- Hello, I'm Jade and it's my life's ambition
to find out the exact amount of time it takes for this ball
to hit the ground when dropped from this bridge.
Simon's down on the ground with a stopwatch
to measure the time.
He's programmed a special type of stopwatch.
Rather than using seconds and microseconds,
this one measures in fractions of a second
to 38 decimal places, to be exact.
All right, let's do it.
(mellow music)
All right, let's see what we got.
(mellow music continues)
2.4450662899032, et cetera, et cetera seconds.
But is that the exact time?
Of course there's a human reaction time we need
to take into account, just like with any measurement.
There's also the processing speed of the computer.
In reality, it's only accurate
up to about three decimal places
but just imagine those things weren't a problem
and we did manage to capture the exact amount of time
it took for the ball to reach the ground.
This number still does not describe
that exact amount of time.
This is only up to 38 decimal places
for this to be the exact number.
It would mean that everything after the 38th digit
is followed by a bunch of zeros,
which as we'll see is extremely unlikely.
Much more likely is that the digits go on.
I could get Simon to keep adding more and more digits
but that doesn't solve the problem.
It makes it more accurate
but it will never be the exact number.
When I say exact, this is what I mean.
Take the number 0.33333333 where the threes repeat forever.
We can't write this number exactly in decimal form
because we can't write infinitely many threes
but when we write it as a fraction, suddenly it's possible.
That's the exact number and that's what I want,
the exact number that describes the time it took
for the ball to hit the ground.
So can I convert my number to a fraction?
Well, this begs the question,
can all numbers be converted to fractions,
a ratio of two integers.
Numbers like this are called rational numbers
and it was thought for a long time that they could.
Say you want to buy some wood but only this much,
what do you tell the cashier?
It's more than one half but less than three quarters.
But luckily we can cut it up into any number of equal parts
that we want.
So you can say, I'd like 13 out of the 25 segments please.
That's pretty handy.
It means we can express complicated numbers like this
by cutting a whole into lots of parts.
You can cut a whole up into so many pieces
that it seems possible to express any fraction
of it as a rational number.
But one day this guy was discovered, the length
of the hypotenuse of a triangle was side length one.
Such a simple number yet try as they might,
no one could express it in terms terms of a fraction
of a whole number.
Here the side is cut into 10 equal parts and the diagonal
into 14 equal parts, but there is some small remainder.
No matter how many equal segments they cut both lengths
into, there was always some small remainder
and so it was discovered that some numbers are irrational.
They just can't be expressed as a fraction.
The digits of an irrational number never end
and can't be written as a repeating pattern.
Okay, but maybe my number can still be converted
into a fraction.
Irrational numbers exist, but maybe I'll be lucky
and mine will be rational.
I mean, what are the odds that it's an irrational number?
Well, actually the odds are high.
Practically 100%.
See the rational and irrational numbers make
up the real number line.
The real number line is like a rubber band.
Only if rubber bands were infinitely long
and infinitely stretchy.
Infinitely long because it never ends.
If we start counting, one, two, three, et cetera,
we won't ever stop.
The real number line extends to infinity
in both the positive and negative directions.
Infinitely stretchy because between any two numbers
there lies infinitely more numbers.
Take the rational numbers one half and one.
Integers are rational numbers
because you can just put them over one.
Between every pair of rational numbers,
there is another rational number.
You can zoom in forever
and there will always be another rational number,
just waiting to be discovered.
So between any two rational numbers,
there are infinitely more rationals.
Notice that you can't do this with the integers.
Between any two integers,
there is not another secret integer waiting.
So you might think
that there are more rational numbers than integers
and that would be a reasonable thought, but wrong.
You see in math, we have a special way to see
if we have the same amount of objects.
If you have this many forks and this many spoons
and you can pair each fork with a spoon so that no forks
or spoons are left over, we know
that there are the same number of forks as spoons.
If some forks are left over,
you know that there are more forks than spoons.
So two sets of objects are the same size
if we can find a one-to-one pairing.
If we cannot find a one-to-one pairing,
one set of objects is bigger than the other.
We can extend this idea to infinite sets.
Take the set of all the natural numbers,
the positive integers
and the set of all the rational numbers.
At first it seems impossible to make a one-to-one pairing.
We'll just keep going down this rabbit hole
and never getting to all the other rationals.
But let's try arranging them like this.
Make a table and write what the numerators will be
along the top from one to infinity
and what the denominators will be along the side.
Now, build the table.
Here will be one over one.
Here will be one over two.
Here will be one over three.
Here will be two over one here will be two over two.
You get the idea.
We will have some rationals that repeat like two over two,
which is the same as one over one
but that doesn't really matter.
We can just cross them out when we recognize them.
If we try to pair the first row with a natural number,
well it goes off to infinity,
so we'll never get to the second row.
But a mathematician named Georg Cantor
had the brilliant idea to count them diagonally.
By doing it this way, you can pair each rational number
with a natural number in a one-to-one correspondence
to any natural number you choose.
You can systematically find
the corresponding rational number.
And remember, if you can place two sets of objects
in a one-to-one pairing, those two sets of objects
are the same size.
Therefore, there are the same amount of rational numbers
as natural numbers.
And so the first lesson about infinite sets,
a subset of a set can be the same size as the whole set.
This pairing with the natural numbers is basically counting.
So we say that the rational numbers
are countably infinite.
So what about the irrational numbers?
Is there a way to make a one-to-one pairing
with them and the natural numbers?
Well, Cantor showed that you can't.
If we try the same trick as with the rational numbers,
you can see why it won't work.
Irrational numbers cannot be expressed as fractions.
So, the whole organizing them by numerator
and denominator doesn't work.
There is no systematic way to count them.
Furthermore, Cantor found that the irrational numbers
are more densely patched together than the rational numbers.
Trying to count the irrationals is like trying
to count the points on a line.
There is always more line
and there are always more irrationals.
Because we cannot count the irrationals,
the size of their infinity is called uncountable infinite.
Cantor found that there are different sizes of infinity.
Remember, if we can't make a one-to-one pairing
between two sets, one set must be bigger than the other
and the uncountable infinity of the irrational numbers
is vastly bigger than the countable infinity
of the rational numbers.
In fact, there are infinitely more irrational numbers
than rational numbers.
The rational numbers make up an infinitesimally small part
of the real numbers.
If we threw a dart onto the real number line,
the chances that it would hit a rational number are so tiny
that they are mathematically zero.
So, this number is most likely an irrational number.
Bummer, that complicates things.
I can't write it as a fraction
but is there some other way to write its exact value?
Can we even write irrational numbers as exact values?
Well look at the square root of two.
That's irrational and we can write that as an exact value.
Or what about this guy?
This is a very famous irrational number, the golden ratio.
It's pretty confusing that the the golden ratio
is irrational but anyway, this number is expressed
in terms of integers and mathematical operations.
When we carry out these operations,
we get the decimal expression of the golden ratio.
We can't express the exact value in decimal form
as the decimals go on forever
but we can express the exact value like this.
I'd be happy with something like that.
Can I convert my number into an arrangement of integers
and mathematical operations?
Well, how did we get this number?
The golden ratio is the number you get
when you divide a line into two parts
so that the whole length divided by the long part
is equal to the long part divided by the short part.
If we make the short part one unit and the long part X,
plug in the numbers to our strange line equation
and rearrange a bit, we get the equation X squared minus X
minus one is equal to zero.
When we solve for X, we get one
plus the square root of five over two.
So this number is the solution to the equation
that describes the golden ratio.
Equations like these are called polynomial equations.
The word polynomial just means many terms.
Polynomial equations have integer coefficients
and use all the tools of algebra,
plus, minus, divide, multiply, and exponentiation.
Numbers that are the solution to polynomial equations
are called algebraic numbers because they are described
and constructed by algebraic means.
So it looks like building a polynomial equation
and then finding its solution is a pretty nifty way
to describe a number.
But can all numbers be expressed
in terms of algebraic operations and integers?
The integer is the most basic,
most fundamental object of mathematics.
If you were to build a number system from scratch,
it's natural you'd start with counting
and therefore with integers.
All the numbers we've dealt with so far
have been expressed in terms of integers.
Rational numbers are just numbers expressed
in terms of the ratio of two integers.
Algebraic numbers are the solutions to polynomials
that have been constructed with integers.
Seeing as integers are the building blocks
of our mathematical world, it makes sense
that all numbers could be expressed in terms of them.
But in a shocking discovery, two of the most famous numbers
were shown not to be related to integers
by any of our algebraic toolkits.
Pi and E.
It doesn't matter how we put them down, flip them,
or reverse them, they will just not be expressed
by algebraic means.
How crazy is that?
We use these numbers so often yet we can't construct them
or even describe them
in terms of our most fundamental mathematics.
Numbers like this are called transcendental
as they transcend algebra.
They're like the wild animals of the number kingdom.
We've managed to capture and tame the algebraic numbers
but the transcendentals, they continue to evade us.
Only a very small portion of numbers
have been proven to be transcendental.
They're listed on one Wikipedia page.
Bummer again.
Maybe my number is not algebraic but also maybe it is.
What are the odds of my number being transcendental?
We can literally build infinitely many polynomial equations
with as many terms as we want.
Any of the infinitely many integer coefficients
and any algebraic operations.
This means that there are infinitely many
polynomial equations and so,
infinitely many algebraic numbers.
If we compare this to the small portion
of proven transcendental numbers,
it's pretty likely that my number will be algebraic, right?
No, it's true that we can build infinitely many
polynomial equations but we are limited by the fact
that their coefficients must be integers.
And how many integers are there?
Countably infinite.
We can only have countably infinitely
many polynomial equations
and therefore, countably infinitely many algebraic numbers.
And how many real numbers are there?
Uncountable infinite.
So just as there are vastly more real numbers
than rational numbers,
there are vastly more real numbers than algebraic numbers.
Do you know what this means?
Most numbers are transcendental.
Most numbers lie beyond integers and algebra.
The numbers we deal with in daily life only represent
a tiny infinitesimal dot of all the numbers that exist.
Here's a nice image to show you which kinds of numbers
fit where in relation to each other.
So, if most real numbers are transcendental,
why is the Wikipedia list so short?
Well, they are the only ones proven to be transcendental.
It turns out that proving a number is transcendental
is a very hard thing to do.
This seems fitting given that they lie beyond
the most natural ways we understand mathematics.
I like how one mathematician put it.
"We can't find the hay in the haystack."
The numbers we're used to working with are the needles
and the tools we have are magnets.
We haven't got a good tool for finding the hay.
Before we get back to describing my number,
I haven't given up yet, I wanna say that it's pretty nuts
that we came to these fascinating conclusions
by analyzing infinities.
It's one of the most surprising
and paradoxical areas of mathematics.
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Okay, back to describing my number.
Where were we?
Oh yeah, this number is
with extreme probability, transcendental.
Bummer.
Okay, fine.
No need to despair.
Maybe I can't write my number with algebra
but maybe I can still describe it, right?
I mean, pi and E are transcendental and I can describe them.
Pi is the ratio
between the circumference and the diameter of a circle.
E is the number in the function E to the power of X
for which the function stays the same
when you differentiate it.
I'll settle for that.
A description with words and symbols.
I implore all the languages of humanity
to describe my number.
Guess what?
All the languages of humanity are made
of a finite number of symbols.
Let's think about what a description is.
A series of words and symbols arrange to convey meaning.
There are a finite number of words in any given language
and an infinite number of ways those words can be combined.
But even if we considered the infinite number
of combinations that can be created,
you could, in theory, list them out.
Each combination could be matched
with a unique natural number.
The set of possible descriptions of numbers
is countably infinite.
So even if we came up with infinitely many descriptions
of numbers using all of the power of language,
we'd still have an uncountable infinity of numbers left
that we could not describe.
It doesn't matter what mathematical tools
you summon or invent, most numbers are
and will always be indescribable.
(gentle jazz music)