theorem
a rule that needs to be proved
In mathematics, a theorem is a statement that has been proven to be true through logical reasoning and mathematical principles. Theorems are considered to be fundamental and important results that contribute to the development of mathematical theories and concepts.
A theorem typically consists of a hypothesis, which is a statement about a certain condition or situation, and a conclusion, which is a statement that follows logically from the hypothesis. The proof of a theorem involves demonstrating that the conclusion is a necessary consequence of the hypothesis, using logical reasoning and mathematical principles.
Theorems can be applied in many different areas of mathematics, including algebra, geometry, calculus, probability theory, and number theory. They are often used to solve problems and make predictions about the behavior of mathematical systems and processes.
Examples of well-known theorems include the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides, and Fermat’s Last Theorem, which states that there are no three positive integers a, b, and c such that a^n + b^n = c^n for any integer value of n greater than 2.
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