Hello.
I'm Professor Von Schmohawk and welcome to Why U.
In the previous lecture
we saw that although the Greeks originally thought that all numbers were rational
they eventually discovered that there were quantities
that could not be represented by rational numbers.
These quantities formed an infinitely large group of numbers, called the irrational numbers.
There are also an infinite number of rational numbers.
This is obvious since the rational numbers include all the integers
and there are an infinite number of integers.
But there are also an infinite number of rational numbers between any two integers.
Take for example the two integers zero and one.
Between zero and one is the rational number one-half.
Between zero and one-half is one-fourth.
Between zero and one-fourth is one-eighth and so on.
You can continue this forever
naming an infinite number of rational numbers between zero and one.
In fact, you can also show that
between any pair of rational numbers there are an infinite number of rational numbers.
You can prove this by starting with two arbitrary rational numbers
and then calculating a number exactly half-way between them.
If we can show that this number must also be rational
then we have proven that between every pair of rational numbers is another rational number.
Likewise, between this number and the two rational numbers on either side
must be two more rational numbers.
We can keep doing this forever
proving that are an infinite number of rational numbers between every pair of rational numbers.
To do this, let's take two arbitrary rational numbers on the number line
n over m, and p over q.
To make it simple, let's assume that both these numbers are positive.
To get the distance between these two numbers
we subtract the smaller number from the larger number.
Now if we add half that distance
to the smaller number
we get the distance from the origin, to the point half-way between the two numbers.
This point must also be a rational number
since it is composed of rational numbers added, subtracted, and multiplied together.
Since rational numbers are closed under all the arithmetic operations
the results of these operations must also be rational.
So we have proven that between any pair of rational numbers is another rational number.
As we saw, this means that there are an infinite number of rational numbers
between any pair of rational numbers
and therefore an infinite number in any region of the number line.
All the points representing rational numbers blend together to look like a solid line
going off to infinity in both the positive and negative directions.
No matter how close we zoom into any region on the number line
we will always find an infinite number of rational numbers.
You might think that this would leave no room for gaps.
However, you can also show that between any pair of rational numbers
there are an infinite number of irrational numbers.
Therefore, there are gaps between every pair of rational numbers
in fact, an infinite number of gaps which are filled by irrational numbers.
Every region of the number line
contains an infinite number of rational and irrational numbers.
We now have two mutually-exclusive collections of numbers.
Each collection can represent some quantities but not others.
However, if we combine the rational and irrational numbers into a single group
then we can represent any type of quantity in the real world.
This collection, composed of rational and irrational numbers, is called the real numbers.
We have seen that there are an infinite number of rational numbers
and when represented on the number line, the points appear to fill in the entire line.
We have also seen that the rational numbers include all the integers.
So if we pick some random location on the number line
what are the chances that this location will correspond to an integer
as opposed to some other number?
Remember that points have zero width.
So if we pick a purely random location
the chances are essentially zero that that location will correspond exactly to an integer.
This is because any finite interval on the number line
contains an infinite number of rational numbers but only a finite number of integers.
Since any interval contains an infinite number of rational numbers
you might think that you would almost certainly pick a rational number.
However, remember that every interval
also contains an infinite number of irrational numbers.
So maybe the chances would be fifty-fifty that you would pick a rational number
as opposed to an irrational number.
However, this is not the case.
The chances are essentially zero that you would pick a rational number.
This is because there are vastly more irrational numbers than rational numbers.
But how can that be true if both collections of numbers are infinite?
Until the late 1800s, mathematicians believed that
nothing could be larger than anything that was infinitely large.
But in 1874, Georg Cantor proved that there were different sizes of infinities.
Rational numbers, integers, whole numbers, and natural numbers were infinite
but a particular type of infinite he called "countable".
Irrational numbers were also infinite, but what he called "uncountable"
a much larger infinity than the countable rational numbers.
Cantor established a hierarchy of infinities.
We saw that real numbers are formed by combining the rational numbers and the irrational numbers.
Since real numbers include irrational numbers which are uncountable
real numbers are also uncountable.
In other words, the infinite collection of real numbers
is the same size infinity as the infinite collection of irrational numbers.
Unlike the rational and irrational numbers
every point on the number line corresponds to some real number.
If you randomly pick a point on the number line
it will always correspond to a real number.
There are no gaps in the real number line.
This means that you can move smoothly between any two points
and never encounter anything but real numbers.
Since there are no gaps, the real numbers form a continuum
a property which is essential for calculus.
So now we know that between any pair of irrational numbers
are an infinite number of rational numbers
and between any pair of rational numbers are an infinite number of irrational numbers.
And don't forget that there are infinitely more irrational numbers than rational numbers.
If this makes your head want to explode then congratulations!
You are on your way to becoming a mathematician.