Class 10 Real Numbers: The Short Chapter That Tests Reasoning

A small chapter with a particular style

Real Numbers is the first chapter in Class 10 Maths and among the smallest. Children treat it as a warm-up and move on. But it is where the paper’s style first appears: a proof, a fixed answer format, and questions where how you wrote it matters as much as the final answer. A student who learns this style early finds the rest of the year easier.

Euclid’s division lemma

The lemma says that for positive integers a and b there exist unique integers q and r with a = bq + r and 0 ≤ r < b. The algorithm uses it repeatedly to find the HCF of two numbers.

1. Divide the larger by the smaller and write a = bq + r.
2. If r is zero, b is the HCF. If not, divide b by r and repeat.
3. Write each line in the a = bq + r form. Marks are given for steps, not only the final HCF.

The same lemma is used to prove statements such as: any positive odd integer is of the form 4q + 1 or 4q + 3. These proofs follow one pattern. Take b equal to 4, list the possible remainders, and show which are even and which are odd. Learn one such proof thoroughly and the others follow.

The Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the factors. Its practical use is finding HCF and LCM by prime factorisation.

• HCF is the product of the smallest power of each common prime factor.
• LCM is the product of the greatest power of each prime factor that appears.
• For two numbers, HCF × LCM equals the product of the numbers. Use this to check your answer, or to find a missing value.

The commonest error is mixing up “smallest power” and “greatest power”. Write both rules at the top of the page when practising. Another form of question asks why a number like 7 × 11 × 13 + 13 is composite; the answer is to take 13 common and show the factors. This shows that the student understands the theorem, not just the procedure.

Irrational numbers and the proof template

The proof that the square root of 2 is irrational is the prototype, and similar proofs are asked for the square roots of 3 or 5, or numbers like 3 + 2√5. The structure never changes:

1. Assume, to the contrary, that the number is rational and equal to p/q, where p and q are co-prime integers and q is not zero.
2. Square and rearrange to show that a prime divides p squared, hence divides p.
3. Substitute p equal to a multiple of that prime and show that it divides q as well.
4. State the contradiction: p and q have a common factor, but they were co-prime. Therefore the assumption is false.

Many students lose marks by dropping the “co-prime” assumption or by not stating the contradiction. Both are explicit marking points. For numbers like 3 + 2√5, first show that the root part is irrational using the standard proof, then argue that a rational number combined with an irrational one cannot give a rational result.

Decimal expansions of rationals

The chapter closes with a rule: a rational number p/q in lowest terms has a terminating decimal expansion if the prime factorisation of q has only 2s and 5s, and a non-terminating repeating expansion otherwise. Questions ask the student to decide without dividing. Reduce the fraction to lowest terms first; students forget this and give a wrong verdict.

Where the marks usually are

The chapter commonly offers a mix of short objective items and one or two longer questions, but weightage and format change with the paper pattern, so check the latest sample paper from the board for the current design. Whatever the pattern, the proof template and the HCF and LCM method are the dependable parts.

A three-day plan

• Day 1: Euclid’s algorithm on six pairs of numbers; the 4q + 1 / 4q + 3 proof.
• Day 2: prime factorisation, HCF, LCM, and the product check.
• Day 3: the irrationality proof written three times without looking, then decimal expansion questions.

For Dwarka and Gurugram families

Students in both Standard and Basic Maths need this chapter. If the child is coming from a weaker Class 9 base, a few early home sessions in this chapter set the habit of writing full steps, which pays off in the proofs of geometry later in the year.

Questions parents ask

Is the chapter really that easy?

The ideas are modest; the writing standard is not. Marks go to the layout and the reasoning stated clearly.

Can the proof be learnt by heart?

It can, and many children do, but a slightly changed number then confuses them. Practise with different numbers so the structure becomes the memory.

Is this chapter the same for ICSE?

ICSE Class 10 Maths has its own sequence of topics and does not follow the CBSE chapter plan, so use the school syllabus to see how much of this applies.