Step by step
The question
Question: prove that √5 is irrational.
1 Assume the opposite
Rational means: it can be written as a fraction of whole numbers.
Irrational means it cannot.
Suppose the opposite: √5 is rational.
Then √5 = a/b.
We cancel the fraction as far as it goes: a and b have no common factor.
Careful!
Without lowest terms there is nothing to break.
Remember it: a and b have no common factor.
2 Work it out
Square both sides: √5 squared is 5, and a/b squared is a squared over b squared.
Times b squared on both sides: a² = 5b².
5 times a whole number: so a squared is a multiple of 5.
5 is a prime number.
When a prime goes into a times a, it goes into a.
So a is a multiple of 5 too: a = 5c, with c a whole number.
Put it into a squared = 5 b squared.
5c squared is 25c squared.
Divide both sides by 5: b² = 5c².
The same as before: b squared is a multiple of 5, so b is a multiple of 5.
3 Find the contradiction
But now a and b are both a multiple of 5.
5 is a common factor.
That breaks lowest terms!
The opposite led to nonsense, so it is false.
√5 is irrational.
4 Check
A calculator: √5 is about 2.23606798, and the digits go on with no repeating pattern.
Even a good fraction like 20/9 squares to about 4.938, close to 5 but not 5.
Right!
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.