Prove 5√3 is irrational

Step by step

The question

Question: prove that 5√3 is irrational.

1 Assume the opposite

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

Suppose the opposite: 5√3 is rational, 5√3 = a/b.

We already know that √3 is irrational.

We proved it.

Careful!

A calculator gives a decimal that stops.

That is only a rounded value.

It cannot prove anything about the exact number.

2 Work it out

Divide both sides by 5: √3 = a/(5b).

a and 5b are whole numbers.

So this is a fraction of whole numbers.

3 Find the contradiction

Then √3 would be rational.

But we know it is irrational!

The opposite led to nonsense, so 5√3 is irrational.

4 Check

A calculator: about 8.66025404, the digits going on with no pattern, like √3's.

Right!

Let's look back.

First: assume the opposite.

Then: work it out.

Then: find the contradiction.

Last: check.