medium difficulty7 min

Proof of irrationality of sqrt(5)

Assume √5 = p/q in lowest terms. After squaring, 5q² = p², so p is divisible by 5. That then forces q to be divisible by 5, which contradicts lowest terms.

In one line: Prime 5 locks both ends of the fraction.

Practice This Concept

Understand it

This proof follows the same contradiction pattern as the proofs for √2 and √3. The only change is the prime number 5. Once both p and q are shown to share factor 5, the assumption of lowest terms collapses.

Example

If p is divisible by 5, write p = 5k and continue until q is also shown to be divisible by 5.

Formula and rule

Assume √5 = p/q in lowest terms, then 5q² = p².

See it worked out

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

Worked example

Let √5 = p/q.

1Then p² = 5q², so p is a multiple of 5.
2Write p = 5k.
3Substituting gives 25k² = 5q², so q² = 5k².
4Therefore q is also a multiple of 5.

Method you can reuse

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

Visual tip

A short contradiction flowchart is enough: assumption, square, divisibility by 5, contradiction.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What does the proof show when √5 = p/q is assumed in lowest terms?

Think first, then show answer

It shows both p and q are divisible by 5, so the assumption cannot be correct.

Common wrong answer

Stopping after finding p divisible by 5 is incomplete because the contradiction comes from both numbers sharing the factor 5.

Don't mix this up

Students copy the earlier proofs but forget to replace the prime with 5 in every divisibility step.

Exam tip

Keep the same proof steps and only change the prime factor to 5.

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 √5 is irrational, or identify the divisibility contradiction in the proof.

1
Enough for 1 mark
Assume √5 = p/q in lowest terms. After squaring, 5q² = p², so p is divisible by 5. That then forces q to be divisible by 5, which contradicts lowest terms.
2
Add for 2 marks
Example: If p is divisible by 5, write p = 5k and continue until q is also shown to be divisible by 5.
3
Add for 3 marks
This proof follows the same contradiction pattern as the proofs for √2 and √3.

MCQ Quiz

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