Mathematical Methods for Physics and Engineering - Matematica.NET

1.7 SOME PARTICULAR METHODS OF PROOFThe essence of the method is to exploit the fact that mathematics is requiredto 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 givethe same answer. Equally, it must not be possible to follow a line of reasoning anddraw a conclusion that contradicts either the input data or any other conclusionbased 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 isnot true, and then use this incorrect assumption and ‘watertight’ reasoning todraw a conclusion that contradicts the assumption. The only way out of theself-contradiction is then to conclude that the assumption was indeed false andtherefore that the proposition is true.It must be emphasised that once a (false) contrary assumption has been made,every subsequent conclusion in the argument must follow of necessity. Proof bycontradiction fails if at any stage we have to admit ‘this may or may not bethe case’. That is, each step in the argument must be a necessary consequence ofresults that precede it (taken together with the assumption), rather than simply apossible consequence.It should also be added that if no contradiction can be found using soundreasoning based on the assumption then no conclusion can be drawn about eitherthe proposition or its negative and some other approach must be tried.We illustrate the general method with an example in which the mathematicalreasoning is straightforward, so that attention can be focussed on the structureof the proof.◮A rational number r is a fraction r = p/q in which p and q are integers with q positive.Further, r is expressed in its lowest terms, any integer common factor of p and q havingbeen divided out.Prove that the square root of an integer m cannot be a rational number, unless the squareroot itself is an integer.We begin by supposing that the stated result is not true and that we can write an equation√ m = r =pqfor integers m, p, q with q ≠1.It then follows that p 2 = mq 2 .But,sincer is expressed in its lowest terms, p and q, andhence p 2 and q 2 , have no factors in common. However, m is an integer; this is only possibleif q =1andp 2 = m. This conclusion contradicts the requirement that q ≠1andsoleadsto the conclusion that it was wrong to suppose that √ m can be expressed as a non-integerrational 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 thestructure associated with this type of proof.33