Writing numbers algebraically
- Let n be an integer. An even number is 2n; an odd number is 2n + 1.
- Consecutive integers: n, n + 1, n + 2. Consecutive even numbers: 2n, 2n + 2. Consecutive odd numbers: 2n + 1, 2n + 3.
- A multiple of k is k × (an integer). To prove something is a multiple of 8, write it as 8 × (an integer expression).
Writing a proof
- Start from the general expressions, simplify, then factorise to show the result. End with a concluding sentence.
- The sum of three consecutive integers: n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1), which is a multiple of 3.
- The difference between the squares of consecutive odd numbers: (2n + 3)² − (2n + 1)² = 4n² + 12n + 9 − 4n² − 4n − 1 = 8n + 8 = 8(n + 1), a multiple of 8.
- (n + 3)² − (n − 3)² = 12n, which is a multiple of 12.
Identities and always-positive expressions
- ≡ means "identically equal": true for every value. Show (x + 2)² − (x − 2)² ≡ 8x by expanding the left side.
- To prove an expression is always positive, complete the square: x² − 6x + 10 = (x − 3)² + 1. A square is never negative, so the expression is at least 1.
- n(n + 1) is always even, because one of two consecutive integers is even.
- One counterexample is enough to disprove a statement. "All prime numbers are odd" is false, because 2 is prime.
Key terms
- Proof
- A logical argument showing a statement is always true.
- Integer
- A whole number.
- Consecutive
- Following one after another.
- Identity
- An equation true for all values, shown by ≡.
- Counterexample
- An example that shows a statement is false.