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Class 9 Maths Chapter 1: Number System Full Revision

This lecture covers the fundamentals of the number system for Class 9 Maths according to NCERT. It explains real numbers, classification of rational and irrational numbers, decimal expansions, locating irrationals on a number line, laws of exponents, and rationalisation.

Class 9
Maths
English · 8 topics
Checked against the video on 25 Oct 2026

Made by AI from the YouTube video Number System Class 9 in One Shot 🔥 | Class 9 Maths Chapter 1 Complete Lecture | Shobhit Nirwan (opens YouTube in a new tab) by Shobhit Nirwan - 9th, then checked against it. All credit for the teaching goes to the creator. Tap a topic’s time to confirm a fact in the video.

Class 9 Maths Chapter 1: Number System Full Revision

English8 topics · 5 MCQs · 12 flashcardsWatch the video (opens YouTube in a new tab)

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In short

This lecture covers the fundamentals of the number system for Class 9 Maths according to NCERT. It explains real numbers, classification of rational and irrational numbers, decimal expansions, locating irrationals on a number line, laws of exponents, and rationalisation.

Classification of Real Numbers

Real numbers are divided into two main categories: rational numbers and irrational numbers.
Rational numbers are further classified into integers and non-integers.
Must knowIntegers consist of positive integers, negative integers, and zero; zero is neither positive nor negative.
Positive integers are also known as natural numbers, denoted by N, which start from 1, 2, 3, 4, 5 and so on.
When zero is included with natural numbers, they form whole numbers, represented by W.

Number Type	Symbol	Examples
Natural Numbers	N	1, 2, 3, 4, ...
Whole Numbers	W	0, 1, 2, 3, ...
Integers	Z or I	..., -2, -1, 0, 1, 2, ...

Rational and Irrational Numbers

Must knowA rational number Q is defined as a number in the form p/q, where p and q are integers and q is not equal to zero.
Decimal expansions that either terminate or repeat are classified as rational numbers.
Decimal expansions that are non-repeating and non-terminating (NRNT) are irrational numbers.
Must knowThe value 22/7 is a rational number, whereas π is an irrational number; 22/7 is only an approximation and not the exact value of π.
A true/false analysis shows that every natural number is a whole number, but not every whole number is a natural number because of zero.
Every integer can be written in the form p/1, making every integer a rational number.

Inserting Numbers Between Two Numbers

To find a single rational number between two rational numbers a and b, use the formula (a + b) / 2.
To insert multiple rational numbers between two rational numbers, multiply the numerator and denominator by a suitable factor to make the denominators identical.
For example, between 1 and 2, writing them as 100/100 and 200/100 allows choosing up to 99 rational numbers such as 101/100, 102/100, etc.
An irrational number between two positive rational numbers a and b can be found using the formula √(ab).
Alternatively, irrational numbers between two numbers can be written directly by creating non-terminating, non-repeating decimal expansions.

Converting Repeating Decimals to p/q Form

To convert a repeating decimal into p/q form, set x equal to the number where the bar is immediately after the decimal point (Equation 1).
Must knowIf one digit is under the bar, multiply both sides by 10; if two digits repeat, multiply by 100; if three digits repeat, multiply by 1000.
Subtracting Equation 1 from Equation 2 cancels out the repeating decimal portion, leaving a simple linear equation to solve for x.
For 0.3 bar, multiplying x = 0.333... by 10 gives 10x = 3.333..., and subtracting gives 9x = 3, so x = 1/3.
If non-repeating digits exist between the decimal point and the bar (like 0.237 with bar on 37), first multiply by 10 to shift the bar right next to the decimal point to form Equation 1.

Properties of Irrational Numbers

Square roots of positive integers that are not perfect squares (such as √2, √3, √5) are irrational.
Cube roots of positive integers that are not perfect cubes are also irrational.
The negative of an irrational number is always an irrational number.
Must knowThe sum, difference, product, and quotient of a non-zero rational number and an irrational number is always irrational.
The sum, difference, product, or quotient of two irrational numbers can be either rational or irrational (cannot say generally).

Representing Irrationals on the Number Line

Pythagoras' theorem states that hypotenuse H equals √(Base² + Perpendicular²).
To construct √2 on a number line, take a base of 1 unit and a perpendicular of 1 unit, giving H = √(1² + 1²) = √2.
An arc drawn from 0 with radius equal to the hypotenuse projects √2 down onto the number line.
To construct √3, use √2 as the base and 1 as the perpendicular, since (√2)² + 1² = 2 + 1 = 3.
To construct √50, take a base of 7 (since √49 = 7) and a perpendicular of 1, because 7² + 1² = 50.

Laws of Exponents and Algebraic Simplification

Must knowWhen multiplying terms with the same base, powers are added: a^m × aⁿ = a^m+n.
When dividing terms with the same base, powers are subtracted: a^m / aⁿ = a^m-n.
A power raised to another power multiplies the exponents: (a^m)ⁿ = a^mn.
Distinct bases with the same exponent combine under multiplication: a^m × b^m = (ab)^m.
Negative exponent rule states that a⁻ⁿ = 1 / aⁿ, and any non-zero number to the power 0 equals 1 (a⁰ = 1).

Rationalisation of Denominators

Rationalisation removes irrational roots from the denominator by multiplying both numerator and denominator by an appropriate conjugate or factor.
For a single term 1/√7, multiply and divide by √7 to get √7 / 7.
For a binomial denominator like 1 / (√7 - √6), multiply numerator and denominator by the conjugate (√7 + √6).
Must knowThe denominator simplifies using the identity (a - b)(a + b) = a² - b², removing the radicals.
For 1 / (√5 + √2), multiplying by (√5 - √2) / (√5 - √2) results in (√5 - √2) / (5 - 2) = (√5 - √2) / 3.

Key terms

Rational Number: A number that can be expressed as p/q where p and q are integers and q is not zero
Irrational Number: A number that cannot be written as p/q, having a non-terminating and non-repeating decimal expansion
Conjugate: An expression formed by changing the sign between two terms, used to rationalise binomial denominators
Whole Numbers: The set of natural numbers including zero (0, 1, 2, 3, ...)

Formulas

Hypotenuse in right triangle: H = √(B² + P²)
Rational number between a and b: (a + b) / 2
Product of powers with same base: a^m × aⁿ = a^m+n
Quotient of powers with same base: a^m / aⁿ = a^m-n
Power of a power: (a^m)ⁿ = a^mn
Negative exponent rule: a⁻ⁿ = 1 / aⁿ
Zero exponent rule: a⁰ = 1

Exam tips and traps

Make sure the bar of repeating digits is immediately after the decimal point before naming the equation as Equation 1.
Do not confuse 22/7 with π; 22/7 is rational because it is in p/q form, while π is irrational.

Quick revision

Read these the night before the exam.

Zero is an integer that is neither positive nor negative.
Every integer is a rational number because it can be written with denominator 1.
Non-terminating, non-repeating decimal expansions represent irrational numbers.
22/7 is a rational number, while π is an irrational number.
The product of a non-zero rational number and an irrational number is always irrational.
The product of two irrational numbers can be either rational or irrational.
The hypotenuse for locating √n on a number line uses √(n-1) as base and 1 as perpendicular.
Any non-zero number raised to the power 0 equals 1.
In rationalisation, multiply the numerator and denominator by the conjugate with the opposite middle sign.

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