Prove √2 + 3 is irrational Step by Step

Step by step

The question

Question: prove that √2 + 3 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: √2 + 3 is rational, √2 + 3 = a/b.

√2 + 3 = ab

We already know that √2 is irrational.

We proved it.

we know: √2 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 √2 + 3 = 4.4142 ✗

2 Work it out

Take away 3 on both sides.

√2 = ab − 3

Same bottom: 3 is 3b over b.

√2 = a − 3bb

The top and the bottom are whole numbers.

So this is a fraction of whole numbers.

a − 3b and b are whole numbers

3 Find the contradiction

Then √2 would be rational.

But we know it is irrational!

so √2 would be rational ✗

The opposite led to nonsense, so √2 + 3 is irrational.

So √2 + 3 is irrational. ∎

4 Check

A calculator: about 4.41421356, the digits going on with no pattern, like √2's.

Right!

√2 + 3 ≈ 4.41421356...

Let's look back.

First: assume the opposite.

Then: work it out.

Then: find the contradiction.

Last: check.

Answer