Proof
Direct Proof, Proof by Exhaustion,
Proof by Contradiction, Counter Examples
Why we study this
Lots of students wonder why we need to prove things. In early geometry, you might think: "I believe the Pythagorean theorem works; I don't need to show why it's true. I'd rather just apply it to solve problems, like how long a ladder needs to be to reach a window." But, maths is totally built on the concept of proof. Without it, a famous rule is just a conjecture instead of a solid fact. Basically, if you skip proof, you're missing the whole point of the subject.
You might have heard of Fermat’s Last Theorem, which says that a^n + b^n = c^n has no whole-number answers if n is bigger than two. Fermat claimed he had the answer way back in 1637, but a real, verified proof didn't show up until 1995 thanks to English mathematician Andrew Wiles.
Plus, some quite simple mathematical conjectures are still unsolved. Look at Goldbach’s Conjecture: the idea that every even number over two is just two prime numbers added together (like 12 is 5 + 7, or 50 is 19 + 31). Try splitting two even numbers for yourself - not too big though, like 32402, which according to AI is 19 + 32383. Even though computers have checked trillions of numbers, no one has found a universal proof yet. Kurt Gödel even showed with his Incompleteness Theorem that some maths truths can’t be proven within their own systems. Don't worry though, you won't have to deal with these impossible tasks in your exams!
Proof is often found to be a difficult topic to revise. This is because it covers such a wide range of possible questions, all of which require different approaches. Make sure you know the difference between the types of proof: direct, proof by contradiction, proof by exhaustion, proof by induction (Further Maths only) and disproof.
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Euclid of Alexandria
c. 300 BC (Antiquity)
Regarded as the patriarch of geometry, Euclid’s 'Elements' pioneered the axiomatic approach, demonstrating that elaborate principles arise from basic, undeniable axioms through careful, step-by-step reasoning.
René Descartes
1596 – 1650 (Early Modern)
Descartes unified the fields of algebra and geometry. He argued that true mathematical insight stems from logical clarity, establishing the framework for modern analytical methods and algebraic verification styles.
Kurt Gödel
1906 – 1978 (20th Century)
Gödel’s groundbreaking Incompleteness Theorems changed logic forever, proving that within any logical framework, there exist certain valid statements that the system’s own rules are unable to verify.
History of Proof
Exam tips
Common Pitfalls
One mishap is thinking a few successful examples constitute a full proof for every case.
You must ensure your argument moves from one line to the next with total transparency, so don't leave out important steps.
Also, unless each step is reversible, don't argue backwards from the statement you are trying to prove! Here is a 'proof' that 2 = 1:
2 = 1
so 1 = 2
Adding both sides, 3 = 3
This is true, so 2 = 1!
[Can you spot the error? This proves that if 2 = 1 then 3 = 3 (which is not very helpful!]
✓
Know your command words in questions: 'prove', 'show', 'verify' all have slightly different meanings . 'Prove...' is usually confined to proofs of formal results (for example that the derivative of x^2 is 2x using first principles). 'Show...' is technically the same as 'prove' but is usually used for specific results and doesn't imply the same level of formality. 'Verify' means you can argue backwards from the given result.
✓
Use the correct logical symbols! Equals or implies? Equals is a verb, as in n^2 - 1 = (n - 1)(n + 1); but 'implies' connects two statements, as in 'Sam is a pig implies Sam has four legs', or 'n - 1 = 2 implies n = 3'.
✓
'Closing the deal' is essential; always conclude by re-affirming the initial statement or stating 'LHS = RHS' or 'QED' (Quod Erat Demonstrandum in Latin)
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Define a general even number as 2n, an odd number as 2n+1 or 2n-1, a rational number is p/q (where n , p and q are integers). A general two-digit number is 10n + m (where m is the units digit and n is the tens digit).
Direct Proof
Define your variables at the start, like setting 'x' as an odd number '2k+1', or defining a two-digit integer as 10m + n, where m and n are single digits, before you start any algebra. Proofs of divibility always involve factorising an general expression, like in the example on the right [text]
Direct proofs must be general, so quoting individual examples is not enough! Sometimes statements are true for a certain set of values, but not others. For example n^2 is always greater than n except when 0 < n < 1.
Proof by Exhaustion
Proof by exhaustion is used when each possible case is proved. Here is an example. [text]
Proof by contradiction
The most well-known example of this is the proof that the square root of 2 is irrational. You will find lots of videos on this on the web - try adapting the proof for the square root of 3, as it's exactly the same structure. Here is an other example also about irrationality. [text]
Disproof by
counter-example
The result often seems possible for n = 1, 2, 3, ... but keep going and you will find the one counter-example you need to disprove the statement. Try looking for not-so-obvious case, such as one negative and one positive. For example, x^2 > x is true provided x > 1, but not true when x = 0.5, and 1/x > 1/x^2 except when x is negative. Try experimenting with a few values until you find the one which disproves the statement. Warning: most statements about prime numbers are generally false!