medium difficulty7 min

Proof of irrationality of sqrt(3)

Assume √3 = p/q in lowest terms. After squaring, 3q² = p², so p is divisible by 3. That then forces q to be divisible by 3, which contradicts lowest terms.

In one line: For √3, the prime 3 repeats its way into both sides.

Practice This Concept

Understand it

The proof uses contradiction just like √2, but the prime number is 3 here. If both numerator and denominator become divisible by 3, the original fraction cannot have been in lowest terms, so the assumption fails.

Example

Write p = 3k after showing p is divisible by 3, then continue until q is also shown to be divisible by 3.

Formula and rule

Assume √3 = p/q in lowest terms, then 3q² = p².

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Let √3 = p/q in lowest terms.

1Then p² = 3q², so p is divisible by 3.
2Write p = 3k.
3Substituting gives 9k² = 3q², so q² = 3k² and q is also divisible by 3.
4That contradicts lowest terms.

Method you can reuse

1Assume √3 = p/q in lowest terms.
2Square both sides.
3Show p is divisible by 3.
4Substitute p = 3k.
5Show q is divisible by 3.
6Reach the contradiction.

Sketch it in your answer

Use the same proof ladder as √2, but replace 2 with 3 in every divisibility step.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What contradiction appears when proving √3 irrational by the fraction method?

Think first, then show answer

Both p and q turn out divisible by 3, so the fraction cannot have been in lowest terms.

Common wrong answer

Claiming that only p being divisible by 3 is enough is wrong; the proof needs the contradiction that both p and q share the factor 3.

Don't mix this up

Students repeat the √2 proof without changing the prime from 2 to 3.

Exam tip

Mention the prime 3 clearly at the divisibility step so the logic stays correct.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Prove √3 is irrational using contradiction, or identify the divisibility step in the proof.

1
Enough for 1 mark
Assume √3 = p/q in lowest terms. After squaring, 3q² = p², so p is divisible by 3. That then forces q to be divisible by 3, which contradicts lowest terms.
2
Add for 2 marks
Example: Write p = 3k after showing p is divisible by 3, then continue until q is also shown to be divisible by 3.
3
Add for 3 marks
The proof uses contradiction just like √2, but the prime number is 3 here.

MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results