Step by step
The question
Question: prove that the sum of an even number and an odd number is odd.
1 Write it with letters
x is even, so x = 2a, where a is a whole number.
That is what even means.
y is odd, so y = 2b + 1, where b is a whole number.
Careful!
An example shows one number.
The letters show every number at once.
2 Put them in
Now put them in: x + y becomes (2a) + (2b + 1).
3 Work it out
Open the brackets: 2a + 2b + 1.
4 Show the form
We take out 2 from all but the 1: 2(a + b) + 1.
a + b is a whole number, because whole numbers added and multiplied stay whole.
2 times a whole number, plus 1: that is the form of an odd number.
So x + y is odd.
The square at the end means: the proof is done.
5 Check
An example: x = 2 and y = 5.
It gives 7.
7 is 2 times 3, plus 1: odd, 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.