A very brief introduction
I suppose everyone knows that
is irrational, and has read the standard textbook proof, with its
we may suppose without loss of generality that
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
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
(of course they're in for a surprise). Since that is a topic on which I have already written at considerable length in my 56page
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
is
1.414... (the calculator displayed value):
>
evalf(sqrt(2), 10);
>
A thoughtful student 
when pressed
 will tell you that 1.414213562
is
not
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
does not have a terminating decimal value. That's an elementary, but serious result. (Aside. What, though,
is
the nonterminating decimal value of
; 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 squareroot 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
is
not
1.414 is simply saying that it is
not
, but (the motivated question) could it be (say)
? Well, let's see:
>
evalf(1414/999);
>
So, it isn't. But could it be
something else
? Some other fraction? Could it be (say)
?:
>
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
be (say)
?:
>
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 Rapproximations
, or in the first year section of my Courses I Teach corner of my web site.
Why
can't
be
? Suppose it was, then
, and then?
,
, and?
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
is
irrational
. In short, by following a wellmotivated 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
(as the old log tables used to show)?

Is
(as ones calculator may show, or, in Maple)?
>
evalf(log[10](2));
>
Then, having easily argued that
does not have a terminating decimal value, move on to easily argue that it is, in fact, irrational. While numbers like (e.g.)
and
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.