Mathematical Proofs
Learn proof techniques and mathematical argumentation
lessons (8)
Introduction to Mathematical Proofs
Learn the fundamentals of mathematical proofs and how to construct logical arguments.
Direct Proof
Learn to construct direct proofs by logically deriving conclusions from given hypotheses using definitions, axioms, and previously proven theorems.
Two-Column Proofs
Learn to write two-column proofs: list each statement on the left and the reason that justifies it on the right, for equations, segments and angles.
Paragraph Proofs
Learn to write a proof as connected sentences with signal words like since and therefore, convert two-column proofs, and check a proof for gaps.
Geometric Proofs
Prove geometric facts step by step with parallel line angles, the triangle angle sum, triangle congruence (SSS, SAS, ASA, AAS), CPCTC and isosceles triangles.
Proof by Contrapositive
Prove an if-then statement by assuming its conclusion is false and showing that its hypothesis must be false too.
Proof by Contradiction
Learn to prove a statement by assuming it is false and showing that this leads to something impossible, with classic proofs such as the irrationality of the square root of 2.
Mathematical Induction
Learn to prove statements for every positive integer with mathematical induction: base case, inductive hypothesis, inductive step and conclusion.
Mathematical proofs provide rigorous demonstrations that statements are true. From direct proofs to proof by contradiction to mathematical induction, understanding proof techniques is essential for advanced mathematics.
What Students Will Learn
- Write direct proofs
- Understand proof by contradiction
- Apply mathematical induction
- Identify proof structures
- Verify proof validity
Frequently Asked Questions
What is a direct proof?
Assume the hypothesis is true and use logical steps to show the conclusion must also be true.
What is proof by contradiction?
Assume the opposite of what you want to prove, then show this leads to a contradiction.
What is mathematical induction?
Prove a base case is true, then prove if it is true for n, it must be true for n+1. This proves it for all positive integers.