Step by step
The question
Question: prove that the difference of two odd numbers is even.
1 Write it with letters
x is odd, so x = 2a + 1, where a is a whole number.
That is what odd means.
y is odd, so y = 2b + 1, where b is a whole number.
A new letter, b: y can be a different number from x.
Careful!
The same letter would make x and y the same number.
Two numbers need two letters.
2 Put them in
Now put them in: x − y becomes (2a + 1) − (2b + 1).
3 Work it out
Open the brackets: 2a + 1 − 2b − 1.
Like terms together: 2a − 2b.
4 Show the form
Every term has a 2 in it.
We take out 2: 2(a − b).
a − b is a whole number, because whole numbers added and multiplied stay whole.
2 times a whole number: so x − y is even.
The square at the end means: the proof is done.
5 Check
An example: x = 3 and y = 5.
It gives -2.
-2 is 2 times -1: even, as the proof says.
Right!
Let's look back.
First: write it with letters.
Then: put them in.
Then: work it out.
Then: show the form.
Last: check.