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# Proof: there's an irrational number between any two rational numbers

Sal proves that when given any two rational numbers, no matter how close, we can find an irrational number that lies between them. Created by Sal Khan.

Video transcript

What I want to do in this video
is prove that between any two rational numbers-- so let's say
that's a rational number there, and then let's say that this
is another rational number that is larger than
this one right over here-- that between any
two rational numbers, you can find an
irrational number. So that number right
over there is irrational. You can find at least
one irrational number. And that's kind of
crazy, because there's a lot of rational numbers. There's an infinite number
of rational numbers. So we're saying between any
two of those rational numbers, you can always find
an irrational number. And we're going to start
thinking about it by just thinking about the
interval between 0 and 1. So if we think about the
interval between 0 and 1, we know that there are
irrational numbers there. In fact, one of them
that might pop out at you is 1 over the square
root of 2, which is the same thing as the
square root of 2 over 2, is equal-- I
shouldn't say equal, is roughly, is approximately
equal to 0.70710678118. And I could just
keep going on and on and on and on and on and on. This thing does not repeat. But the important point is,
it's clearly between 0 and 1. So I could write 1 over
the square root of 2 is clearly between 0 and 1. So the way that I'm going
to prove that there's an irrational number between
any two rational numbers is I'm going to start with
this set of inequalities, and I'm going to
manipulate it so I end up with an r1 over here
and an r2 over here. And then from 1 over
the square root of 2, I would have manipulated
this to construct that irrational-- at least
one of the irrational numbers that's between those
two rational ones. So instead of making this
an interval between 0 and 1, let's make this an
interval between 0 and the difference
between these two numbers. So the distance between
r1 and r2 is r2 minus r1. So let's multiply
both sides of this-- or all three parts
of this inequality, I guess I could say, by
r2 times r2 minus r1. So let's do that. So if you multiply this,
0 times r2 minus r1, well you're just still
going to have 0 there, is less than-- And we know
that r2 is greater than r1, so r2 minus-- let me make
it clear what we're doing. We're going to multiply
everything times r2 minus r1. r2, we're assuming,
is greater than r1, so this thing right over here
is going to be greater than 0. So if you multiply
the different sides of an inequality by
something greater than 0, you don't switch the inequality. So 0 times that is 0, 1 over
the square root of 2 times that is going to be 1
over the square root of 2 times r2 minus r1. And then that's going
to be less than-- well, 1 times that is just
going to be r2 minus r1. And now, we just have to kind
of shift everything over. So let's add r1 to
all sides of this. So if we add something to
all parts of the inequality, then that's also not going
to change the inequality. So we're going to
add r1 over here. We can add r1 over here. And we can add r1 over there. And so on the
left-hand side, we have r1 is less than r1 plus--
let me just copy and paste all of this so I don't have
to keep changing colors. Whoops, that's not
what I wanted to do. Let me do this. There you go. All right. That should be pretty good. So copy and paste that. r1 plus this, plus that--
let me write the plus down-- plus that, is less
than-- that one is a different shade of
blue-- is less than-- well, what's r1 plus r2 minus r1? Well, that's just
going to be r2. So I've just shown
you that you give me any two rational
numbers, and I'm assuming r2 is greater
than r1, I have just constructed an
irrational number that's going to be between those
two rational numbers. You take r1, you take the
lower of the rational numbers, and to that you add
1 over square root of 2 times the difference
between those two rational numbers,
and you are going to get this right over here
is an irrational number. You're saying hey, how do
I know that this thing-- how can I be satisfied that
this thing is irrational? Well, we've already seen. You take the product of an
irrational and a rational, you get an irrational number. You take the sum of
an irrational number and a rational number, you
get an irrational number. So we've constructed
an irrational number that's between
these two rationals.