Integers

Integers Definition

Integers are the number zero (0), a positive natural number (1, 2, 3, …), or the negation of a positive natural number (−1, −2, −3, …). Integers do not include fractions or decimals.

Integers Types

1. Positive Integers

Numbers greater than zero.
Example: 1, 2, 10, 500

2. Negative Integers

Numbers less than zero.
Example: -1, -7, -100

3. Zero

Zero is neither positive nor negative.

Absolute Value of an Integer

Absolute value means distance from zero.

| -5 | = 5
| 7 | = 7

It is always positive.

Properties of Integers

Let’s understand each property operation-wise.

1️⃣ Closure Property of Integers

👉 A set is closed under an operation if performing that operation on two integers gives another integer.

✔ Closure Under Addition

Example:

• 5 + (-3) = 2 ✔
• (-4) + (-6) = -10 ✔

Result is always an integer.

✔ Closure Under Subtraction

Example:

• 7 − 9 = -2 ✔
• (-5) − (-2) = -3 ✔

Result is always an integer.

✔ Closure Under Multiplication

Example:

• 6 × (-2) = -12 ✔
• (-3) × (-5) = 15 ✔

Result is always an integer.

❌ Not Closed Under Division

Example:

• 8 ÷ 2 = 4 ✔
• 7 ÷ 2 = 3.5 ✘ (Not an integer)

Therefore, integers are not closed under division.

2️⃣ Commutative Property of Integers

👉 If changing the order does not change the result.

✔ Commutative Under Addition

a + b = b + a

Example:

• 4 + (-2) = (-2) + 4

❌ Not Commutative Under Subtraction

Example:

• 5 − 2 = 3
• 2 − 5 = -3

Results are different ❌

✔ Commutative Under Multiplication

a × b = b × a

Example:

• 3 × (-4) = (-4) × 3

❌ Not Commutative Under Division

Example:

• 8 ÷ 4 = 2
• 4 ÷ 8 = 1/2

Different results ❌

3️⃣ Associative Property of Integers

👉 If grouping does not change the result.

✔ Associative Under Addition

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

Example:
(2 + 3) + 4 = 2 + (3 + 4)

❌ Not Associative Under Subtraction

Example:
(5 − 3) − 2 = 0
5 − (3 − 2) = 4

Different results ❌

✔ Associative Under Multiplication

(a × b) × c = a × (b × c)

Example:
(2 × 3) × 4 = 2 × (3 × 4)

❌ Not Associative Under Division

Example:
(8 ÷ 4) ÷ 2 = 1
8 ÷ (4 ÷ 2) = 4

Different results ❌

4️⃣ Identity Property of Integers

Additive Identity

a + 0 = a

0 is the additive identity.

Example:
5 + 0 = 5

Multiplicative Identity

a × 1 = a

1 is the multiplicative identity.

Example:
7 × 1 = 7

No Identity for Subtraction and Division

There is no special number that keeps all integers unchanged in subtraction or division.

5️⃣ Distributive Property (Very Important)

Multiplication distributes over addition and subtraction.

a × (b + c) = a×b + a×c
a × (b − c) = a×b − a×c

Example:
3 × (5 − 2) = 3×5 − 3×2
= 15 − 6
= 9

Real-Life Applications of Integers

Integers are used in everyday life:

1. Temperature

• 10°C (above zero)
• -5°C (below zero)

2. Bank Transactions

• +500 (deposit)
• -200 (withdrawal)

3. Elevation

• +300 m (above sea level)
• -50 m (below sea level)

4. Profit and Loss

-₹500 (loss)

+₹2000 (profit)