A proof is a step-by-step logical argument that shows a math statement is true in every case it covers. It starts from given facts and uses definitions, postulates and statements already proved, with a reason for each step. A statement that has been proved is called a theorem.
Every proof travels from what you are given to what you must prove, and every step on the way needs a reason someone else can check.
Formal definition and notation
A proof of a statement P ⇒ Q is a finite sequence of statements, starting from the hypothesis P and accepted axioms, in which each statement follows by a valid rule of logic from earlier ones, and the last statement is the conclusion Q. Common methods are direct proof, indirect proof (contradiction), and proof by cases; common formats are two-column, paragraph and flowchart proofs.
Symbol
Read it as
Meaning
∴
therefore
marks the conclusion of a step or of the proof
∎ or Q.E.D.
end of proof
Latin quod erat demonstrandum, which was to be shown
Checking many examples is not a proof. n2 + n + 41 is prime for n = 0, 1, …, 39, but at n = 40 it equals 1,681 = 412.
One counterexample is enough to disprove a general statement; a proof is needed to establish one.
Reasons allowed in a geometry proof: given information, definitions, postulates (accepted without proof), algebra properties, and previously proved theorems.
A direct proof goes forward from the given; an indirect proof assumes the opposite of the conclusion and reaches a contradiction.
Formats differ, but the logic is the same: two-column, paragraph and flowchart proofs all need a reason for every step.
A theorem is a statement that has been proved; the proof is the argument that proves it. The Pythagorean theorem is the statement; its many proofs are the arguments.
Proofvs.Conjecture
A conjecture is a guess based on patterns that has not been proved yet. Once someone proves it, it becomes a theorem.
Quick check: 3 questions
Question 1 of 3
Which of these can be used as a reason in a geometry proof?
Show answer and explanation
Answer: C. Definitions, postulates, properties and proved theorems are valid reasons. Appearance, a few examples and the conclusion itself are not.
Question 2 of 3
What does one counterexample show about a statement that says all cases work?
Show answer and explanation
Answer: B. A single case where the hypothesis holds but the conclusion fails proves an all statement false. More examples cannot rescue it, and a false statement can never be a theorem.
Question 3 of 3
To prove ‘the product of two even numbers is even,’ which start is best?
Show answer and explanation
Answer: B. Writing them as 2a and 2b uses the definition of even, and 2a · 2b = 2(2ab) finishes the proof. Examples prove nothing general, assuming the conclusion is circular, and plain a and b do not show evenness.
Proof: frequently asked questions
Why do we need proofs if examples work?
Because a pattern can hold for many examples and still fail. The expression n2 + n + 41 gives primes for every n from 0 to 39 and then fails at 40. A proof covers every case at once, including the ones nobody has checked, so once a statement is proved it can be used with complete confidence.
What are the types of proofs in geometry?
By format: two-column proofs list statements and reasons side by side, paragraph proofs explain the same steps in sentences, and flowchart proofs show the steps as connected boxes. By method: direct proofs reason forward from the given, indirect proofs assume the conclusion is false and reach a contradiction, and coordinate proofs place the figure on a grid and use algebra.
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