Algebraic proof

In a nutshell

Algebraic proof involves using algebraic notation and manipulation to prove a mathematical statement is true. Start by finding an algebraic expression to represent different types of numbers, like even numbers, odd numbers or multiples of five. Then, use these expressions inside a proof to show whether something is true. It is also possible to disprove a statement by using a counterexample that shows a statement to be false.

Write expressions for proof

Write expressions to represent different types of numbers using algebraic notation.

Examples

Here are some examples, for any integer, $n$ .

DESCRIPTION	EXPRESSION
an even number	$2n$
an odd number	$2n+1$
a multiple of $3$	$3n$
an even multiple of $5$	$5(2n)=10n$
an odd mutliple of $7$	$7(2n+1)=14n+7$

Algebraic proof

Use algebraic expressions within a mathematical statement to prove something is true.

Example 1

Prove that the sum of two odd numbers is even.

Answer

First write expressions for the two odd numbers. For any integers, $m$ and $n$ , write the odd numbers as

$2m+1 \space and \space 2n+1$

Perform the calculation to proof the result

$\begin{aligned}2m+1 + 2n+1 &= 2m+2n+2 \\&=2(m+n+1)\end {aligned}$

As $m$ and $n$ are integers, $m+n+1$ is also an integer. Two multiplied by any integer gives an even number.

Therefore, $\underline{2(m+n+1)}$ is an even number.

Disproof by counter example

It is possible to disprove a statement by finding an example to show that the statement is false.

Example 2

Give a counter example to disprove the statement 'the product of two prime numbers is always odd'.

Answer

Pick two prime numbers which - when multiplied - would give an even answer. As two is the only even prime number, multiply this number with any other prime number.

$2 \times 5 = 10$

$10$ is not an odd number so

The statement 'The product of two prime numbers is always odd' is false.