Irrational Numbers Class 10: Meaning, Proofs & Exam Guide (Real Numbers 1.3)

If the words “prove that √2 is irrational” make you nervous, you’re in exactly the right place. Irrational numbers class 10 (Section 1.3 of Real Numbers, “Revisiting Irrational Numbers”) looks scary because it’s one of the first places you’re asked to prove something instead of just calculate it. The good news: there are really only three ideas here, and once they click, every question in this section starts to look the same. Let’s make it easy.

By the end of this guide you’ll know what an irrational number actually is, the one tool that unlocks all the proofs, exactly how the √2 proof works in plain English, and the common traps that cost students easy marks. If you want the bird’s-eye view of the whole chapter first, start with our Real Numbers Class 10 chapter guide and then come back here for the deep dive on 1.3.

What Section 1.3 is really about

In Class 9 you met irrational numbers and even placed them on the number line — but you never proved they were irrational. Section 1.3 — Revisiting Irrational Numbers — fixes that, and it’s the part of the irrational numbers topic in NCERT Class 10 that trips most students up. It uses one powerful idea from earlier in the chapter (the Fundamental Theorem of Arithmetic) to prove that numbers like √2, √3 and √5 genuinely cannot be written as neat fractions.

So this section is proof-heavy, not calculation-heavy. That’s actually great news: proofs follow a fixed recipe. Learn the recipe once and you can reproduce it for √2, √3, √5, or a surd you’ve never seen before.

Idea 1 — What makes a number “irrational”

A quick refresher, because every proof depends on this definition.

• A rational number can be written as a fraction p/q, where p and q are integers and q ≠ 0 (e.g. 3, −7, ½, 0.75, 0.333…).
• An irrational number cannot be written that way. Its decimal expansion is non-terminating and non-repeating — it never stops and never settles into a repeating block.

Exam-ready shortcut: √(a perfect square) is rational (√9 = 3). √(not a perfect square) is irrational (√8, √12). Spotting this instantly saves time in MCQs — and you’ll meet plenty of them in our MCQ practice for Class 10 Maths.

Rational or irrational? Quick reference:

A fast way to test whether you’ve truly got this distinction is a short mixed “rational or irrational?” set — the kind of instant-feedback drilling you get from topic practice on examfront, which keeps serving them until the pattern becomes automatic.

🎬 Watch the 30-second reel: Can you find the irrational number?

Idea 2 — The one tool behind every proof

Before proving anything, Section 1.3 gives you a helper result. In plain words:

If a prime number p divides a², then that same prime p must divide a.

Why does this matter? Because it comes straight from the Fundamental Theorem of Arithmetic — the fact that every number has exactly one prime factorisation. When you square a number, you don’t create any new prime factors; you just double the ones already there. So if a prime shows up in a², it had to be hiding in a all along.

Quick sense-check: 6 = 2 × 3, and 6² = 36 = 2² × 3². The primes in 36 are still just 2 and 3 — nothing new appeared. That’s the whole idea. (Prime factorisation feels shaky? It’s worth a quick refresh from the Real Numbers chapter guide before moving on.)

You’ll use this result (with p = 2, then p = 3, then p = 5) in almost every proof in this section. Memorise it as your master key — this single result opens every “prove it’s irrational” proof in the section.

Idea 3 — Why √2 is irrational (the proof, in plain English)

This is the famous proof and a very common exam question, and it’s your first real taste of proof by contradiction in Class 10. The idea: assume the opposite of what you want, then follow the logic until it reaches a contradiction.

Why does proof by contradiction work? If assuming something leads to an impossible situation, then the assumption itself must have been false. So when we assume “√2 is rational” and it forces something impossible, the only way out is that √2 is not rational. That’s the whole engine — no magic.

Here’s the reasoning, step by step in everyday language (not a formula dump):

1. Assume the opposite. Suppose √2 is rational. Then we can write √2 = a/b, where a and b are integers with no common factor — the fraction is fully reduced, i.e. in lowest terms (you may also see this called coprime). CBSE uses all three phrasings for the same idea, so recognise them.
2. Square both sides. That gives 2 = a²/b², so a² = 2b². This tells us a² is even.
3. Use the master key. Since 2 (a prime) divides a², it must divide a. So a is even — write a = 2c.
4. Substitute back. Then (2c)² = 2b² → 4c² = 2b² → b² = 2c². So b² is even too, which means b is even.
5. Spot the contradiction. Now both a and b are even — they share a factor of 2. But in step 1 we said they had no common factor. That’s impossible.
6. Conclude. The only thing that could be wrong is our assumption. So √2 is not rational — it is irrational. ∎

Notice the shape: assume rational → reach “both share a factor” → contradiction → therefore irrational. That’s the reusable recipe.

The √2 proof at a glance

Assume √2 is rational
        ↓
Write √2 = a/b   (lowest terms)
        ↓
Square:  a² = 2b²
        ↓
a is even
        ↓
b is even
        ↓
Both divisible by 2
        ↓
Contradiction  (they had NO common factor)
        ↓
√2 is irrational

How to remember the order

Five beats, easy to recall under exam pressure:

A → S → E → E → C
Assume · Square · a Even · b Even · Contradiction

How to write it in the exam (board answer format)

Students always ask, “exactly how do I prove root 2 is irrational, Class 10 board style?” Here’s the clean layout examiners expect:

Prove that √2 is irrational.

Assume, to the contrary, that √2 is rational.
Then √2 = a/b, where a, b are integers, b ≠ 0,
and a, b are coprime (lowest terms).

Squaring:   2 = a²/b²
        ⇒   a² = 2b²                    … (1)
        ⇒   2 divides a²  ⇒  2 divides a
Let a = 2c.

Substitute in (1):
    (2c)² = 2b²  ⇒  4c² = 2b²  ⇒  b² = 2c²
        ⇒   2 divides b²  ⇒  2 divides b

So 2 divides both a and b.
But a and b are coprime — contradiction.

∴ our assumption is wrong.
Hence, √2 is irrational.

Write those lines in that order and you tick every box in the marking scheme.

Students rarely get √2 wrong — they lose marks by skipping the justification steps (like why a is even). Working through model answers with step-by-step feedback, the way examfront’s practice companion does it, trains you to write exactly the lines the examiner is looking for.

🎬 Watch the 30-second reel: √2 is irrational — where is the contradiction?

Same recipe: √3, √5 and √p

To prove √3 is irrational, run the identical recipe with the prime 3 instead of 2: assume √3 = a/b (coprime) → 3b² = a² → 3 divides a → a = 3c → b² = 3c² → 3 divides b → both share 3 → contradiction → √3 is irrational.

Swap in 5 and you’ve proved √5 is irrational. In general, √p is irrational for any prime p. If you can do √2, you can do all of them — only the number changes.

Because the method is identical and only the prime changes, this is a perfect topic for spaced revision. A quick chapter quiz — like the ones built into examfront — lets you rotate through √2, √3 and √5 so the recipe sticks under exam pressure. You’ll also spot these framed as reasoning questions in chapter-wise previous year questions.

🎬 Watch the 30-second reel: What is the product of √2 × √8?

Combinations: rational ± irrational, and rational × irrational

The second half of 1.3 tests whether you understand two facts:

• Rational ± irrational = irrational (e.g. 5 − √3, 2 + √7)
• Non-zero rational × irrational = irrational (e.g. 3√2, √5⁄2)

The proof trick is short and always the same: assume the whole expression is rational, then rearrange to isolate the irrational part.

Example — show 5 − √3 is irrational: Assume 5 − √3 is rational. Rearranging gives √3 = 5 − (that rational number). But “rational minus rational” is rational, which would make √3 rational — and we already proved it isn’t. Contradiction → 5 − √3 is irrational.

Example — show 3√2 is irrational: Assume 3√2 is rational. Then √2 = (that rational)/3, which is rational — contradicting that √2 is irrational. So 3√2 is irrational.

Same engine every time: isolate the surd, and the contradiction appears.

Mixing up these rules (“is 5 + √3 rational or irrational?”) is a classic slip. A tool that flags exactly this kind of rule-confusion — like the mistake identification on examfront — helps you fix it before the exam, not after.

🎬 Watch the 30-second reel: Evaluate (2 + √3) + (2 − √3)

How this section shows up in the exam

Section 1.3 is proof-based, so most “irrational numbers proof” Class 10 questions appear as short “prove that … is irrational” tasks and as reasoning-style MCQs (e.g. “Which of these is irrational?”). The three things examiners reward:

• Stating the assumption clearly (“Let us assume, to the contrary…”).
• Justifying each step, especially why a number is even/divisible.
• Naming the contradiction at the end and concluding.

Note for editors: exact marks/weightage for Real Numbers and this section should be confirmed against the official CBSE blueprint before publishing any figures — see the unit-wise weightage page. No weightage numbers were invented here.

Knowing your weak spots matters more than doing 50 random sums. A strength & weakness view — the kind examfront builds automatically — shows whether you’re losing marks on the definition, the proof steps, or the combination rules, so your revision targets the real gap. For focused practice, our important questions for Class 10 Maths collection is a good next stop.

Quick common traps (full breakdown in the mistakes section)

• Thinking √4 or √9 is irrational — perfect-square roots are rational.
• Writing √2 = a/b but forgetting to say a and b are coprime — the whole proof depends on it.
• Saying “a² is even so a² is divisible by 2, therefore… “ but skipping why a itself is even.
• Assuming irrational + irrational is always irrational (√2 + (−√2) = 0, which is rational!).

Fixing these four traps alone can lift your Real Numbers score. Track your progress on them over time with the progress dashboard on examfront so you can see the improvement week by week.

Once 1.3 feels solid, keep the momentum going into the next chapter with our Polynomials Class 10 guide, or grab every key result in one place from the Class 10 Maths formula sheet.

Frequently Asked Questions

What is an irrational number in Class 10? A real number that cannot be written as p/q (integers, q ≠ 0). Its decimal never terminates and never repeats. Examples: √2, √3, √5, π.

How do you prove √2 is irrational? By contradiction: assume √2 = a/b in lowest terms, show both a and b turn out even, which contradicts “lowest terms.” So √2 is irrational.

Is √4 irrational? No — √4 = 2 is a whole number, so it’s rational. Only roots of non-perfect-squares are irrational.

Why is 5 − √3 irrational? If it were rational, rearranging would force √3 to be rational, which is false. So 5 − √3 is irrational.

Which theorem is used to prove roots are irrational? The result “if a prime p divides a², then p divides a,” which comes from the Fundamental Theorem of Arithmetic.

Related Resources

• Real Numbers Class 10 — full chapter guide
• Class 10 Maths — all formulas
• CBSE Class 10 Maths — previous year questions
• Chapter-wise PYQs — Class 10 Maths
• Class 10 Maths — full chapter list

Ready to make Section 1.3 automatic?

You’ve got the recipe — now lock it in. Try a short set of Real Numbers proofs and reasoning MCQs, and let examfront point you to your exact weak spot so your revision actually moves the needle.

Start practising Real Numbers on examfront →