Writing numbers in algebra
- If n is an integer: 2n is always even, and 2n + 1 is always odd.
- Consecutive integers: n, n + 1, n + 2. Consecutive even numbers: 2n, 2n + 2. Consecutive odd numbers: 2n + 1, 2n + 3.
- A multiple of 3 can be written 3n. A square number can be written n².
Writing a proof
- Write the numbers in algebra, form the expression, then simplify it into a form that shows the result. To show something is a multiple of 4, get it into the form 4(…).
- The sum of three consecutive integers is n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1), which is always a multiple of 3.
- (2n + 1)² = 4n² + 4n + 1 = 4(n² + n) + 1, so the square of an odd number is always 1 more than a multiple of 4.
Identities and counterexamples
- To prove an identity (≡), expand and simplify one side until it matches the other.
- One counterexample is enough to show a statement is false. 'n² + n + 41 is always prime' is false: when n = 40, it equals 1681 = 41 × 41.
- Checking some examples is not a proof. A proof must work for every value.
Key terms
- Proof
- An argument that shows a statement is always true.
- Counterexample
- One example that shows a statement is false.
- Identity
- A statement that is true for all values, shown with ≡.
- Consecutive
- Following one after another, such as n and n + 1.
- Integer
- A whole number: positive, negative or zero.