: \(a \times (b + c) = a \times b + a \times c\)
Reason (R): This property holds for all integers, including negative ones.

(a) Both true, R explains A.

(b) Both true, R doesn't explain A.

(c) A true, R false.

(d) A false, R true.

Answer: (a) — Both true. The distributive property holds for all integers. R explains the scope of A (it's not limited to positive integers).

Frequently Asked Questions

What are the sign rules for multiplying integers?

Positive times positive gives positive. Positive times negative gives negative. Negative times positive gives negative. Negative times negative gives positive. In summary, same signs give positive and different signs give negative products. NCERT Class 7 Ganita Prakash Part II Chapter 2 explains these rules.

Why does negative times negative equal positive?

One way to understand this is through patterns. Minus 3 times 3 equals minus 9, minus 3 times 2 equals minus 6, minus 3 times 1 equals minus 3, minus 3 times 0 equals 0. Following the pattern (adding 3 each time), minus 3 times minus 1 equals 3, which is positive. NCERT Class 7 explains this pattern approach.

How do you divide integers with different signs?

When dividing integers, the sign rules are the same as multiplication. Positive divided by positive is positive. Positive divided by negative is negative. Negative divided by positive is negative. Negative divided by negative is positive. NCERT Class 7 Chapter 2 covers integer division.

What is the multiplicative identity for integers?

The multiplicative identity is 1 because any integer multiplied by 1 gives the same integer. For example, minus 7 times 1 equals minus 7. Also, any integer multiplied by 0 gives 0 and multiplied by minus 1 gives its additive inverse. NCERT Class 7 Maths covers these properties.

Can you divide by zero in integers?

No, division by zero is undefined for integers and all numbers. You cannot divide any integer by zero because no number multiplied by zero gives a non-zero result. NCERT Class 7 Ganita Prakash Part II reminds students that division by zero is not allowed in mathematics.

Frequently Asked Questions — Chapter 2

What is Multiplication and Division of Integers in NCERT Class 7 Mathematics?

Multiplication and Division of Integers is a key concept covered in NCERT Class 7 Mathematics, Chapter 2: Chapter 2. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.

How do I solve problems on Multiplication and Division of Integers step by step?

To solve problems on Multiplication and Division of Integers, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 7 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.

What are the most important formulas for Chapter 2: Chapter 2?

The essential formulas of Chapter 2 (Chapter 2) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.

Is Multiplication and Division of Integers important for the Class 7 board exam?

Multiplication and Division of Integers is part of the NCERT Class 7 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.

What mistakes should students avoid in Multiplication and Division of Integers?

Common mistakes in Multiplication and Division of Integers include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.

Where can I find more NCERT practice questions on Multiplication and Division of Integers?

End-of-chapter NCERT exercises for Multiplication and Division of Integers cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 2, and solve at least one previous-year board paper to consolidate your understanding.

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