Mathematical Methods for Physics and Engineering : A Comprehensive Guide

1.7 SOME PARTICULAR METHODS OF PROOF

The essence of the method is to exploit the fact that mathematics is required

to be self-consistent, so that, for example, two calculations of the same quantity,

starting from the same given data but proceeding by different methods, must give

the same answer. Equally, it must not be possible to follow a line of reasoning and

draw a conclusion that contradicts either the input data or any other conclusion

based upon the same data.

It is this requirement on which the method of proof by contradiction is based.

The crux of the method is to assume that the proposition to be proved is

nottrue, and then use this incorrect assumption and ‘watertight’ reasoning to

draw a conclusion that contradicts the assumption. The only way out of the

self-contradiction is then to conclude that the assumption was indeed false and

therefore that the proposition is true.

It must be emphasised that once a (false) contrary assumption has been made,

every subsequent conclusion in the argumentmustfollow of necessity. Proof by

contradiction fails if at any stage we have to admit ‘this may or may not be

the case’. That is, each step in the argument must be anecessaryconsequence of

results that precede it (taken together with the assumption), rather than simply a

possibleconsequence.

It should also be added that if no contradiction can be found using sound

reasoning based on the assumption then no conclusion can be drawn about either

the proposition or its negative and some other approach must be tried.

We illustrate the general method with an example in which the mathematical

reasoning is straightforward, so that attention can be focussed on the structure

of the proof.

A rational numberris a fractionr=p/qin whichpandqare integers withqpositive.

Further,ris expressed in its lowest terms, any integer common factor ofpandqhaving

been divided out.

Prove that the square root of an integermcannot be a rational number, unless the square

root itself is an integer.

We begin by supposing that the stated result isnottrue and that wecanwrite an equation

√

m=r=

p

q

for integersm, p, qwith q=1.

It then follows thatp^2 =mq^2 .But,sinceris expressed in its lowest terms,pandq,and
hencep^2 andq^2 , have no factors in common. However,mis an integer; this is only possible
ifq=1andp^2 =m. This conclusion contradicts the requirement thatq=1andsoleads
to the conclusion that it was wrong to suppose that

√

mcan be expressed as a non-integer
rational number. This completes the proof of the statement in the question.

Our second worked example, also taken from elementary number theory,

involves slightly more complicated mathematical reasoning but again exhibits the

structure associated with this type of proof.