A proof is a chain of logical deductions from axioms, definitions, and previously proven statements to a conclusion. It is the standard of truth in mathematics. A statement is not accepted as true until it is proved.

Proofs take many forms. Direct proof derives the conclusion step by step. Proof by contradiction assumes the opposite and shows it leads to an impossibility. Mathematical induction proves statements about all natural numbers by proving a base case and an inductive step. Geometric proofs use diagrams and axioms.

Writing proofs is a skill developed through practice. It requires clarity, rigor, and creativity. The first proofs students encounter are often in geometry, but proof is central to all branches of mathematics. A proof not only establishes truth but explains why something is true.

• Logical argument establishing truth
• Forms: direct, contradiction, induction
• Built from axioms and prior results
• Explains why a statement is true