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.