Prove √3 - 1 is irrational Step by Step

Step by step

The question

Question: prove that √3 − 1 is irrational.

1 Assume the opposite

Rational means: it can be written as a fraction of whole numbers.

rational = a fraction ab of whole numbers

Suppose the opposite: √3 − 1 is rational, √3 − 1 = a/b.

√3 − 1 = ab

We already know that √3 is irrational.

We proved it.

we know: √3 is irrational

Careful!

A calculator gives a decimal that stops.

That is only a rounded value.

It cannot prove anything about the exact number.

Careful: not √3 − 1 = 0.7321 ✗

2 Work it out

Add 1 on both sides.

√3 = ab + 1

Same bottom: 1 is b over b.

√3 = a + bb

The top and the bottom are whole numbers.

So this is a fraction of whole numbers.

a + b and b are whole numbers

3 Find the contradiction

Then √3 would be rational.

But we know it is irrational!

so √3 would be rational ✗

The opposite led to nonsense, so √3 − 1 is irrational.

So √3 − 1 is irrational. ∎

4 Check

A calculator: about 0.73205081, the digits going on with no pattern, like √3's.

Right!

√3 − 1 ≈ 0.73205081...

Let's look back.

First: assume the opposite.

Then: work it out.

Then: find the contradiction.

Last: check.

Answer