PROFESSOR: The well ordering principle
is one of those facts in mathematics
that's so obvious that you hardly notice it.
And the objective of this brief introduction
is to call your attention to it.
We've actually used it already.
And in subsequent segments of this presentation,
I'll show lots of applications of it.
So here's a statement of the well ordering principle.
Every nonempty set of nonnegative integers
has a least element.
Now this is probably familiar.
Maybe you haven't even thought about it.
But now that I mentioned it, I expect it's a familiar idea.
And it's pretty obvious too if you
think about it for a minute.
Here's a way to think about it.
Given an nonempty set of integers, you could ask,
is 0 the least element in it?
Well, if it is, then you're done.
Then you could say, is 1 the least element in it?
And if it is, you're done.
And if it isn't, you could say 2, is 2 the least element?
And so on.
Given that it's not empty, eventually you're
to hit the least element.
So, if it wasn't obvious before, there
is something of a hand-waving proof of it.
But I want to get you to think about this well
ordering principle a little bit because it's not-- there are
some technical parts of it that matter.
So for example, suppose I replace nonnegative integers
by nonnegative rationals.
And I asked does every nonempty set of nonnegative rationals
have a least element?
Well, there is a least nonnegative rational, namely 0.
But not every nonnegative set of rationals has a least element.
I'll let you think of an example.
Another variant is when, instead of talking
about the nonnegative integers, I just
talk about all the integers.
Is there a least integer?
Well, no, obviously because minus 1 is not the least.
And minus 2 is not the least.
And there isn't any least integer.
We take for granted the well ordering principle just
all the time.
If I ask you, what was the youngest age of an MIT graduate
well, you wouldn't for a moment wonder whether there
was a youngest age.
And if I asked you for the smallest number of neurons
in any animal, you wouldn't wonder
whether there was or wasn't a smallest number of neurons.
We may not know what it is.
But there's surely a smallest number of neurons
because neurons are nonnegative integers.
And finally, if I ask you what was the smallest number of US
coins that could make $1.17, again, we
don't have to worry about existence
because the well ordering principle knocks that
off immediately.
Now for the remainder of this talk,
I'm going to be talking about the nonnegative integers
always, unless I explicitly say otherwise.
So I'm just going to is the word number
to mean nonnegative integer.
There's a standard mathematical symbol
that we use to denote the nonnegative integers.
It's that letter N at the top of the slide
with a with a diagonal double bar.
These are sometimes called the natural numbers.
But I've never been able to understand or figure out
whether 0 is natural or not.
So we don't use that phrase.
Zero is included in N, the nonnegative integers.
And that's what we call them in this class.
Now, I want to point out that we've actually
used the well ordering principle already
without maybe not noticing it, even
in the proof that the square root of 2 was not rational.
That proof began by saying, suppose the square root of 2
was rational, that is, it was a quotient of integers m over n.
And the remark was that you can always express a fraction
like that in lowest terms.
More precisely, you can always find positive numbers m and n
without common factors, such that the square root of 2
equals m over n.
If there's any fraction equal to the square root of 2,
then there is a lowest terms fraction m
over n with no common factors.
So now we can use well ordering to come up
with a simple, and hopefully very clear and convincing,
argument for why every fraction can
be expressed in lowest terms.
In particular, let's look at numbers m and n
such that the square root of 2 is equal to m over n--
that fraction.
And let's just choose the smallest numerator that works.
Find the smallest numerator m, such that squared of 2
is equal to m over n.
Well, I claim that that fraction, which
uses the smallest possible numerator,
has got to be in lowest terms because suppose
that m and n had a common factor c that was greater
than 1-- a real common factor.
Then you could replace m over n by m over c,
the numerator is a smaller numerator that's
still an integer, and n over c.
The denominator is still an integer.
And we have a numerator that's smaller than m
contradicting the way that we chose m in the first place.
And this contradiction, of course,
implies that m and n have no common factors.
And therefore, as claimed, m over n is in lowest terms.
And of course, the way I formulated
this was for our application of the fraction that was
equal to the square root of 2.
But this proof actually shows that any rational number,
any fraction, can be expressed in lowest terms.