# Introduction to rational and irrational numbers | Algebra I | Khan Academy

December 3, 2019

So let’s talk a little bit
about rational numbers. And the simple way to think
about it is any number that can be represented as
the ratio of two integers is a rational number. So for example, any integer
is a rational number. 1 can be represented as 1/1 or
as negative 2 over negative 2 or as 10,000/10,000. In all of these cases, these are
all different representations of the number 1,
ratio of two integers. And I obviously can
have an infinite number of representations
of 1 in this way, the same number over
the same number. The number negative 7 could be
represented as negative 7/1, or 7 over negative 1, or
negative 14 over positive 2. And I could go on, and
on, and on, and on. So negative 7 is definitely
a rational number. It can be represented as
the ratio of two integers. But what about things
that are not integers? For example, let us imagine–
oh, I don’t know– 3.75. How can we represent that as
the ratio of two integers? Well, 3.75, you
could rewrite that as 375/100, which is the
same thing as 750/200. Or you could say, hey,
3.75 is the same thing as 3 and 3/4– so let
me write it here– which is the same
thing as– that’s 15/4. 4 times 3 is 12, plus 3 is
15, so you could write this. This is the same thing as 15/4. Or we could write this as
negative 30 over negative 8. I just multiplied the
numerator and the denominator here by negative 2. But just to be clear,
this is clearly rational. I’m giving you multiple
examples of how this can be represented as
the ratio of two integers. Now, what about
repeating decimals? Well, let’s take
maybe the most famous of the repeating decimals. Let’s say you have 0.333, just
keeps going on and on forever, which we can denote by
putting that little bar on top of the 3. This is 0.3 repeating. And we’ve seen–
and later we’ll show how you can convert
any repeating decimal as the ratio of two integers–
this is clearly 1/3. Or maybe you’ve seen things like
0.6 repeating, which is 2/3. And there’s many, many,
many other examples of this. And we’ll see any
repeating decimal, not just one digit repeating. Even if it has a million
digits repeating, as long as the pattern
starts to repeat itself over and over and
over again, you can always represent that as
the ratio of two integers. So I know what you’re
probably thinking. Hey, Sal, you’ve
just included a lot. You’ve included all
of the integers. You’ve included all of finite
non-repeating decimals, and you’ve also included
repeating decimals. What is left? Are there any numbers
that are not rational? And you’re probably
guessing that there are, otherwise people
wouldn’t have taken the trouble of trying to
label these as rational. And it turns out– as you
can imagine– that actually some of the most famous
numbers in all of mathematics are not rational. And we call these numbers
irrational numbers. And I’ve listed there
just a few of the most noteworthy examples. Pi– the ratio of
the circumference to the diameter of a circle–
is an irrational number. It never terminates. It goes on and on and on
forever, and it never repeats. e, same thing– never
terminates, never repeats. It comes out of continuously
compounding interest. It comes out of
complex analysis. e shows up all over the place. Square root of 2,
irrational number. Phi, the golden ratio,
irrational number. So these things that
really just pop out of nature, many of these
numbers are irrational. Now, you might say, OK,
are these irrational? These are just these
special kind of numbers. But maybe most
numbers are rational, and Sal’s just picked out
some special cases here. But the important thing to
realize is they do seem exotic, and they are exotic
in certain ways. But they aren’t uncommon. It actually turns out
that there is always an irrational number between
any two rational numbers. Well, we could go on and on. There’s actually
an infinite number. But there’s at least one,
so that gives you an idea that you can’t
really say that there are fewer irrational numbers
than rational numbers. And in a future
video, we’ll prove that you give me two rational
numbers– rational 1, rational 2– there’s going to be
at least one irrational number between those, which
is a neat result, because irrational
numbers seem to be exotic. Another way to think about it–
I took the square root of 2, but you take the square root
of any non-perfect square, you’re going to end up
with an irrational number. You take the sum
of an irrational and a rational number– and
we’ll see this later on. We’ll prove it to ourselves. The sum of an irrational
and a rational is going to be irrational. The product of an
irrational and a rational is going to be irrational. So there’s a lot, a lot, a
lot of irrational numbers out there.