Step by step
The question
Question: prove that √5 + 2 is irrational.
1 Assume the opposite
Rational means: it can be written as a fraction of whole numbers.
Suppose the opposite: √5 + 2 is rational, √5 + 2 = a/b.
We already know that √5 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
Take away 2 on both sides.
Same bottom: 2 is 2b over b.
The top and the bottom are whole numbers.
So this is a fraction of whole numbers.
3 Find the contradiction
Then √5 would be rational.
But we know it is irrational!
The opposite led to nonsense, so √5 + 2 is irrational.
4 Check
A calculator: about 4.23606798, the digits going on with no pattern, like √5's.
Right!
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.