Integers - Properties of addition, subtraction, multiplication, and division of integers

Hello, future mathematicians! Welcome to today’s fun and interactive math lesson.

In Class 6, you learned about Integers—that giant family of numbers containing positive numbers ($1, 2, 3...$), zero ($0$), and negative numbers ($-1, -2, -3...$). Now, in Class 7, we take a step further.

Think of mathematical operations (Addition, Subtraction, Multiplication, and Division) as a game. Just like any board game, integers follow specific rules and behaviors when we perform these operations. In mathematics, we call these rules Properties.

Understanding these properties will make your calculations super fast, super easy, and help you avoid silly mistakes in your exams! Let’s dive in.

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1. Properties of Addition and Subtraction

Imagine integers living in a special closed kingdom called Integer Land. Let me introduce you to the rules of addition and subtraction in this kingdom!

A. Closure Property

Analogy: If two citizens of Integer Land get together to add or subtract, do they stay in Integer Land, or do they transform into aliens (fractions/decimals)?

Under Addition: When you add any two integers, the result is *always* an integer.

$$\text{Integer} + \text{Integer} = \text{Integer}$$

Example: $5 + (-8) = -3$ ($-3$ is an integer!)

Verdict: Addition is Closed for integers.

Under Subtraction: When you subtract any two integers, the result is *always* an integer.

$$\text{Integer} - \text{Integer} = \text{Integer}$$

Example: $(-4) - 7 = -11$ ($-11$ is an integer!)

Verdict: Subtraction is Closed for integers.

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B. Commutative Property

Analogy: "Commute" means to move around. Does changing the order of numbers change the final answer?

Under Addition: You can add integers in any order you like!

$$a + b = b + a$$

Example: Let’s test with $a = -3$ and $b = 5$.

Left Hand Side (LHS): $(-3) + 5 = 2$

Right Hand Side (RHS): $5 + (-3) = 2$

LHS = RHS!

Verdict: Addition is Commutative for integers.

Under Subtraction: Order matters a lot!

$$a - b \neq b - a$$

Example: Let’s test with $a = 5$ and $b = 3$.

LHS: $5 - 3 = 2$

RHS: $3 - 5 = -2$

$2 \neq -2$

Verdict: Subtraction is NOT Commutative for integers.

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C. Associative Property

Analogy: "Associate" means to form groups with friends. If three integers are adding up, does it matter who pairs up first?

Under Addition: Grouping does not change the sum.

$$(a + b) + c = a + (b + c)$$

Example: Let $a = -2$, $b = 3$, $c = -5$

LHS: $[(-2) + 3] + (-5) = 1 + (-5) = -4$

RHS: $(-2) + [3 + (-5)] = (-2) + (-2) = -4$

LHS = RHS!

Verdict: Addition is Associative for integers.

Under Subtraction:

$$(a - b) - c \neq a - (b - c)$$

Verdict: Subtraction is NOT Associative for integers.

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D. Additive Identity and Additive Inverse

Additive Identity ($0$): Adding $0$ to any integer keeps its identity unchanged!

$$a + 0 = a = 0 + a$$

Example: $(-9) + 0 = -9$

Additive Inverse: The opposite sign of a number. When added together, they give $0$.

$$a + (-a) = 0$$

Example: Additive inverse of $7$ is $-7$, because $7 + (-7) = 0$.

Example: Additive inverse of $-12$ is $12$, because $(-12) + 12 = 0$.

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2. Properties of Multiplication

Multiplication of integers is simply repeated addition, but it comes with some powerful shortcut properties!

A. Closure Property

When you multiply any two integers, the result is always an integer.

$$a \times b = \text{Integer}$$

Example: $(-4) \times (-5) = 20$ (an integer!)

Verdict: Integers are Closed under multiplication.

B. Commutative Property

Changing the order of factors does not change the product.

$$a \times b = b \times a$$

Example: $(-6) \times 4 = -24$ and $4 \times (-6) = -24$

Verdict: Multiplication is Commutative for integers.

C. Associative Property

When multiplying three integers, you can group them in any way.

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

Example: $[(-2) \times 3] \times (-4) = (-6) \times (-4) = 24$

$ (-2) \times [3 \times (-4)] = (-2) \times (-12) = 24$

Verdict: Multiplication is Associative for integers.

D. Multiplicative Identity and Zero Property

Multiplicative Identity ($1$): Multiplying any integer by $1$ leaves it unchanged.

$$a \times 1 = a = 1 \times a$$

Multiplication by Zero ($0$): Any integer multiplied by zero becomes zero!

$$a \times 0 = 0 = 0 \times a$$

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E. The Super Hero Property: Distributive Property!

Teacher's Secret Tip: This is the most important property in Class 7! It helps you break down big multiplication problems into small, easy steps.

Over Addition:

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

Over Subtraction:

$$a \times (b - c) = (a \times b) - (a \times c)$$

Real-World Analogy: Imagine a superhero $a$ visiting a house where friends $b$ and $c$ live together. Superhero $a$ must shake hands with both $b$ and $c$ individually!

Example: Find $12 \times 105$ using properties.

Write $105$ as $(100 + 5)$.

$12 \times (100 + 5) = (12 \times 100) + (12 \times 5)$

$= 1200 + 60 = 1260$.

*See how easy that was without long multiplication?*

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3. Properties of Division

Division is the reverse of multiplication. Let's see how integers behave under division:

Closure Property: NO!

Example: $(-3) \div (-6) = \frac{-3}{-6} = \frac{1}{2}$ (This is a fraction, not an integer!).

Commutative Property: NO!

Example: $(-8) \div 2 = -4$, but $2 \div (-8) = -\frac{1}{4}$. They are not equal.

Associative Property: NO!

Grouping changes the result completely.

Division Rules with $0$ and $1$:

Any integer divided by $1$ gives the same integer: $a \div 1 = a$.

Zero divided by any non-zero integer gives zero: $0 \div a = 0$.

WARNING: Division of an integer by zero ($a \div 0$) is Not Defined!

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Master Summary Table

Property	Addition	Subtraction	Multiplication	Division
Closure	Yes ($\checkmark$)	Yes ($\checkmark$)	Yes ($\checkmark$)	No ($\times$)
Commutative	Yes ($\checkmark$)	No ($\times$)	Yes ($\checkmark$)	No ($\times$)
Associative	Yes ($\checkmark$)	No ($\times$)	Yes ($\checkmark$)	No ($\times$)
Identity	$0$	None	$1$	None

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Practice Time! (3 Solved Questions)

Let's test your understanding with three classic exam-style problems. Try solving them yourself before reading the step-by-step solutions!

Question 1

Find the product using suitable properties:

$$8 \times 53 \times (-125)$$

Detailed Solution:

Step 1: Look for pairs that give clean round numbers (like $10, 100, 1000$). Here, we know $8 \times 125 = 1000$.

Step 2: Use the Commutative Property ($a \times b = b \times a$) to swap positions:

$$8 \times 53 \times (-125) = 53 \times [8 \times (-125)]$$

Step 3: Multiply inside the brackets:

$$8 \times (-125) = -1000$$

Step 4: Now multiply $53$ by $-1000$:

$$53 \times (-1000) = -53000$$

Answer: $-53,000$

*(Property used: Commutative and Associative properties of multiplication)*

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Question 2

Simplify the following expression using the Distributive Property:

$$(-26) \times 72 + (-26) \times 28$$

Detailed Solution:

Step 1: Notice that $(-26)$ is common in both terms. This fits the form $(a \times b) + (a \times c)$.

Step 2: Apply the Distributive Property in reverse: $a \times (b + c)$

Here, $a = -26$, $b = 72$, and $c = 28$.

$$(-26) \times 72 + (-26) \times 28 = (-26) \times [72 + 28]$$

Step 3: Solve the expression inside the brackets first:

$$72 + 28 = 100$$

Step 4: Multiply the remaining terms:

$$(-26) \times 100 = -2600$$

Answer: $-2,600$

*(Property used: Distributive property of multiplication over addition)*

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Question 3

Verify the Associative Property of Addition for the given integers:

$$a = -5, \quad b = 3, \quad c = -8$$

Detailed Solution:

Goal: Verify if $(a + b) + c = a + (b + c)$

Step 1: Calculate Left Hand Side (LHS)

$$\text{LHS} = (a + b) + c$$

$$\text{LHS} = [(-5) + 3] + (-8)$$

$$\text{LHS} = (-2) + (-8) = -10$$

Step 2: Calculate Right Hand Side (RHS)

$$\text{RHS} = a + (b + c)$$

$$\text{RHS} = (-5) + [3 + (-8)]$$

$$\text{RHS} = (-5) + (-5) = -10$$

Step 3: Compare LHS and RHS

$$\text{LHS} = \text{RHS} = -10$$

Conclusion: Since LHS = RHS, the Associative Property of Addition is verified for the given integers!

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Keep Practicing!

Great job working through this tutorial! Remember, math is just like learning a sport—the more you practice applying these rules, the faster and better you'll get. Keep up the brilliant work!