The real number line is the continuous line whose points are the real numbers.
In this video we are going to explore the current thinking on the following 3 properties related to the real number line:
The completeness of the real numbers
The cardinality of the continuum
The Lebesgue measure of the rational and irrational numbers
These properties are interesting in themselves,
but they are also important because they will allow us to talk precisely about Zeno’s paradox, which is the topic of my next video.
Try to imagine a continuous line whose points are the integers.
It’s impossible, in part because the ordinary ordering on the integers is not dense.
In other words, there is a gap between adjacent integers.
Now try to imagine a continuous line whose points are the rational numbers.
In between any pair of rational numbers you can always find another rational number,
so we’d say that the ordinary ordering on the rational numbers is dense.
But nevertheless there are still gaps between the rational numbers, at each irrational number.
And so we say that the rational numbers are incomplete
and we formalize this idea by saying that the rationals do not contain all limits.
More precisely, a sequence of numbers converges to a limit,
if (and only if) its elements eventually come and remain arbitrarily close to each other.
Such convergent sequences are called Cauchy sequences.
And the rational numbers are incomplete because some Cauchy sequences of rational numbers have limits which are not rational.
But what about the real numbers?
Well, every Cauchy sequence of real numbers converges to a real number.
So the real numbers are complete.
In other words, the real number line is continuous.
And for that reason, the real numbers are the numbers of choice in calculus, and many other branches of mathematics.
Now, how many points make up the real number line?
In other words, how many real numbers are there?
Georg Cantor gave us a precise answer to this question.
First, to make any progress at answering this question we have to make one important assumption.
We must assume that there exists a complete set of natural numbers.
And we say that the cardinality of that set equals aleph-naught.
Which loosely means that there are aleph-naught natural numbers.
Any set that can be put in a one-to-one correspondence with the set of natural numbers also has a cardinality of aleph-naught.
For example, we can put the set of even natural numbers in a one-to-one correspondence with the set of natural numbers
by pairing each natural number with 2 times itself.
Therefore the set of even natural numbers has a cardinality of aleph-naught.
The set of integers can also be put in a one-to-one correspondence with the set natural numbers
so it also has a cardinality of aleph-naught.
And the set of positive rational numbers, which is tabulated here,
with all positive numerators and denominators considered and duplicates greyed out,
can also be put in a one-to-one correspondence with the set of natural numbers
so it too has a cardinality of aleph-naught.
And if we throw 0 in the list and have each number followed by it’s negative,
we have a one-to-one correspondence between the set of all rational numbers and the set of natural numbers,
so the set of rational numbers also has a cardinality of aleph-naught.
We say that any set which has a cardinality of aleph-naught is countably infinite.
But we can’t say that about the set of real numbers.
There’s no way to neatly lay the real numbers out.
We can’t do anything better than listing them randomly.
And if you give me any random list of real numbers,
I can name a real number whose tenth digit is different from the first number’s,
and whose hundredth digit is different from the second number’s,
and whose thousandth digit is different from the third number’s, and so on.
Which means that I can name a number that is different from every number on that list,
so the complete set of real numbers cannot be listed.
The cardinality of the real numbers must be greater than aleph-naught.
So we say that it is uncountably infinite.
But how big is it?
For the moment, let’s work in binary.
With 1 digit, we can uniquely describe 2 numbers, 2 to the power of 1.
With 2 digits, we can uniquely describe 4 numbers, 2 to the power of 2.
With 3 digits, we can uniquely describe 8 numbers, 2 to the power of 3, and so on.
And since aleph-naught digits are required to describe the real numbers,
there must be 2 to the aleph-naught real numbers.
The real number line is composed of 2 to the aleph-naught points.
Now, let’s conclude by answering this question:
Points have 0 length, so how can we assemble a collection of 0 length objects to produce a line having length?
To answer this question, we’re going to make use of geometric series.
Let’s assume that this geometric series sums to s.
Multiplying it by r, we get rs equals r + r^2 + r^3 and so on.
And subtracting the bottom from the top, we get s-rs=1.
Factoring out the s and solving for s, we get s = 1/1-r.
And so this infinite series equals r/1-r.
So for example, when r equals 1/11, we get 0.1 equals this infinite series.
Now, imagine a ribbon whose length is 0.1.
We can cut that ribbon such that the first piece has a length of 1/11,
the second piece has a length of 1/11 squared,
the third piece has a length of 1/11 cubed, and so on.
These pieces get incredibly small, so you can’t see them on the screen, but they’re there.
There are aleph-naught pieces of ribbon, each with a non-zero length. 
Now, earlier I showed you that there are aleph-naught rational numbers.
So we can assign a piece ribbon to each rational number.
And then we can cover each rational number by its assigned ribbon.
Now, there may be some overlapping ribbon, so we can conclude
that the rational numbers take up at most 0.1 length on the real number line.
But our choice of a ribbon length of 0.1 was arbitrary.
If we chose r to be 1/101, we’d have a ribbon length of 0.01
and we would conclude that the rational numbers take up at most 0.01 length on the real number line.
And if we chose r to be 1/1001, we’d have a ribbon length of 0.001,
and we would conclude that the rational numbers take up at most 0.001 length on the real number line.
So how long are the rational numbers?
Well, they must have a length less than 0.1, less than 0.01, less than 0.001, and so on
and the only length less than all numbers in this sequence is 0.
The rational numbers must have a length of 0.
Or better put, the rational numbers have measure zero.
Which means that the length, the measure, of the real number line must come from the irrational numbers.
The real number line is continuous,
it is composed of 2 to the power of aleph-naught points,
and its length, it’s measure, comes from the irrational numbers.
I’m Ryan O’Connor, and thanks for watching.
