Page 22 - 'Blast_Into_Math

Blast into Math! Pure mathematics: the proof of the pudding is in the eating

2.4 Ready? Set? Prove!

The power of mathematics comes from the infallibility of its proofs. Once a theorem is proven it is like
an indestructible brick. We can use the theorem to prove bigger and bigger theorems, like building a
great math tower, or we can use the theorem for specific purposes here and there, like filling a hole in
a wall. Because a proof is an unbreakable logical argument, learning to prove mathematical statements

not only strengthens your mathematics but also improves your ability to make logical arguments. A
strong command of logic and proof is advantageous for law, politics, world affairs, and everyday human
interactions. There are three basic types of proofs.

• Direct Proof. In a direct proof you make true statements that follow directly from the
hypotheses, definitions, and previously proven theorems until you reach the conclusion. This
is the most common type of proof and often the simplest.
• Contradiction. To avoid confusion you should always begin a proof by contradiction by
stating that your proof is by contradiction. Next you assume that the conclusion is false but
the hypothesis is true. Then you use the hypothesis and the assumption that the conclusion

is false to show that something impossible happens. This proves that the assumption that the
conclusion was false caused the impossibility so it must be false. This false assumption was
that the conclusion was false, which means that the conclusion must be true.

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