Noticing and guessing
Mathematicians start by exploring examples, noticing patterns, and making a conjecture: a statement they believe is true but haven’t proven.
From examples to conjecture
Example: Add two odd numbers.
- 3 + 5 = 8
- 7 + 9 = 16
- 11 + 21 = 32
Conjecture: The sum of two odd numbers is always even.
This kind of thinking, where you generalize from examples to a general statement, is inductive reasoning.
Testing conjectures
Test with many and varied examples:
- Small and large numbers
- Negatives, zero, fractions (if they fit the conjecture)
- Unusual cases
Counterexamples
One counterexample proves a conjecture false.
| Conjecture | Counterexample |
|---|---|
| All prime numbers are odd. | 2 |
| n² + n + 41 is always prime. | n = 40 gives 1,681 = 41² |
| The product of two numbers is always bigger than each. | 0.5 × 4 = 2 |
| Every quadrilateral with 4 equal sides is a square. | a rhombus |
The n² + n + 41 example is famous: it gives primes for n = 0 to 39, then fails. Many examples aren’t the same as a proof!
From conjecture to proof
A proof uses logic to show a statement is always true.
Prove: the sum of two odd numbers is even.
- Any odd number can be written 2a + 1 (a an integer).
- (2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1)
- That’s 2 times an integer, so it’s even. ∎
Revising conjectures
When a counterexample appears, you can refine the conjecture.
- "Multiplying makes a number bigger" → "Multiplying a positive number by a number greater than 1 makes it bigger." ✓
Famous conjectures
- Goldbach’s conjecture (1742): every even number greater than 2 is the sum of two primes. Checked for enormous numbers, but still unproven.
- Fermat’s Last Theorem: no whole numbers satisfy aⁿ + bⁿ = cⁿ for n > 2. Stated in 1637, proven by Andrew Wiles in 1994.
Worked examples
Example 1: Make a conjecture: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16. The sum of the first n odd numbers is n².
Example 2: Find a counterexample: "All multiples of 4 are multiples of 8." 12.
Example 3: Conjecture: "The square of any number is greater than the number." Counterexample? 0.5² = 0.25 (or 1² = 1).
Common mistakes
- Treating a conjecture as proven after a few examples.
- Testing only similar examples.
- Thinking a counterexample is "just one case". One is enough to disprove.
Try it yourself
- Make a conjecture about the sum of two even numbers.
- Find a counterexample: "All numbers ending in 7 are prime."
- Conjecture: "The sum of three consecutive integers is divisible by 3." Test it with three examples.
- Prove the conjecture in question 3.
Answers: 1) It’s always even. 2) 27 3) e.g. 1 + 2 + 3 = 6, 4 + 5 + 6 = 15, 9 + 10 + 11 = 30 4) n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1).
Quick recap
- A conjecture comes from patterns (inductive reasoning).
- One counterexample disproves it.
- A proof shows it’s always true.