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Name: Denis
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How do mathematicians find theorems like the Pythagorean theorem? Is it just random playing numbers of something they thought of?

Denis, There are many situations that can lead to mathematical theorems. It can result from noticing a pattern and working out details to see what the pattern actually is. It can result from needing to discover how several things relate and then looking for this relationship. It can result from just being curious about something that originally seemed unimportant. Theorems are seldom originally intended to be theorems. This cannot happen until the mathematician knows an important relationship exists. In all cases, a good imagination is probably the most necessary factor for creating a theorem.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College

The development of mathematical theorems depends upon a set of ELEMENTS or DEFINITIONS of the terms involved (these could be: points, lines, angles, numbers, etc...etc.) and a set of RULES or OPERATIONS of these elements that defines how the elements behave with respect to one another. Some new RELATION may follow from these elements and operations -- THEOREMS. Just "how" a mathematician "discovers" these theorems is quite diverse. Some of the new relationships may be "obvious" -- others may be quite "obscure". Mathematical PROOF consists in deriving the new relations from the DEFINITIONS and OPERATIONS so that all of the ELEMENTS and RULES are not violated. Sometimes the chain of logic is simple and straightforward, and other times the chain of logic can cover dozens of pages.

In some cases the mathematician may "see" the result as being "true" and then set about showing (PROVING) that the result follows logically from the definitions and rules. In other cases the mathematician may notice some regularity in the behavior of the elements and then set about proving that the observation is a logical consequence of the definitions and rules of operation.

For the mathematician the objective is to show that the new relation (THEOREM) is a necessary consequence of the logical relation between the ELEMENTS/DEFINITIONS and the RULES/OPERATIONS. Establishing this relationship can be easy or tricky, obvious or obscure. A couple of examples: 1. There are an infinite number of prime numbers (prime numbers are divisible those that are divisible only by the number itself and the number "1").

2. Every even number greater than "2" is the sum of two prime numbers -- examples: 4 = 3+1, 14 = 11 + 3, etc. Just showing examples is not sufficient. One has to show that the result is a necessary consequence of the "counting" numbers and the rules of arithmetic.

Vince Calder

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