Types of Numbers in Mathematics

 Numbers are the foundation of mathematics. They help us count, measure, and perform calculations. Different types of numbers exist based on their properties and usage. Let’s explore them in detail.

1. Natural Numbers (N)

Definition:

Natural numbers are the basic counting numbers starting from 1. They do not include zero or negative numbers.

Set Representation:

N={1,2,3,4,5,…}N={1,2,3,4,5,…}

Example:

  • 5, 23, 100 are natural numbers.
  • 0, -2, 3.5 are not natural numbers.

2. Whole Numbers (W)

Definition:

Whole numbers include all natural numbers along with zero.

Set Representation:

W={0,1,2,3,4,…}W={0,1,2,3,4,…}

Example:

  • 0, 7, 34 are whole numbers.
  • -1, 2.3, -5 are not whole numbers.

3. Integers (Z)

Definition:

Integers include all positive and negative whole numbers, including zero.

Set Representation:

Z={…,−3,−2,−1,0,1,2,3,…}Z={…,−3,−2,−1,0,1,2,3,…}

Example:

  • -5, 0, 12 are integers.
  • 3.7, ½, √2 are not integers.

4. Rational Numbers (Q)

Definition:

A number is rational if it can be expressed in the form p/q, where p and q are integers, and q ≠ 0.

Set Representation:

Q={pq∣p,q∈Z,q≠0}Q={qp​∣p,q∈Z,q=0}

Example:

  • 2/3, -5/4, 0.75 are rational numbers (can be written as fractions).
  • √2, π are not rational numbers.

5. Irrational Numbers (I)

Definition:

Numbers that cannot be expressed as fractions (p/q) are called irrational numbers. Their decimal expansion never terminates and never repeats.

Example:

  • √2 = 1.4142135… (Non-repeating, non-terminating decimal)
  • π = 3.141592653… (Never-ending decimal)

6. Real Numbers (R)

Definition:

All rational and irrational numbers together form the set of real numbers.

Set Representation:

R=Q∪IR=Q∪I

Example:

  • 2, -3.5, π, √7 are real numbers.
  • √(-4), 3 + 2i (Complex numbers) are not real numbers.

7. Prime Numbers

Definition:

prime number has exactly two factors: 1 and itself.

Example:

  • 2, 3, 5, 7, 11, 13, 17, … are prime numbers.
  • 4, 6, 9, 12 are not prime (they have more than two factors).

8. Composite Numbers

Definition:

composite number has more than two factors.

Example:

  • 4 (Factors: 1, 2, 4), 9 (Factors: 1, 3, 9), 15 (Factors: 1, 3, 5, 15) are composite numbers.
  • 7, 13, 17 are not composite (they are prime).

9. Even and Odd Numbers

Even Numbers: Divisible by 2. (e.g., 2, 4, 6, 8, …)

Odd Numbers: Not divisible by 2. (e.g., 1, 3, 5, 7, …)

10. Complex Numbers

Definition:

Numbers in the form a + bi, where a, b are real numbers, and i = √(-1) (imaginary unit).

Example:

  • 3 + 2i, -5i, 7 - 4i are complex numbers.
  • 2, -5, π are not complex numbers (they are real numbers).

MCQs on Types of Numbers

1. Which of the following is a natural number?

a) -3

b) 0

c) 5

d) 3.5

✅ Answer: c) 5

2. What is the smallest whole number?

a) 1

b) 0

c) -1

d) 2

✅ Answer: b) 0

3. Which of the following is NOT an integer?

a) -7

b) 0

c) 2.5

d) 10

✅ Answer: c) 2.5

4. Which of the following is a rational number?

a) π

b) √3

c) 7/9

d) √5

✅ Answer: c) 7/9

5. Which number is irrational?

a) 4/5

b) √2

c) -3

d) 10

✅ Answer: b) √2

6. What is the only even prime number?

a) 3

b) 2

c) 5

d) 7

✅ Answer: b) 2

7. Which of the following is a composite number?

a) 11

b) 13

c) 15

d) 17

✅ Answer: c) 15

8. Which number is both rational and real?

a) 1.75

b) √7

c) π

d) i (imaginary number)

✅ Answer: a) 1.75

9. What is the sum of an even and an odd number?

a) Always even

b) Always odd

c) Sometimes even, sometimes odd

d) Always prime

✅ Answer: b) Always odd

10. Which of the following is NOT a real number?

a) 4

b) -7

c) 5i

d) 2.5

✅ Answer: c) 5i

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