* Statement: A theorem expresses a specific idea or relationship.
* Proven: It's not just a guess or an assumption; it's been demonstrated to be true through a series of logical steps.
* Mathematical reasoning: The proof uses accepted axioms, definitions, and previously proven theorems to establish the validity of the statement.
Think of it like a building block in mathematics:
* Axioms: Basic, unproven truths that form the foundation.
* Definitions: Clear and precise explanations of terms.
* Theorems: Proven statements built upon axioms and definitions, adding to the structure of mathematical knowledge.
Examples of theorems:
* Pythagorean theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
* Fundamental theorem of calculus: The relationship between differentiation and integration.
In essence, a theorem is a powerful tool that allows mathematicians to confidently build upon existing knowledge and discover new truths.