# Theorem

## Characteristics Of A Theorem

The initial premises on which all theorems are based are called axioms. An axiom, or postulate, is a basic fact which is not subject to formal proof. For example, the statement that there is an infinite number of even integers is a simple axiom. Another is that two points can be joined to form a line. When developing a theorem, mathematicians choose axioms, which seem most reliable based on their experience. In this way, they can be certain that the theorems are proved as near to the truth as possible. However, absolute truth is not possible because axioms are not absolutely true.

To develop theorems, mathematicians also use definitions. Definitions state the meaning of lengthy concepts in a single word or phrase. In this way, when we talk about a figure made by the set of all points which are a certain distance from a central point, we can just use the word circle.

## Resources

### Books

Dunham, William. Journey Through Genius. New York: Wiley, 1990.

Kline, Morris. Mathematics for the Nonmathematician. New York: Dover, 1967.

Lloyd, G.E R. Early Greek Science: Thales to Aristotle. New York: W. W. Norton, 1970.

Newman, James R., ed. The World of Mathematics. New York: Simon and Schuster, 1956.

Paulos, John Allen. Beyond Numeracy. New York: Knopf, 1991.

Perry Romanowski

## KEY TERMS

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Axiom

—A basic statement of fact that is stipulated as true without being subject to proof.

Deductive reasoning

—A type of logical reasoning that leads to conclusions which are undeniably true if the beginning assumptions are true.

Definition

—A single word or phrase that states a lengthy concept.

Pythagorean theorem

—An idea suggesting that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. It is used to find the distance between two points.