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Indirect Proof in Algebra and Geometry

Proof by contradiction.

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A Proof of Innocence

Indirect Proof in Algebra and Geometry

Recall: An indirect proof is a way to prove something true or not true by assuming something that something in the given statement is false.

Credit: Charles Yu
License: CC BY-NC 3.0

No one can stay mad at this "culprit" for long! [Figure1]

Let’s say that you are babysitting at someone’s home for their little boy and his dog. While you are in the living room, watching the child, you hear a loud crash and rush over to the scene of the crime. The family’s beloved porcelain vase is broken! How will you explain to the homeowners that this was not your fault? Well, you could prove your innocence with indirect reasoning. To prove that you are not guilty, you must first assume that you are guilty. If you had broken the vase, you would have been in the dining room, the room in which the vase was broken. But you’ve never been to the dining room before in your entire life and their child can vouch for that! Phew, case closed. Due to the fact that you were not in the dining room and that the vase was broken in that room, you can prove to the homeowners that you were not guilty. Also, because you were watching their child at the time the vase broke, he could not have been the culprit either...So, who is? Unless an intruder of some sort had broken in, the likely culprit would have been the only other being in the house, the dog. Although it is likely that the dog broke the vase, one cannot say with certainty that he did, only that neither you nor the child broke the vase.

The homeowners return home and you explain to them the dilemma. They are so astounded by your indirect reasoning that they decide to give you an additional 15 dollars!

Creative Applications

1. Why might it be useful to prove something indirectly rather than proving something to be true?

2. Consider this given statement: A triangle cannot have two obtuse angles. Would you use an indirect proof to prove this or a two-column proof? Why?

3. When do you use indirect reasoning in your life?

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Image Attributions

  1. [1]^ Credit: Charles Yu; License: CC BY-NC 3.0

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