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WBB • Class 7 • Mathematics (গণিত প্রভা) • Ch 4
Estimated Time: 55 Mins
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Addition, Subtraction, Multiplication and Division of Integers

Welcome to Chapter 4 "Addition, Subtraction, Multiplication and Division of Integers" of the West Bengal Board (WBBSE) Class 7 Mathematics curriculum. Built upon the TargetExams Gold-Standard 5-Step Pedagogy System, this comprehensive guide covers the integer continuum, directional numbers, absolute values, the geometry of additive inverses, sign laws of multiplication and division, the undefined nature of division by zero, and the strategic deployment of the Distributive Law in algebraic problem-solving.

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🌍 Why Negative Times Negative Equals Positive: The Mathematics of Reality

Why does multiplying two negative debts turn into positive wealth?

For millennia, ancient counting systems relied strictly on natural numbers ($1, 2, 3, ...$). But when commercial trade, debt, and temperature extremes emerged, mathematicians needed a formal language for directional opposites. In 628 CE, the Indian astronomer-mathematician Brahmagupta formally codified the arithmetic of integers, defining positive numbers as "Fortunes" and negative numbers as "Debts".

Imagine a video camera filming water draining from a reservoir at $4\text{ liters per minute}$ ($-4\text{ L/min}$). If you rewind the video $5\text{ minutes}$ into the past ($-5\text{ min}$), the water level in the reservoir was $20\text{ liters}$ higher! In mathematical terms: $(-4) \times (-5) = +20\text{ L}$. From aviation altimeters and submarine sonar to digital banking and rocket flight algorithms, integer sign laws keep our modern world functioning without error.

Why This Chapter Matters

Welcome to Chapter 4 "Addition, Subtraction, Multiplication and Division of Integers" of the West Bengal Board (WBBSE) Class 7 Mathematics curriculum. Built upon the TargetExams Gold-Standard 5-Step Pedagogy System, this comprehensive guide covers the integer continuum, directional numbers, absolute values, the geometry of additive inverses, sign laws of multiplication and division, the undefined nature of division by zero, and the strategic deployment of the Distributive Law in algebraic problem-solving.

Before You Begin (Prerequisites)

  • Understanding of Natural Numbers ($\mathbb{N}$) and Whole Numbers ($\mathbb{W}$).
  • Mastery of four basic arithmetic operations (addition, subtraction, multiplication, division).
  • Familiarity with the horizontal number line and geometric distance.

What You Will Learn (Core Objectives)

  • Define the set of Integers ($\mathbb{Z}$), locate values on the number line, and compute absolute values ($|x|$).
  • Perform multi-step integer additions and subtractions using additive inverses ($a - b = a + (-b)$).
  • Apply the universal sign laws of multiplication ($+ \times + = +$, $- \times - = +$, $+ \times - = -$) across multiple factors.
  • Execute integer divisions and rigorously justify why division by zero is mathematically undefined.
  • Utilize the Commutative, Associative, and Distributive Properties to evaluate complex numerical expressions efficiently.

Chapter Roadmap & Progression

1 1. The Integer Continuum, Direction...
2 2. Addition and Subtraction on the...
3 3. Integer Multiplication & The Uni...
4 4. Integer Division & The Unique Pr...
5 5. Algebraic Laws: Commutative, Ass...

Complete Concept Guide (100% Curriculum Coverage)

1. The Integer Continuum, Directional Numbers & Absolute Value

1. The Intuition

Every measurement in physics and everyday commerce requires an anchor point ($0$). Elevation above sea level is positive ($+8{,}848\text{ m}$ for Mt. Everest), while depth below sea level is negative ($-10{,}994\text{ m}$ for the Mariana Trench). Similarly, a financial profit is $+₹500$, while a loss or debt is $-₹500$. Directional signs ($+$ and $-$) provide the algebraic orientation necessary to model real-world phenomena.

2. Formal Concept & Structure

The set of Integers ($\mathbb{Z}$) consists of positive integers, zero, and negative integers:

$$\mathbb{Z} = \{..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...\}$$

  • The Zero Origin ($0$): Zero is neither positive nor negative; it serves as the neutral origin of the coordinate continuum.
  • Number Line Directionality: Any number lying to the right is strictly greater than any number lying to its left. Hence, $-15 < -4 < 0 < +6$.
  • Absolute Value ($|x|$): The geometric distance of a number from the origin $0$ on the number line. Because distance is scalar, absolute value is always non-negative.
Absolute Value Axiom:
$$|+a| = a \quad \text{and} \quad |-a| = a$$ Example: $|+14| = 14$ and $|-14| = 14$.
3. Concrete Worked Example

Problem: Arrange the following integers in ascending order and evaluate the sum of their absolute values: $-24, +11, 0, -8, -35, +18$

Step 1 (Order comparison): More negative numbers lie further left on the number line.

$-35 < -24 < -8 < 0 < +11 < +18$

Ascending Order: $\mathbf{-35, -24, -8, 0, +11, +18}$

Step 2 (Absolute values sum):

$|-35| + |-24| + |-8| + |0| + |+11| + |+18| = 35 + 24 + 8 + 0 + 11 + 18 = \mathbf{96}$

Answer: Order: $-35, -24, -8, 0, 11, 18$; Sum of absolute values $= \mathbf{96}$.

4. Pitfall & Examiner Trap
⚠️ Common Trap: Comparing Negative Integers by Magnitude
Students often assume that because $30 > 10$, then $-30 > -10$. This is false!
Correct Principle: On the negative axis, greater absolute magnitude denotes a smaller value: $\mathbf{-10 > -30}$. Being ₹10 in debt is better than being ₹30 in debt.
5. Why This Matters in Life

Cryogenic laboratory freezers ($-80^\circ\text{C}$), submarine navigation systems, civil engineering ground excavations, and stock market loss indicators all rely directly on directional negative integers.

2. Addition and Subtraction on the Number Line & Additive Inverses

1. The Intuition

Walking along a number line illustrates sign operations intuitively: adding means stepping forward, while subtracting means stepping backward. If you subtract a negative direction, you are stepping backward while facing backward—which moves you forward! Hence, subtracting a negative is algebraically identical to adding a positive ($a - (-b) = a + b$).

2. Formal Concept & Structure

Additive Inverse: Two numbers whose sum is zero are called additive inverses of each other. The additive inverse of $a$ is $-a$, since $a + (-a) = 0$.

Rules for Integer Addition:

  • Adding Integers with Like Signs: Add their absolute values and attach the common sign:
    $(+9) + (+6) = +(9 + 6) = +15$
    $(-9) + (-6) = -(9 + 6) = -15$
  • Adding Integers with Unlike Signs: Subtract the smaller absolute value from the larger, and attach the sign of the larger absolute value:
    $(+16) + (-9) = +(16 - 9) = +7$
    $(-24) + (+11) = -(24 - 11) = -13$

The Golden Law of Subtraction:

Subtraction Transformation Formula: To subtract an integer, add its additive inverse to the first integer.
$$\mathbf{a - b = a + (-b)}$$ $$\mathbf{a - (-b) = a + b}$$
3. Concrete Worked Example

Problem: Simplify: $(-50) - (-28) + (-15) - (+12)$

Step 1 (Convert subtractions to additions of additive inverses):

$-(-28) \rightarrow +28$, and $-(+12) \rightarrow +(-12)$.

Expression becomes: $(-50) + (+28) + (-15) + (-12)$

Step 2 (Group by like signs):

Negative terms: $(-50) + (-15) + (-12) = -(50 + 15 + 12) = -77$

Positive term: $+28$

Step 3 (Final resolution): $(-77) + (+28) = -(77 - 28) = -49$

Answer: $\mathbf{-49}$

4. Pitfall & Examiner Trap
⚠️ Danger Trap: Confusing Addition of Negatives with Multiplication
Students often memorize "minus minus becomes plus" and erroneously compute $(-6) + (-4) = +10$.
Correction: Adding two debts deepens debt: $(-6) + (-4) = -10$. The "minus times minus is plus" rule applies only to multiplication and division.
5. Why This Matters in Life

If a bank account is overdrawn at $-\$100$ and the customer service department waives (subtracts) a $-\$35$ penalty fee, the new balance is $-\$100 - (-\$35) = -\$65$.

3. Integer Multiplication & The Universal Laws of Signs

1. The Intuition

Multiplication is repeated addition: $4 \times (-3) = (-3) + (-3) + (-3) + (-3) = -12$. But why does $(-4) \times (-3) = +12$? Consider maintaining algebraic consistency under the Distributive Law: $(-4) \times [3 + (-3)] = (-4) \times 0 = 0$. Expanding: $(-4 \times 3) + [(-4) \times (-3)] = 0 \implies -12 + [(-4) \times (-3)] = 0$. What must be added to $-12$ to yield $0$? Exactly $+12$! Thus, the product of two negative integers must be positive.

2. Formal Concept & Sign Matrix
1st Term Operation 2nd Term Product Sign Axiomatic Rule
$+$ (Positive)$\times$$+$ (Positive)$\mathbf{+}$ (Positive)Like signs produce Positive
$-$ (Negative)$\times$$-$ (Negative)$\mathbf{+}$ (Positive)Like signs produce Positive
$+$ (Positive)$\times$$-$ (Negative)$\mathbf{-}$ (Negative)Unlike signs produce Negative
$-$ (Negative)$\times$$+$ (Positive)$\mathbf{-}$ (Negative)Unlike signs produce Negative
General Parity Rule for Multiple Negative Factors:
• If the count of negative factors is even, the final product is positive ($+$).
• If the count of negative factors is odd, the final product is negative ($-$).
3. Concrete Worked Example

Problem: Evaluate: $(-3) \times (-2) \times (+5) \times (-4) \times (-1)$

Step 1 (Count negative factors):

There are 4 negative factors ($-3, -2, -4, -1$).

Since 4 is an even integer, the overall product sign is positive ($+$).

Step 2 (Multiply absolute values):

$3 \times 2 \times 5 \times 4 \times 1 = 6 \times 20 \times 1 = 120$

Step 3 (Combine sign and magnitude): $+120$

Answer: $\mathbf{+120}$ or $\mathbf{120}$

4. Pitfall & Examiner Trap
⚠️ Examiner Trap: $-x^2$ versus $(-x)^2$
$-6^2 = -(6 \times 6) = -36$ (exponent applies solely to the base 6).
Conversely, $(-6)^2 = (-6) \times (-6) = +36$ (parentheses force the negative sign to be squared).
5. Why This Matters in Life

Digital image filtering, acoustic noise-cancellation (where an inverted sound wave multiplies and cancels background noise), and game physics engines utilize this sign math constantly.

4. Integer Division & The Unique Properties of Zero

1. The Intuition

Division is the inverse operation of multiplication. Asking "$(-40) \div (-8) = ?$" is equivalent to asking "What number multiplied by $-8$ gives $-40$?". Since $(+5) \times (-8) = -40$, the result is $+5$. Division preserves the identical sign relationships established in multiplication.

2. Formal Concept & Properties of Zero

Division Sign Rules:

  • $(+a) \div (+b) = +(a \div b)$ and $(-a) \div (-b) = +(a \div b)$ (Like signs yield positive).
  • $(+a) \div (-b) = -(a \div b)$ and $(-a) \div (+b) = -(a \div b)$ (Unlike signs yield negative).
Cardinal Rules of Zero in Division:
• Zero divided by any non-zero integer is zero: $\mathbf{0 \div a = 0}$ (for $a \ne 0$).
• Division of any number by zero is Undefined: $\mathbf{a \div 0 = \text{Undefined}}$. If $a \div 0 = k$, then $0 \times k = a$, which has no real solution for any non-zero $a$.

Non-Closure under Division: Integers are closed under addition, subtraction, and multiplication, but not closed under division, because the quotient of two integers is not necessarily an integer (e.g., $(-9) \div 4 = -2.25 \notin \mathbb{Z}$).

3. Concrete Worked Example

Problem: Simplify: $[(-96) \div (-12)] \div [(-32) \div (+8)]$

Step 1 (Resolve bracketed expressions):

Left bracket: $(-96) \div (-12) = +(96 \div 12) = +8$ (Like signs $\rightarrow$ positive)

Right bracket: $(-32) \div (+8) = -(32 \div 8) = -4$ (Unlike signs $\rightarrow$ negative)

Step 2 (Final quotient):

$(+8) \div (-4) = -(8 \div 4) = -2$

Answer: $\mathbf{-2}$

4. Pitfall & Examiner Trap
⚠️ Pitfall: Assuming Division is Associative
$[(-32) \div 8] \div 2 = (-4) \div 2 = -2$
But $(-32) \div [8 \div 2] = (-32) \div 4 = -8$
Since $-2 \ne -8$, division is neither associative nor commutative.
5. Why This Matters in Life

In software engineering, attempting to divide by zero triggers critical runtime exceptions that can shut down trading systems or launch pads without proper error handling.

5. Algebraic Laws: Commutative, Associative & Distributive Properties

1. The Intuition

Evaluating $(-48) \times 98$ by long multiplication is slow and prone to errors. Using the Distributive Law transforms it into mental arithmetic: $(-48) \times (100 - 2) = (-48 \times 100) - (-48 \times 2) = -4800 - (-96) = -4800 + 96 = -4704$. Algebraic properties are calculation multipliers that make arithmetic effortless.

2. Formal Concept & Laws
  • Commutative Property:
    Addition: $a + b = b + a$
    Multiplication: $a \times b = b \times a$
    (Does NOT hold for subtraction or division)
  • Associative Property:
    Addition: $(a + b) + c = a + (b + c)$
    Multiplication: $(a \times b) \times c = a \times (b \times c)$
  • Identities:
    Additive Identity: $a + 0 = 0 + a = a$
    Multiplicative Identity: $a \times 1 = 1 \times a = a$
  • Distributive Property of Multiplication:
    Over Addition: $\mathbf{a \times (b + c) = (a \times b) + (a \times c)}$
    Over Subtraction: $\mathbf{a \times (b - c) = (a \times b) - (a \times c)}$
3. Concrete Worked Example

Problem: Use the Distributive Property to evaluate: $(-45) \times 72 + (-45) \times 28$

Step 1 (Factor out common multiplier $(-45)$):

$= (-45) \times [72 + 28]$

Step 2 (Evaluate sum in bracket):

$= (-45) \times 100 = \mathbf{-4500}$

Answer: $\mathbf{-4500}$

4. Pitfall & Examiner Trap
⚠️ Common Trap: Distributing Negative Signs Across Subtraction
When expanding $(-5)(y - 6)$, writing $-5y - 30$ is a frequent mistake.
Correct: $(-5) \times y - (-5) \times 6 = -5y - (-30) = \mathbf{-5y + 30}$.
5. Why This Matters in Life

Modern graphics processing units (GPUs) execute billions of parallel vector calculations per second by decomposing complex matrix multiplication using the Distributive Law.

Key Formulas, Identities & Theorems

Absolute Value
$$|+a| = a, \quad |-a| = a$$
Distance from origin is always non-negative
Additive Inverse Subtraction
$$a - b = a + (-b), \quad a - (-b) = a + b$$
Subtracting is adding the additive inverse
Multiplication of Like Signs
$$(+) \times (+) = +, \quad (-) \times (-) = +$$
Product is positive
Multiplication of Unlike Signs
$$(+) \times (-) = -, \quad (-) \times (+) = -$$
Product is negative
Distributive Law over Addition
$$a \times (b + c) = (a \times b) + (a \times c)$$
Foundation for algebraic expansion
Zero Division Axiom
$$a \div 0 = \text{Undefined}$$
Division by zero is strictly prohibited

Conceptual Solved Examples & Case Studies

Example 1
Simplify using the Distributive Property: $(-32) \times 75 + (-32) \times 25$
Step-by-Step Solution:
Factor out common multiplier $(-32)$: $= (-32) \times [75 + 25]$
$= (-32) \times 100 = \mathbf{-3200}$
Answer: $\mathbf{-3200}$
Example 2
Evaluate the multi-step expression: $[(-120) \div (-6)] \div (-5)$
Step-by-Step Solution:
Step 1: $(-120) \div (-6) = +20$
Step 2: $(+20) \div (-5) = \mathbf{-4}$
Answer: $\mathbf{-4}$
Example 3
An elevator was at basement level 3 ($-3$). It moved up 8 floors, then descended 4 floors. What is its final floor position?
Step-by-Step Solution:
Initial floor $= -3$
Movement: $(-3) + 8 - 4 = 5 - 4 = \mathbf{+1}$ (First Floor).
Answer: $\mathbf{+1}$ (1st Floor).

Common Misconceptions & Examiner Traps

Common Misconception

Applying multiplication sign rules to addition.

Scientific Reality & Correction

$(-a) + (-b) = -(a + b)$, whereas $(-a) \times (-b) = +(ab)$. Do not confuse these two operations.

Common Misconception

Claiming that division by zero equals zero.

Scientific Reality & Correction

$0 \div a = 0$ (for $a \ne 0$), but $a \div 0$ is mathematically undefined.

Common Misconception

Confusing $-a^2$ with $(-a)^2$.

Scientific Reality & Correction

$-5^2 = -25$, while $(-5)^2 = +25$.

Visual Learning & Conceptual Map

WBBSE Mathematics Foundations: The Integer Continuum & Sign Rule Compass

Origin alignment, directional vectors, and the universal laws of signs
-6 -5 -4 -3 -2 -1 0 Origin (Neutral) +1 +2 +3 +4 +5 +6 ← Negative Integers (Decreasing Value) Positive Integers (Increasing Value) →
Like Signs
$(+) \times (+) = \mathbf{+}$
$(+6) \times (+3) = +18$
Like Signs
$(-) \times (-) = \mathbf{+}$
$(-6) \times (-3) = +18$
Unlike Signs
$(+) \times (-) = \mathbf{-}$
$(+6) \times (-3) = -18$
Unlike Signs
$(-) \times (+) = \mathbf{-}$
$(-6) \times (+3) = -18$

Chapter Summary & 10 Key Takeaways

Takeaway 1
Integers Set ($\mathbb{Z}$): Comprises positive integers, zero, and negative integers ($\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$).
Takeaway 2
Absolute Value ($|x|$): Non-negative distance of an integer from zero ($|-a| = a$).
Takeaway 3
Additive Inverse: The unique number $-a$ such that $a + (-a) = 0$.
Takeaway 4
Subtraction Rule: $a - b = a + (-b)$ and $a - (-b) = a + b$.
Takeaway 5
Multiplication Sign Rules: Like signs yield positive ($+$); unlike signs yield negative ($-$).
Takeaway 6
Parity of Negative Factors: Even count of negative factors $\rightarrow$ Positive; Odd count $\rightarrow$ Negative.
Takeaway 7
Division by Zero: $0 \div a = 0$ ($a \ne 0$), but $a \div 0$ is strictly undefined.
Takeaway 8
Distributive Property: $a \times (b + c) = ab + ac$ and $a \times (b - c) = ab - ac$.

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.

1
Evaluate: $(-22) - (-35) + (-13)$
Reveal Answer & Explanation
Answer: $(-22) + 35 + (-13) = 35 + [(-22) + (-13)] = 35 + (-35) = \mathbf{0}$
Convert $-(-35)$ into $+35$.
2
Determine the product: $(-1) \times (-1) \times (-1) \times ... \text{ (99 times)}$
Reveal Answer & Explanation
Answer: $\mathbf{-1}$ — Since the exponent 99 is odd, the product is $-1$.
Count the number of negative factors: odd count results in negative.
3
Use the Distributive Property to compute: $(-25) \times 104$
Reveal Answer & Explanation
Answer: $(-25) \times (100 + 4) = (-2500) + (-100) = \mathbf{-2600}$
Express $104$ as $(100 + 4)$ and distribute $(-25)$.
4
A submarine was submerged at a depth of $450\text{ m}$ below sea level ($-450\text{ m}$). It rises $180\text{ m}$. What is its new depth?
Reveal Answer & Explanation
Answer: $-450 + 180 = \mathbf{-270\text{ m}}$ (270 meters below sea level).
Add $+180$ to the initial negative elevation.
5
Evaluate: $[(-72) \div 8] \div (-3)$
Reveal Answer & Explanation
Answer: $[(-72) \div 8] = -9$; then $(-9) \div (-3) = \mathbf{+3}$
Evaluate the inner bracket first: unlike signs yield negative, then like signs yield positive.
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