Prove if n^2 is even, then n is even Step by Step

Step by step

The question

Question: prove that if n² is even, then n is even.

1 Turn it around

An if-then statement means the same as its contrapositive: if not Q, then not P.

We saw it with truth tables.

if P, then Q ≡ if not Q, then not P

Turned around, it says: if n is odd, then n² is odd.

Starting from n is much easier: we know how to write it with letters.

turned around: if n is odd, then n² is odd

Careful!

That swaps the two parts.

It is the converse, a different statement.

Turned around means: swap the parts and put not on both.

Careful: not if n is even, then n² is even ✗

2 Write it with letters

n is odd, so n = 2a + 1, where a is a whole number.

n = 2a + 1

That is what odd means.

odd = 2 × a whole number + 1

3 Put them in

Now put them in: n² becomes (2a + 1)².

n² = (2a + 1)²

4 Work it out

(2a + 1)² means the bracket times itself.

= (2a + 1)(2a + 1)

Every part times every part: 4a² + 2a + 2a + 1.

= 4a² + 2a + 2a + 1

Like terms together: 4a² + 4a + 1.

= 4a² + 4a + 1

5 Show the form

We take out 2 from all but the 1: 2(2a² + 2a) + 1.

= 2(2a² + 2a) + 1

2a² + 2a is a whole number, because whole numbers added and multiplied stay whole.

2a² + 2a is a whole number

2 times a whole number, plus 1: that is the form of an odd number.

So n² is odd.

The turned statement is true, so the first one is true too: if n² is even, then n is even.

The square at the end means: the proof is done.

So: if n² is even, then n is even. ∎

6 Check

Try n = 2.

Then n² is 4: even.

n = 2: n² = 4, even

And 2 is even, as the statement says.

Right!

and 2 is even

Let's look back.

First: turn it around.

Then: write it with letters.

Then: put them in.

Then: work it out.

Then: show the form.

Last: check.

Answer