Proving Two Things Are Equal
If asked to do this in an exam it might be as simple as rearranging one side of an equation to show that it is the same as the other.
Example: Prove (2w+2)² ≡ 4(w² + 2w + 1)
Expand the brackets on the left and then simplify them to show they equal to the right hand side.
(2w + 2)(2w +2) ≡ 4w² + 8w + 4 ≡ 4(w² + 2w + 1)
NOTE: notice the equals sign had 3 lines. This means that two things are equal for all values, not just certain values like in a normal equation.
Often though you are asked to prove that something is odd or even. For this you should remember the following:
- Any even number can be written 2n (anything multiplied by 2 is even)
- Any odd number can be written 2n + 1 therefore
- Consecutive numbers can be written and n, n+1, n+2 etc
- Consecutive even numbers can be written 2n, 2n + 2, 2n + 4 etc
Example: w is an even number. Explain why (w – 1)(w + 1) will always be odd.
Expand the brackets
w² + w – w -1 = w² – 1
Any even number multiplied by itself is even. Therefore when you take one away from any value of w squared you will be left with an odd number.
Disproving By Example
You might be asked to disprove something. You just have to find one value where the statement doesn’t hold up.
Example: Liz says that when m > 1, m2 + 2 is never a multiple of 3.
Give a counter example to show that she is wrong.
Keep plugging in numbers until you find something an example that doesn’t work.
Let m = 5 then m2 + 2 = 27, which is a multiple of 3!
So Liz was talking rubbish.