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I was talking with my geometry teacher the other day and we discussed inductive and deductive reasoning. He wanted me to find out exactly what they are and find an example just to see if I could do it. Can you help me answer this question?"Deductive reasoning" refers to the process of concluding that something must be true because it is a special case of a general principle that is known to be true. For example, if you know the general principle that the sum of the angles in any triangle is always 180 degrees, and you have a particular triangle in mind, you can then conclude that the sum of the angles in your triangle is 180 degrees.
Deductive reasoning is logically valid and it is the fundamental method in which mathematical facts are shown to be true.
"Inductive reasoning" (not to be confused with "mathematical induction" or and "inductive proof", which is something quite different) is the process of reasoning that a general principle is true because the special cases you've seen are true. For example, if all the people you've ever met from a particular town have been very strange, you might then say "all the residents of this town are strange". That is inductive reasoning: constructing a general principle from special cases. It goes in the opposite direction from deductive reasoning.
Inductive reasoning is not logically valid. Just because all the people you happen to have met from a town were strange is no guarantee that all the people there are strange. Therefore, this form of reasoning has no part in a mathematical proof.
However, inductive reasoning does play a part in the discovery of mathematical truths. For example, the ancient geometers looked at triangles and noticed that their angle sums were all 180 degrees. After seeing that every triangle they tried to build, no matter what the shape, had an angle sum of 180 degrees, they would have come to the conclusion that this is something that is true of every triangle. Then they would have looked for a way to prove it using deductive reasoning; that is, deduce it as a consequence of other known general properties of triangles.
In summary, then: inductive reasoning is part of the discovery process whereby the observation of special cases leads one to suspect very strongly (though not know with absolute logical certainty) that some general principle is true. Deductive reasoning, on the other hand, is the method you would use to demonstrate with logical certainty that the principle is true.
Both are necessary parts of mathematical thinking. If you just started with the known properties of triangles and played around with them aimlessly using deductive reasoning, it is unlikely you would discover the fact that the angle sum is always 180 degrees (though if you did happen to discover it that way, you'd know it for certain). However, by noticing that it's true in all the examples you've ever seen, inductive reasoning leads you to suspect that this fact is true. Then, once your suspicions have given you a target and a direction for your deductive reasoning, you construct your rigorous logical proof using deductive reasoning.
The "inductive reasoning" mentioned above is nothing to do with the "principle of induction", which says that if you know something is true for the number 1, and if whenever it is true for one number it is also true for the next number, it is then true for every positive integer. Although this principle is a form of reasoning that gets you to a general principle from some individual cases (which is the reason for the name "induction"), it does so in a precise and logically valid way that is really a form of deductive reasoning if viewed in the correct way. When people refer to an "inductive proof", they generally mean a proof that uses the (logically valid) principle of induction, rather than meaning a form of (logically invalid) inductive reasoning in the sense described above.
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