Knowledge organisers

Real Numbers: CBSE Class 10 Maths knowledge organiser

Everything to know about real numbers 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

Prime factorisation: Writing a number as a product of primes; it is unique apart from the order.
HCF: The highest common factor: the largest number that divides each of the numbers exactly.
LCM: The lowest common multiple: the smallest number that each of the numbers divides exactly.

Key formulas

Fundamental Theorem of Arithmetic: every composite number is a product of primes, uniquely apart from order
HCF = product of the smallest power of each common prime; LCM = product of the greatest power of each prime
For two positive integers	\(\text{HCF}(a,b)\times\text{LCM}(a,b)=a\times b\)
\(\sqrt2,\sqrt3,\sqrt5\) and \(\sqrt p\) (\(p\) prime) are irrational: prove by contradiction (\(p\mid a^2\Rightarrow p\mid a\))

Worked example

Find HCF and LCM of \(96\) and \(360\).
\(96 = 2^5 \times 3\), \(360 = 2^3 \times 3^2 \times 5\). HCF \(= 2^3 \times 3 = 24\); LCM \(= 2^5 \times 3^2 \times 5 = 1440\). Check: \(24 \times 1440 = 34560 = 96 \times 360\).

Common mistakes

Using \(\text{HCF} \times \text{LCM} = \text{product}\) for three numbers.
Taking the highest power for HCF (or the lowest for LCM).
In irrationality proofs, forgetting to state that \(a\) and \(b\) are coprime, or not stating the contradiction clearly.
Leaving a “prime factorisation” with a composite factor such as \(77\) or \(143\).

You should be able to…

Write any number as a product of primes and find the HCF and LCM of two numbers quickly and accurately.
Handle every standard board question: HCF/LCM word problems, the HCF × LCM rule, and the short 'show that it is irrational' proofs.
Write complete 5-mark and case-study answers, with the prime factorisation shown, a clear final sentence and units, so no step mark slips away.
Crack the trickiest questions: finding unknown powers, listing all possible number pairs and spotting which sums and products of surds stay irrational.

The printable sheet

Real Numbers knowledge organiser for CBSE Class 10 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Real Numbers knowledge organiser (CBSE Class 10 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 10 Maths topics: Polynomials · Pair of Linear Equations in Two Variables · Quadratic Equations · Arithmetic Progressions · Triangles · Coordinate Geometry · Introduction to Trigonometry · Some Applications of Trigonometry · Circles · Areas Related to Circles · Surface Areas and Volumes · Statistics · Probability · All CBSE Class 10 Maths organisers