Step by step
The question
Question: prove that √2 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: √2 is rational.
Then √2 = 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: √2 squared is 2, and a/b squared is a squared over b squared.
Times b squared on both sides: a² = 2b².
2 times a whole number: so a squared is even.
And when a squared is even, a is even.
We proved that by contrapositive.
So a is even too: a = 2c, with c a whole number.
Put it into a squared = 2 b squared.
2c squared is 4c squared.
Divide both sides by 2: b² = 2c².
The same as before: b squared is even, so b is even.
3 Find the contradiction
But now a and b are both even.
2 is a common factor.
That breaks lowest terms!
The opposite led to nonsense, so it is false.
√2 is irrational.
4 Check
A calculator: √2 is about 1.41421356, and the digits go on with no repeating pattern.
Even a good fraction like 17/12 squares to about 2.007, close to 2 but not 2.
Right!
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.