A Classical Beginning 7

All the theorems and corollaries in this chapter are unified by a grand universal theorem,

asserting that

k

√

n is irrational unless n is itself a perfect integer kth power, meaning that

n = r

k

for some integer r. This is equivalent to saying that all the exponents in the prime

factorization of n are multiples of k.

Mathematical Habits

State claims explicitly. Do not allow ambiguity in your mathematical claims and the-

orem statements. Distinguish between similar but inequivalent statements. Formulate

your claim to state exactly what you intend.

Know exactly what you are trying to prove. Before embarking on an argument or

proof, get completely clear about the meaning and content of the claim that is to be

proved.

Insist on proof. Be prepared to prove essentially every mathematical statement you

make. When challenged, be prepared to give further and more detailed justification.

Do not make assertions that you cannot back up. Instead, make weaker statements

that you can prove. Some of the exercises ask you to “prove your answer,” but this is

redundant, because, of course, you should always prove your answers in mathematical

exercises.

Try proof by contradiction. When trying to prove a statement, imagine what it would

be like if the statement were false. Write, “Suppose toward contradiction that the

statement is false, ...” and then try to derive a contradiction from your assumption.

If you succeed, then you will have proved that the statement must be true.

Try to prove a stronger result. Sometimes a difficult or confusing theorem can be

proved by aiming at the outset to prove a much stronger result. One overcomes a

distracting hypothesis simply by dispensing with it, by realizing it as a distraction, an

irrelevant restriction. In the fully general case, one sometimes finds one’s way through

with a general argument.