Ever wonder how people determine that certain constructions are impossible? Did mathematicians get together and take turns trying to solve problems that others couldn't? The answer is no. To "prove" something, by definition, is to use deductive reasoning to deduce things.
"Deductive Reasoning" is a way of reaching a conclusion that follows from a set of accepted reasons. If these reasons are true, then the conclusion must be true also. Proofs use this kind of reasoning.
For example, the proof of the inability to construct a Heptagon is as follows:
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STATEMENTS
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REASONS
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| 1. To construct a regular heptagon you need to inscribe seven segments inside a circle. | 1.Given. |
| 2. Each of these segments is an irrational number. | 2. A conclusion from calculus or complicated mathematical geometry. |
| 3.Constructing a Heptagon is Impossible. | 3.It is impossible to construct an irrational number. Therefore, it is impossible to construct a regular heptagon. |
I know that this seems insufficient to prove it impossible, but this is really all there is to it. If you can figure out what needs to happen to make the construction work and then prove that that can't be done, then the construction is impossible assuming there is no other way to construct it. (In the case of the Heptagon, there is no other construction, which can be proved by a different proof). In many cases there are alternate ways to do constructions, and in case of their inability to complete the problem- that remains unproven.
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