Number System – Part 1

The Number System is the fundamental concept in mathematics. It forms the backbone of Arithmetic and appears in almost every competitive exam, including SSC, RRB, Banking, Teaching, Defence, Police Constable, and State-level examinations. Understanding the classification of numbers and their properties helps in solving questions quickly and accurately.

Introduction to Number System

The Number System is the method of representing numbers in a structured way. Every number we use in day-to-day life belongs to some category such as natural numbers, whole numbers, integers, rational numbers, irrational numbers, or real numbers.

Classification of Numbers

Numbers are broadly classified into the following major types:

• Real Numbers (R)
• Imaginary Numbers
• Complex Numbers
• Rational Numbers (Q)
• Irrational Numbers (Q’)
• Integers (Z)
• Whole Numbers (W)
• Natural Numbers (N)

Real Numbers

Real Numbers include all numbers that can be represented on the number line.
They include:

Natural numbers
Whole numbers
Integers
Rational numbers
Irrational numbers

Examples:
2, 0, −8, 3/5, 0.75, √2, 3.14…

Real numbers form the largest set of commonly used numbers.

Imaginary Numbers

Imaginary numbers are numbers which cannot be placed on the number line because they involve the square root of a negative number.

Example:

• √–5
• √–14

These numbers use the imaginary unit i, where:

i² = –1

Imaginary numbers are mainly used in advanced mathematics and engineering.

Complex Numbers

A complex number is the combination of a real number and an imaginary number.

Format:a + bi

Where:

• a = real part
• b = coefficient of imaginary part

Examples:
5 + 2i
7 – 3i

Every real number is also a complex number (because we can write 5 as 5 + 0i).

Natural Numbers (N)

Natural numbers are counting numbers beginning from 1.
Examples:

1, 2, 3, 4, 5…

Whole Numbers (W)

Whole Numbers include all natural numbers and zero.
Examples:

0, 1, 2, 3, 4, 5…

Zero is introduced in the set of whole numbers, which makes it bigger than natural numbers.

Integers (Z)

Integers include:

• Positive numbers
• Negative numbers
• Zero

Examples:
… −5, −4, −3, −2, −1, 0, 1, 2, 3, 4 …

Integers are divided into:

Even Numbers

Numbers divisible by 2
Examples: −4, 0, 2, 4, 6…

Odd Numbers

Numbers not divisible by 2
Examples: −5, −3, 1, 3, 5…

Properties of Zero (0)

Zero has special properties:

• 0 is neither positive nor negative
• 0 is an even number
• Any number × 0 = 0
• 0 ÷ any number = 0
• Any number ÷ 0 = undefined / not possible

Zero is the central point separating positive and negative integers.

Prime Numbers

Prime numbers have exactly two factors: 1 and the number itself.

Examples:
2, 3, 5, 7, 11, 13, 17…

Important facts:

• 2 is the only even prime number
• 1 is neither prime nor composite
• All prime numbers > 3 are in the form 6n ± 1
• Prime numbers are infinite

 Composite Numbers

Composite numbers are numbers that have more than two factors.

Examples:
4, 6, 8, 9, 10, 12…

Co-Prime Numbers

Two numbers are called co-prime if their HCF = 1, even though they may not be prime.

Examples:
15 and 23
8 and 9
11 and 16

Twin Prime Numbers

Twin prime numbers are pairs of prime numbers whose difference is 2.

Examples:
(3, 5)
(11, 13)
(17, 19)

Rational Numbers (Q)

A number is rational if it can be expressed in the form: p/q, where q ≠ 0

Rational numbers include:

• Integers (5 = 5/1)
• Fractions (3/7)
• Terminating decimals
• Non-terminating repeating decimals

Examples:
3/5, 0.75, 0.333…, −2, 4

Irrational Numbers (Q’)

Irrational numbers cannot be written in p/q form.
They are:

• Non-terminating
• Non-repeating

Examples:
√2, √3, √11, π (3.14159265…)

These numbers cannot be expressed exactly in decimal form.

Decimal Numbers

Decimal numbers are of three types:

Terminating Decimals

Decimal ends after a few digits.
Examples:
0.25, 2.14, 5.6

Non-Terminating Repeating

Digit or group of digits repeats.
Examples:
0.666…, 0.727272…

Non-Terminating Non-Repeating

These are irrational numbers.
Examples:
π = 3.141592…
√3 = 1.732050…

Representation of Numbers on Number Line

Every real number can be represented on a number line.
This includes:

• Integers
• Fractions
• Decimals
• Square roots (√2, √5 etc.)

Positive numbers lie on the right side of 0, and negative numbers lie on the left side.

Example:
To represent 13/5, mark between 2 and 3 on the number line.

Important Standard Forms in Number System

Square of an Odd Number

The square of any odd number is always in the form: 8n + 1

Example:
5² = 25
25 = 8(3) + 1

Prime Numbers above 3

All prime numbers greater than 3 are in the form: 6n + 1 or 6n – 1

Example:
5 = 6(1) – 1
7 = 6(1) + 1
11 = 6(2) – 1

Hierarchy of Numbers

According to standard classification:

Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real

Meaning:

• Every Natural number is Whole
• Every Whole number is Integer
• Every Integer is Rational
• Every Rational number is Real

But the reverse is not always true.

Frequently Asked MCQs (Exam Based)

Here are some solved MCQs

Q1. What does each point on a number line represent?

Real Number

Q2. Which is NOT a pair of twin primes?

(131, 133)

Q3. Which of the following is irrational?

√169 – √196 (as explained in material)

Q4. How many prime numbers are between 1 and 100?

25

Q5. 1 is:

Neither prime nor composite

Q6. Zero (0) is:

An even number

Q7. The square of an odd integer is of the form:

8n + 1