Algebraic proof · Year 13
In Year 13 you get fluent with: proof by algebra.
An algebraic proof shows that a statement is true for every number, not just for the examples you have tried. Write general numbers with letters — 2n for an even number, 2n + 1 for an odd one — then use algebra to show the result always has the form you need.
Key facts
- Even numbers: 2n. Odd numbers: 2n + 1. Consecutive integers: n, n + 1, n + 2.
- A multiple of k can be written as k × (an integer).
- To prove a number is even, show it equals 2 × (an integer).
- Checking examples is not a proof — but one counterexample is enough to show a statement is false.
- An identity (≡) is true for every value; prove it by expanding one side until it matches the other.
Worked examples
Example 1
Prove that the product of two odd numbers is odd.
Answer: (2a + 1)(2b + 1) = 2(2ab + a + b) + 1