A very brief introduction

I suppose everyone knows that sqrt(2) is irrational, and has read the standard textbook proof, with its we may suppose without loss of generality that m/n is in reduced form... . I hate the standard textbook presentation of the proof. I don't hate the proof ; rather I hate the presentation .

Sadly (in my view) youngsters don't get a chance to think sqrt(2) before being exposed to its brutal irrationality proof that rambles on about "we may suppose that ... may be expressed in reduced form" (even G. H. Hardy - an early hero of mine - is guilty of it in his classic A Mathematician's Apology ). Youngsters should be given an opportunity to attempt to find a 'fraction' - a term that youngsters probably prefer over 'rational number' - that possibly equals sqrt(2) (of course they're in for a surprise). Since that is a topic on which I have already written at considerable length in my 56-page Number theorising with Talented Youth (available from the Talented Youth corner of my web site) then I will not pursue it here.

Briefly, how do I think irrationality (as a subject to be contemplated) can/should be introduced to youngsters? From my experience, many youngsters think that sqrt(2) is 1.414... (the calculator displayed value):

> evalf(sqrt(2), 10);



A thoughtful student - when pressed - will tell you that 1.414213562 is not sqrt(2) because, when you square it, you get a decimal that ends in a 4, and so it can't be 2. More pressing (that could be motivated by by further computations like:)

> evalf(sqrt(2), 40); # this "really ends" in '7'



will lead - with proof (the crux!) - to the conclusion that sqrt(2) does not have a terminating decimal value. That's an elementary, but serious result. (Aside. What, though, is the non-terminating decimal value of sqrt(2) ; what are its digits? Once, when I was a student in London, I sat with a group of other students in the company of C. A. Rogers. Someone asked him which question he would most like to have answered. He replied: to know the decimal expansion of the square-root of two . We all knew what he meant. Many years later, in June 1986, I travelled to London to attend the UCL meeting marking C.A.R.'s retirement. Looking back over his life, Rogers remarked on his good fortune to have had a good teacher at school, and he mentioned the friendly competition that he had later enjoyed with Wolfgang Schmidt.)

Next, in my view (based on teaching experience ) the discussion can continue on this line of observation (followed by appropriate questioning): numbers with terminating decimal values are simply fractions with a very particular type of denominator (powers of 10), and so saying (e.g) that sqrt(2) is not 1.414 is simply saying that it is not 1414/1000 , but (the motivated question) could it be (say) 1414/999 ? Well, let's see:

> evalf(1414/999);



So, it isn't. But could it be something else ? Some other fraction? Could it be (say) 47321/33461 ?:

> evalf(sqrt(2), 10);
evalf(47321/33461, 10);



No. And why not? Resorting to computation we see they aren't equal, as they differ on the last decimal point:

> evalf(sqrt(2), 11);
evalf(47321/33461, 11);



But could sqrt(2) be (say) 4478554083/3166815962 ?:

> evalf(sqrt(2), 20);
evalf(4478554083/3166815962, 20);




Again, no. But why not? And this is the point at which one can start asking for reasons independent of computation. Of course it's also the time to start asking: where are those fantastic fractions coming from? It's the sort of thing - if you don't know about this - you may read about in the Talented Youth corner of my web site, or in my ICTMT4 Plymouth 1999 talk on L- and R-approximations , or in the first year section of my Courses I Teach corner of my web site.

Why can't 4478554083/3166815962 be sqrt(2) ? Suppose it was, then sqrt(2) = 4478554083/3166815962 , and then?

sqrt(2)^2 = (4478554083/3166815962)^2 , 2 = (4478554083/3166815962)^2 , and?

3166815962^2 = 4478554083^2 (and then what?)

Impossible since... (e.g.: the LHS ends in '8', the RHS ends in '9'; the LHS is even, the RHS is odd). That's not the end of it, of course. But it all eventually leads to having a proof that sqrt(2) is irrational . In short, by following a well-motivated path, one may lead young minds to a basic understanding of the fundamental mathematical concept of irrationality. If we are interested in teaching, then this - or something like it - is what we should be doing. Incidentally, a much faster route into irrationality may be given by asking questions like:

  • Is log[10](2) = .3010 (as the old log tables used to show)?
  • Is log[10](2) = .3010299957 (as ones calculator may show, or, in Maple)?

> evalf(log[10](2));



Then, having easily argued that log[10](2) does not have a terminating decimal value, move on to easily argue that it is, in fact, irrational. While numbers like (e.g.) sqrt(2) and log[10](2) are both irrational - the latter having the easier irrationality proof - these two numbers are fundamentally quite different : the former is an algebraic number, while the latter is a transcendental number. But I am getting ahead of myself. It is time I moved on to the next section.

Contact details 

After August 31st 2007 please use the following Gmail address: jbcosgrave at gmail.com

This page was last updated 18 February 2005 15:10:03 -0000