Algebraic proof
Checking a few numbers is not a proof, because the next number might break the rule. A proof uses algebra to show something is true for every whole number. A single counterexample is enough to show a statement is false.
Part 1 of 3: Learn it
In short
- Even numbers are 2n, odd numbers are 2n + 1, consecutive integers are n, n + 1, n + 2.
- To show a multiple of k, get the expression into the form k × (something).
- One counterexample disproves a statement.
Where this is in your specification
Spec points: DfE A6
| Board | Topic: Proof and iteration |
|---|---|
| DfE content | A6, A20 |
| AQA 8300 | Algebra |
| Edexcel 1MA1 | Algebra |
| OCR J560 | 6 Algebra |
| Eduqas C300 | HA6, HA20 |
| Cambridge IGCSE 0580 | E2.5 (solving by iteration is not in 0580) |
Writing numbers in algebra
| Number | Algebra (n an integer) |
|---|---|
| Any even number | 2n |
| Any odd number | 2n + 1 |
| Two consecutive integers | n and n + 1 |
| Two consecutive even numbers | 2n and 2n + 2 |
| A multiple of 5 | 5n |
| Two different odd numbers | 2a + 1 and 2b + 1 |
Use different letters when the numbers are not linked. Writing two odd numbers as 2n + 1 and 2n + 1 would make them the same number.
Writing the proof
- Write each number in algebra.
- Expand and simplify.
- Factorise to show the result: 3(n + 1) is a multiple of 3, 2(...) + 1 is odd.
- Finish with a sentence that says what you have shown.
Identities
The sign ≡ means the two sides are equal for every value of x. To show (x + 4)² − 16 ≡ x(x + 8), expand the left side, x² + 8x + 16 − 16 = x² + 8x, and factorise it into the right side.
Counterexamples
To show a statement is false, find one case where it fails and say why it fails. "The square of a number is always bigger than the number" is false: 0.5² = 0.25, which is smaller than 0.5.
Give a counterexample to "all prime numbers are odd".
Show the answer
2 is prime and even.
Part 2 of 3: See it worked
Worked examples
Example 1
Prove that the sum of three consecutive integers is always a multiple of 3.
- Let the integers be n, n + 1 and n + 2
- Sum: n + n + 1 + n + 2 = 3n + 3
- = 3(n + 1)
Answer: 3(n + 1) is 3 times an integer, so the sum is always a multiple of 3.
Example 2
Prove that (2n + 1)² − (2n − 1)² is a multiple of 8 for any integer n.
- (2n + 1)² = 4n² + 4n + 1
- (2n − 1)² = 4n² − 4n + 1
- Subtract: 4n² + 4n + 1 − 4n² + 4n − 1 = 8n
Answer: 8n is 8 times an integer, so it is always a multiple of 8.
Common mistakes
- Testing a few numbers and calling it a proof.
- Using the same letter for two numbers that do not have to be equal.
- Losing a sign when subtracting a bracket: −(4n² − 4n + 1) is −4n² + 4n − 1.
- Stopping at the algebra without a sentence that says what it shows.
Show that (n + 3)² − (n + 1)² is always a multiple of 4.
Show the answer
n² + 6n + 9 − n² − 2n − 1 = 4n + 8 = 4(n + 2), which is a multiple of 4.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Give a counterexample to "all prime numbers are odd".
2 is prime and even.
Show that (n + 3)² − (n + 1)² is always a multiple of 4.
n² + 6n + 9 − n² − 2n − 1 = 4n + 8 = 4(n + 2), which is a multiple of 4.
Prove that the product of two odd numbers is odd.
(2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1, which is one more than an even number, so it is odd.
Practise this topicAlgebraic proof, with exam-style questions
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