Let's continue our discussion of
common sets of numbers
we just discussed a moment ago.
The set of all integers, which is notated
with a capital Z,
and the set of all rational numbers.
In other words the set of all fractions,
which is notated with a capital Q.
So, is there anything beyond the domain
of the rational numbers?
Anything that's not a fraction
that we can think of?
Well, one of the most famous numbers of
them all is actually not a rational number
The number "pi".
Pi has some sort of quirky properties,
right?
We know the decimal expansion
begins 3.1415... and so forth.
One of the interesting aspects of pi is
that decimal expansion goes on forever,
so it's infinite
and it's also non-repeating,
or you could say non-recurring.
So, the distribution of the numbers
in the decimal expansion of pi
are, effectively, random.
I can't write pi as the ratio of
one whole number (one integer)
over another.
So if a number can't be written
in this fashion,
if it can't be expressed
in rational form
then it's called irrational.
So pi is an irrational number.
There's a number that's not in Q,
let's label that
in our Venn diagram.
Somewhere out here is the number pi.
You might know that the number "e",
also sort of a special number in math
and science, is 2.7... etc, etc,
has also this nice property
that its decimal expansion
is non-repeating and infinite,
so it's likewise an irrational number.
So e is also out here.
In fact, there is an infinite set of
irrational numbers.
How can I just argue that there
is an infinite set of irrationals?
One fact is that if you take
the square root of a prime number,
you get an irrational result.
So it's decimal expansion is
non-repeating and infinite.
Number two, is that the number
of primes is infinite.
Now, prime, just to remind you
is a whole number
greater than or equal to 2
whose only divisors are 1 and itself.
So the square root of a prime
is irrational
and the number of primes is infinite.
Now, this is not necessarily obvious,
right?
We don't know that sort of a-priori.
In fact, I'm including a little
bonus lecture where I proof a la Euclid
that there's an infinite number of primes.
If you put these two together,
then you can show that there is
an infinite set of irrationals.
Let's check that out.
If there's an infinite number of primes
and the square root of any prime
is irrational
then there's the reason again that
there's an infinite number
of irrational numbers.
A slightly different justification.
Let's call this justification number one,
that there's an infinite set
of irrationals.
Justification number two,
from sort of a "complexity" perspective,
that there's an infinite number
of irrationals.
What I've said before is that
the decimal expansion of
an irrational number is non-repeating
and infinite.
So we can go in reverse here,
because if I am just constructing
infinite non-repeating decimal expansions
I could ask, how many are possible?
Now, that's a big counting problem,
but as it turns out
there's an infinite number of such
decimal sequences that I can construct.
For instance, let's say I just start by
flipping a coin or asking a computer
to compute a super-random number
and according to the outcomes,
these coin flips, or whatever,
I fill in some number.
How many different such
decimal expansions are possible?
There is an infinite number of ways
I can express a non-repeating
infinite decimal expansion.
So there are an infinite number of
irrational numbers, because there's
an infinite number of such
decimal expansions I could construct.
This leads us to our next set of numbers,
and the most important
set of numbers of them all,
which is the reals.
The reals consist of the rational numbers
let's just say "plus", kind of informally,
the irrational numbers.
Take note here.
The reals, without a doubt,
are the most important and
commonly encountered number system
in any applied science field.
So, when you do calculus,
you're working with the reals.
When you do linear algebra,
differential equations, you're basically
working with the reals.
Sometimes other sets,
but generally speaking
it's the reals or a subset of the reals.
In proper notation, we would say
it's really the union of those two sets.
So, now, I can add another
concentric circle to my Venn diagram,
which is showing the sequence of
common sets of numbers.
Now I've got the reals as the
outermost shell here.
If I graph the real line, as it's called,
I get a straight line, importantly though,
with no gaps.
On the other hand, if I plot all the
rational numbers on the line,
I get lots of gaps.
In fact, it's a well-known
mathematical result
that between any two rational numbers
lies an irrational number.
The reals then give us kind of this
full-fledged closed domain,
in the sense there're no gaps.
In other words, when I plot the reals,
I get, importantly,
a continuum of numbers.
So, if we plot the reals,
we get a continuum.
Now, what happens if I sort of track
the motion of an object
or a particle in space?
Let us imagine we see a particle moving
and we track it,
and it goes along this path here.
It's starting in the lower left
moving to the right
and as it goes, say, time elapses.
So, the way I experience the motion of
that particle is a little bit debatable,
but generally the accepted interpretation
is that I experience that motion
as a continuous process.
So, therefore, it would be of use
as mathematicians and scientists
to use a model that replicates
that continuous behavior.
So, in other words, we would probably
like to use, generally speaking,
a number system that is a continuum.
Now, you've heard of the
space-time continuum, right?
I won't get too new agey here, but we
generally experience the real world
as a space-time continuum,
and the number system we want to use
to model that is a continuum.
Now, similarly,
just to add to that notion,
for those of us that have maybe had
a little bit of calculus or seen functions
and, why not?
We can think of a function...
Usually in calculus we'd like to study
functions that are continuous, right?
Those have kind of nice
mathematical properties.
So, if we take a function
that's continuous,
maybe as a model for the motion
of the particle, and we use it
in conjunction with the domain
for that function that's also continuous,
together, we get a continuous model.
And that, generally speaking,
is gonna be a good description
of real world phenomena.