Knowledge organisers

Number System: CBSE Class 9 Maths knowledge organiser

Everything to know about number system on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Rational number: A number that can be written as p/q, where p and q are integers and q ≠ 0.
Irrational number: A real number whose decimal never ends and never repeats, such as √2 or π.
Rationalising: Rewriting a fraction so there is no surd in the denominator.

Key formulas

\(\mathbb N\subset\mathbb W\subset\mathbb Z\subset\mathbb Q\subset\mathbb R\); \(\frac pq\) terminates iff \(q\) (lowest terms) has only the primes 2, 5
Recurring decimal, e.g.	\(0.\overline{47}=\frac{47}{99},\) \(0.1\overline6=\frac{15}{90}=\frac16\)
Surds	\(\sqrt{ab}=\sqrt a\sqrt b,\) \((\sqrt a+\sqrt b)(\sqrt a-\sqrt b)=a-b\)
Rationalise	\(\frac1{\sqrt a+\sqrt b}=\frac{\sqrt a-\sqrt b}{a-b}\)
Exponents	\(a^ma^n=a^{m+n},\) \((a^m)^n=a^{mn},\) \(a^{-m}=\frac1{a^m},\) \(a^{\frac pq}=(\sqrt[q]a)^p\)

Worked example

Write \(2.\overline{3}\) as a fraction.

\(x = 2.333\ldots\), \(10x - x = 21\), so \(x = \dfrac{21}{9} = \dfrac73\).

Common mistakes

\(\sqrt{a + b} = \sqrt a + \sqrt b\) (false: \(\sqrt{9 + 16} = 5 \neq 7\)).
Calling a value used for \(\pi\) in a question (such as a fraction) equal to \(\pi\): it is only a rational approximation; \(\pi\) is irrational.
\(a^{-m} = -a^m\) (it is \(\dfrac1{a^m}\)), and \(a^m \times a^n = a^{mn}\) (it is \(a^{m+n}\)).
In \(0.1\overline{6}\), treating the whole of \(16\) as repeating.

You should be able to…

Classify numbers, spot terminating decimals, convert recurring decimals and simplify simple surds and powers.
Rationalise denominators, simplify expressions with exponents and find rationals between two numbers.
Prove a number is irrational and work through surd and exponent problems of several parts.
Handle conjugate surds, exponent equations and mixed reasoning about rational and irrational numbers.

The printable sheet

Number System knowledge organiser for CBSE Class 9 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Number System knowledge organiser (CBSE Class 9 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 9 Maths topics: Introduction to Polynomials · Sequences and Progressions · Exploring Algebraic Identities · Linear Equations in Two Variables · Coordinate Geometry · Introduction to Euclid's Geometry: Axioms and Postulates · Lines and Angles · Triangles – Congruence Theorems · 4-gons (Quadrilaterals) · Circles · Area and Perimeter · Surface Area and Volume · Statistics · Introduction to Probability · All CBSE Class 9 Maths organisers