UBSE • Class 7 • Mathematics • Ch 1

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Integers

In Class 7 Mathematics, Chapter 10 "Operations with Integers" establishes complete mastery over signed number arithmetic. Aligned with the 2026–27 NCERT Ganita Prakash curriculum, this master material explores addition and subtraction using the Token Model, proves why multiplying two negatives yields a positive, analyzes Brahmagupta's historic rules of fortune and debt, unpacks integer division, and applies algebraic properties (closure, commutativity, associativity, distributivity).

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🌡️ Have You Ever Wondered?

Why does a negative times a negative make a positive?

If you have a debt of Rs. 10 and you take 3 such debts, you have $3 \times (-10) = -\text{Rs. } 30$. That feels completely natural. But why on earth should $(-3) \times (-10)$ become $+\text{Rs. } 30$? How can multiplying two debts turn into wealth?

Think of a video recording of water draining out of a tank at $2\text{ liters per minute}$ ($-2$). If you play the video backwards into the past ($-3\text{ minutes}$), what do you see? The water level in the tank was higher by $6$ liters! Mathematically: $$(-2) \times (-3) = \mathbf{+6}$$

More than $1,400$ years ago, the great Indian mathematician Brahmagupta formulated this exact principle: "The product of two debts is a fortune." Signed numbers are not an arbitrary invention—they are the exact language of physical balance, temperature drops, and financial flow.

Why This Chapter Matters

Before You Begin (Prerequisites)

Representation of integers on a number line (positive to right of zero, negative to left).
Additive inverse: for every integer $a$, there exists $-a$ such that $a + (-a) = 0$.
Whole number multiplication as repeated addition ($3 \times 4 = 4 + 4 + 4$).

What You Will Learn (Core Objectives)

Add and subtract integers with like and unlike signs using magnitude rules and the Token Model.
Explain why the product of two negative integers is positive using number patterns.
Multiply and divide positive and negative integers with 100% sign accuracy.
Determine the sign of a product containing multiple negative factors.
Apply closure, commutative, associative, and distributive properties to simplify complex integer calculations.

Chapter Roadmap & Progression

1 1. Adding and Subtracting Integers:...

2 2. Multiplication of Integers & The...

3 3. Division of Integers & Core Alge...

Complete Concept Guide (100% Curriculum Coverage)

1. Adding and Subtracting Integers: Signs & Zero Pairs

1. The Intuition (The Token Model)

Imagine green tokens for $+1$ and red tokens for $-1$. When one green token meets one red token, they cancel each other out completely to form a zero pair: $$(+1) + (-1) = \mathbf{0}$$

2. The Two Addition Rules

Same Signs: Add their magnitudes and keep the common sign. $$\mathbf{(+4) + (+3) = +7}, \quad \mathbf{(-4) + (-3) = -7}$$
Different Signs: Subtract the smaller magnitude from the larger magnitude, and attach the sign of the number with the larger magnitude. $$\mathbf{(-9) + (+4) = -5}, \quad \mathbf{(+9) + (-4) = +5}$$

Subtraction Rule: To subtract an integer, add its additive inverse! $$\mathbf{a - b = a + (-b)}, \quad \mathbf{a - (-b) = a + b}$$

3. Concrete Worked Example

Example: Evaluate: $(-18) - (-25) + (-12)$

Step 1: Convert subtraction to addition of opposite: $(-18) + 25 + (-12)$

Step 2: Group negatives together: $[(-18) + (-12)] + 25 = (-30) + 25$

Step 3: Different signs → $30 - 25 = 5$, sign of $30$ is negative: $\mathbf{-5}$.

4. Pitfall & Examiner Trap

⚠️ Trap: Two Negatives in Addition vs. Multiplication
Students often say "two negatives make a positive" and write $(-5) + (-3) = +8$.
Reality: Two debts combined make a BIGGER debt: $(-5) + (-3) = \mathbf{-8}$! The phrase "two negatives make a positive" applies ONLY to multiplication and subtraction of a negative.

5. Why This Matters in Life

In high-altitude mountaineering, if night temperature is $-15^\circ\text{C}$ and daytime temperature is $+4^\circ\text{C}$, the temperature change is $4 - (-15) = 19^\circ\text{C}$.

2. Multiplication of Integers & The Pattern of Negatives

1. The Intuition (The Descending Pattern)

Observe the multiplication table of $-3$ counting downward:

$3 \times (-3) = -9$
$2 \times (-3) = -6 \quad (+3)$
$1 \times (-3) = -3 \quad (+3)$
$0 \times (-3) = 0 \quad (+3)$
$(-1) \times (-3) = \mathbf{+3} \quad (+3)$
$(-2) \times (-3) = \mathbf{+6} \quad (+3)$

To keep the consistent pattern of increasing by $+3$ at each step, the product of two negatives must be positive!

2. Brahmagupta's Master Multiplication Rules

Factor 1	Factor 2	Sign of Product	Example
Positive ($+$)	Positive ($+$)	Positive ($+$)	$4 \times 5 = +20$
Positive ($+$)	Negative ($-$)	Negative ($-$)	$4 \times (-5) = -20$
Negative ($-$)	Positive ($+$)	Negative ($-$)	$(-4) \times 5 = -20$
Negative ($-$)	Negative ($-$)	Positive ($+$)	$(-4) \times (-5) = +20$

Rule for Multiple Factors:

If the count of negative integers is EVEN, the product is POSITIVE.
If the count of negative integers is ODD, the product is NEGATIVE.

3. Concrete Worked Example

Example: Find the product: $(-2) \times (-3) \times (-4) \times (-5)$

Step 1: Count the negative signs: there are $4$ negative signs (an even number).

Step 2: Therefore, the final sign is Positive ($+$).

Step 3: Multiply magnitudes: $2 \times 3 \times 4 \times 5 = 6 \times 20 = \mathbf{+120}$.

4. Pitfall & Examiner Trap

⚠️ Trap: Odd Count of Negatives
In $(-1) \times (-1) \times (-1)$, students see three negatives and write $+1$.
Rule: Three is an odd number! Negative $\times$ Negative $=$ Positive ($+1$), and $(+1) \times (-1) = \mathbf{-1}$.

5. Why This Matters in Life

In 7th century India, Brahmagupta established these algebraic laws in his treatise Brahmasphutasiddhanta, paving the way for modern global mathematics.

3. Division of Integers & Core Algebraic Properties

1. Division Sign Rules

Division is the inverse of multiplication. Therefore, the sign rules for division are identical to multiplication:

$\frac{\text{Positive}}{\text{Positive}} = \mathbf{Positive}$ → $20 \div 4 = 5$
$\frac{\text{Negative}}{\text{Negative}} = \mathbf{Positive}$ → $(-20) \div (-4) = \mathbf{+5}$
$\frac{\text{Negative}}{\text{Positive}} = \mathbf{Negative}$ → $(-20) \div 4 = \mathbf{-5}$
$\frac{\text{Positive}}{\text{Negative}} = \mathbf{Negative}$ → $20 \div (-4) = \mathbf{-5}$

Division by Zero: For any integer $a$, $a \div 0$ is meaningless and undefined! However, $0 \div a = 0$ (for $a \ne 0$).

2. Properties of Integers Across Operations

Property	Addition	Subtraction	Multiplication	Division
Closure	Yes	Yes	Yes	No (e.g. $3 \div 2 \notin \mathbb{Z}$)
Commutative	$a+b = b+a$	No	$ab = ba$	No
Associative	$(a+b)+c = a+(b+c)$	No	$(ab)c = a(bc)$	No
Distributive	$a \times (b + c) = ab + ac$ (Multiplication over Addition)

3. Concrete Worked Example (Mental Distributivity)

Example: Evaluate $(-26) \times 72 + (-26) \times 28$

Step 1: Notice common factor $(-26)$: $(-26) \times (72 + 28)$

Step 2: Add in bracket: $72 + 28 = 100$

Step 3: Multiply: $(-26) \times 100 = \mathbf{-2,600}$

4. Pitfall & Examiner Trap

⚠️ Trap: Assuming Integers are Closed Under Division
Integers are closed under $+$, $-$, and $\times$ (the result is always an integer). But integers are NOT closed under division because $5 \div 2 = 2.5$, which is a fraction, not an integer!

5. Why This Matters in Life

Submarine depth navigation uses negative integers; calculating dive descent rate over time relies on dividing signed integers: $(-300\text{ m}) \div 15\text{ min} = -20\text{ m/min}$.

Visual Learning & Conceptual Map

Integer Sign Matrix for Multiplication and Division

Like signs produce positive; unlike signs produce negative

$(+) \times (+)$

$+$ POSITIVE

$(+) \times (-)$

$-$ NEGATIVE

$(-) \times (+)$

$-$ NEGATIVE

$(-) \times (-)$

$+$ POSITIVE

Chapter Summary & 10 Key Takeaways

Takeaway 1

Addition Rules: Same signs add magnitudes and keep common sign; different signs subtract magnitudes and keep sign of larger magnitude.

Takeaway 2

Subtraction as Additive Inverse: $a - b = a + (-b)$ and $a - (-b) = a + b$.

Takeaway 3

Multiplication Signs: Like signs yield positive ($+$); unlike signs yield negative ($-$). Even count of negatives is positive; odd count is negative.

Takeaway 4

Brahmagupta's Law: A debt times a debt is a fortune ($(-) \times (-) = (+)$).

Takeaway 5

Division: Same sign rules as multiplication; division by zero is undefined.

Takeaway 6

Properties: Integers are closed under $+$, $-$, $\times$ (not $\div$); multiplication is commutative, associative, and distributes over addition.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

Evaluate: $(-30) \div [(-13) + (-2)]$

Reveal Answer & Explanation

Answer: $+2$
Inside bracket: $(-13) + (-2) = -15$. Then $(-30) \div (-15) = +2$.

What is the sign of the product of 18 negative integers and 5 positive integers?

Reveal Answer & Explanation

Answer: Positive ($+$)
There are $18$ negative integers (an even number). An even number of negative factors always yields a positive product.

Evaluate using the distributive property: $(-49) \times 18$

Reveal Answer & Explanation

Answer: $-882$
$(-49) \times (20 - 2) = (-49) \times 20 - (-49) \times 2 = -980 - (-98) = -980 + 98 = -882$.

An elevator descends into a mine shaft at the rate of $6\text{ m/min}$. If the descent starts from $10\text{ m}$ above ground level, how long will it take to reach $-350\text{ m}$?

Reveal Answer & Explanation

Answer: $60\text{ minutes}$ (or $1\text{ hour}$)
Total vertical distance $= 10 - (-350) = 360\text{ m}$. Time $= 360 \div 6 = 60\text{ minutes}$.

Is $a - b = b - a$ true for integers? Give an example.

Reveal Answer & Explanation

Answer: No, subtraction is not commutative for integers.
Take $a = 5, b = 3$: $5 - 3 = 2$, but $3 - 5 = -2$. $2 \ne -2$.

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