What is a Theorem in Mathematics?

What is a Theorem in Mathematics?

In mathematics, a theorem is a significant, proven statement that is based on axioms, definitions, and previously established results, such as other theorems, propositions, or lemmas. Theorems are central to mathematical theory, representing important truths that have broad implications and are often complex or deep results.

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A theorem typically requires a rigorous proof to demonstrate its truth, and once proven, it is accepted as a fundamental truth within its mathematical framework. Theorems are the main achievements in mathematical work and are used to derive additional results, sometimes as corollaries.

Key Points About Theorems

  1. Central Importance: Theorems are major results that form the core of mathematical theories.
  2. Require Rigorous Proof: Theorems must be proven through logical deduction and rely on accepted axioms, definitions, and previously proven statements.
  3. Broad Implications: Theorems often lead to corollaries, providing further results or applications within mathematics.

A classic example of a theorem is the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem has widespread applications in geometry, trigonometry, and many other fields.

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