Class 9 Maths Chapter 1: Number Systems

Number Systems is Chapter 1 of NCERT Class 9 Maths. It builds on the natural numbers, whole numbers and integers you already know and extends them to rational numbers, irrational numbers and the complete set of real numbers. These notes explain each idea simply, list the key results and laws of exponents, and walk you through fresh solved examples for CBSE exam practice.

What you will learn

• How natural numbers, whole numbers, integers, rational numbers and irrational numbers fit inside the real numbers
• How to tell from the decimal expansion whether a number is rational or irrational
• How to convert a recurring decimal into the form p/q
• How to add, multiply and rationalise the denominator of expressions with square roots
• The laws of exponents for real numbers, including rational powers

Key Concepts and Formulas

The Family of Numbers

Every new set of numbers in this chapter contains the earlier ones. Think of them as boxes placed one inside another, with the real numbers as the biggest box.

Natural numbers N = {1, 2, 3, 4, ...}
Whole numbers   W = {0, 1, 2, 3, ...}
Integers        Z = {..., −2, −1, 0, 1, 2, ...}
Rational        Q = p/q, p and q integers, q ≠ 0
N ⊂ W ⊂ Z ⊂ Q ⊂ R

• Whole numbers are the natural numbers together with zero.
• Between any two rational numbers there are infinitely many rational numbers.

Irrational Numbers and Real Numbers

A number that cannot be written in the form p/q is called irrational. The Pythagoreans in ancient Greece were the first to realise that such numbers exist.

• Examples of irrational numbers: √2, √3, √5, √11, π, and 0.1010010001...
• Real numbers (R): all rational and all irrational numbers taken together
• Every real number corresponds to a unique point on the number line, and every point on the number line represents a unique real number.
• If n is a natural number that is not a perfect square, then √n is irrational.
• Note that 22/7 is only an approximation of π; π itself is irrational.

Decimal Expansions

Looking at the decimal form is the quickest way to classify a number.

• Terminating decimal (for example 0.625): the number is rational.
• Non-terminating recurring decimal (for example 0.454545...): the number is rational.
• Non-terminating non-recurring decimal (for example 1.41421356...): the number is irrational.
• To convert a recurring decimal to p/q, let x equal the decimal, multiply by 10, 100 or 1000 (depending on how many digits repeat), and subtract to remove the repeating part.

Operations on Real Numbers

Rational and irrational numbers behave differently when you combine them. The identities below hold for positive real numbers a and b.

√(ab) = √a × √b
√(a/b) = √a ÷ √b
(√a + √b)(√a − √b) = a − b
(a + √b)(a − √b) = a² − b
(√a + √b)² = a + 2√(ab) + b

• Rational + irrational is always irrational; a non-zero rational × irrational is also irrational.
• Irrational + irrational or irrational × irrational can be rational or irrational (for example √3 × √3 = 3).

Rationalising the Denominator

When the denominator contains a square root, we multiply the numerator and denominator by a suitable expression so that the denominator becomes rational. This expression is usually the conjugate of the denominator.

For 1/√a        multiply by √a/√a
For 1/(a + √b)  multiply by (a − √b)/(a − √b)
For 1/(√a − √b) multiply by (√a + √b)/(√a + √b)

Laws of Exponents for Real Numbers

For a real number a > 0 and rational numbers p and q, the following laws hold. Here a^p means "a raised to the power p".

a^p × a^q = a^(p + q)
(a^p)^q = a^(pq)
a^p ÷ a^q = a^(p − q)
a^p × b^p = (ab)^p
a^0 = 1
a^(−p) = 1/a^p
a^(1/n) = ⁿ√a
a^(m/n) = (ⁿ√a)^m

Solved Examples

Example 1: Express 0.272727... (27 repeating) in the form p/q.

Let x = 0.272727...
100x = 27.272727...
100x − x = 27
99x = 27
x = 27/99
∴ x = 3/11

Example 2: Express 1.3333... in the form p/q.

Let x = 1.3333...
10x = 13.3333...
10x − x = 13.3333... − 1.3333...
9x = 12
∴ x = 12/9 = 4/3

Example 3: Rationalise the denominator of 1/(√5 − √3).

1/(√5 − √3) × (√5 + √3)/(√5 + √3)
= (√5 + √3)/((√5)² − (√3)²)
= (√5 + √3)/(5 − 3)
∴ Answer = (√5 + √3)/2

Example 4: Simplify (√7 + √2)(√7 − √2).

(√a + √b)(√a − √b) = a − b
(√7 + √2)(√7 − √2) = 7 − 2
∴ Answer = 5, which is rational

Example 5: Find the value of 125^(2/3).

125 = 5³
125^(2/3) = (5³)^(2/3)
= 5^(3 × 2/3)
= 5²
∴ 125^(2/3) = 25

Example 6: Simplify 2^(1/2) × 2^(3/2).

2^(1/2) × 2^(3/2) = 2^(1/2 + 3/2)
= 2^2
∴ Answer = 4

Example 7: Find two irrational numbers between 0.2 and 0.3.

Choose decimals that never end and never repeat:
0.2101001000100001...
0.2202002000200002...
∴ Both lie between 0.2 and 0.3 and are irrational

Common Mistakes to Avoid

• Do not write √(a + b) = √a + √b. For example √(9 + 16) = 5, but √9 + √16 = 7.
• 3.14 and 22/7 are rational approximations of π; never call π rational in an answer.
• While converting recurring decimals, multiply by 10 for one repeating digit, 100 for two, and 1000 for three. Choosing the wrong power is the most common error.
• After rationalising, always simplify the final answer and cancel common factors; examiners deduct marks for unsimplified answers.

Quick Revision

N ⊂ W ⊂ Z ⊂ Q ⊂ R
(a + √b)(a − √b) = a² − b
(√a + √b)(√a − √b) = a − b
a^(m/n) = (ⁿ√a)^m
a^(−p) = 1/a^p

• Rational numbers have terminating or non-terminating recurring decimals.
• Irrational numbers have non-terminating non-recurring decimals.
• Use the conjugate of the denominator to rationalise.

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