Rationality of numbers, so this is the last topic of this
video. Prove that the square root of 5 plus the square root of 3 is irrational.
Now I often get students very confused about this because it,
you know, you just think, “Oh well I know that square root of 5 is irrational
and square root of 3 is irrational, so if I add them it's going to be irrational.”
That logic’s wrong, right? I mean square root of 2 is irrational and then negative
square root of 2 is irrational,
so if I add those together I get 0,
which is definitely a rational number, it's an integer.
So you have to be a little bit careful, it's not just enough to say that each of these terms
is individually irrational,
you need to say something about the sum
of these two terms. So you can add two irrational numbers and get something rational,
that is possible. Not likely, but it's possible.
Give this a shot, pause the video, see if you can determine this.
If you can't figure this out, you should at least know what kind of proof technique to use.
That at least counts for something.
Now that you're back, hopefully you've at least approached this by
using a contradiction argument.
Proving that something is irrational is difficult
because it's a “not” condition, right? Being irrational means you're not rational,
because of that “not” condition, you usually want to
proceed by contradiction or contrapositive something like this.
Here we don't have an implication so contrapositive isn’t a possibility,
and to be honest with you, contradiction is just always stronger anyway,
so I always like to use contradiction.
So I'm going to do a proof by contradiction, we're going to assume that this number is rational,
and see where we can go from there.
So I'm going to assume that this number, square root of 5 plus square root of 3
is equal to some x, and that x is rational.
Now what am I going to do? Well it seems natural to want to square,
I want to try to get rid of these square roots, so I'm going to square both sides, and I'm left with this thing.
Now it's a little bit ugly and I still have a square root, but
instead of having two square roots I only have one.
So it seems…it stands to reason that if I
isolate for that square root, like I've done here,
and I square again, I'm going to get rid of the square root.
I'm going to be left with something to deal with.
So I square again, and I get this polynomial here.
I'm going to isolate by bringing it over and now I'm left with
0 is equal to x to the power 4, minus 16x squared plus 4x.
Now what can I do? Well if I think about this - so again x was a given rational number…
oh there should be no x, here. It should just be plus 4,
I apologize for that typo.
So now I have this plus 4, okay?
So by the Rational Roots Theorem, I know that the
only possible roots have numerator that's a factor of 4,
and the denominator that's a factor of x,
or a factor the coefficient of x to the 4 which is 1.
This only leaves me with
6 roots: plus minus 1, plus minus 2, plus minus 4.
Now what you have to do - so I didn't do this I just said it,
a quick check shows that none of these work. You have to actually plug these in
and make sure that none of these roots are actually roots of this thing, right?
It's possible that the square root of 5 plus the square root of 3 is equal to 2,
or 4, or minus 1, it's possible, a priori, right?
So make sure that none of these roots actually work, and then we have a contradiction.
Well why? Well we assumed that x,
right, which is a rational number, is a root of the polynomial that I'm given by
x to the 4 minus 16x squared plus 4. Get rid of this x.
But of course it's not, right, none of these work
and so this polynomial - this rational number,
or this number that we assumed was rational, doesn't actually satisfy this polynomial,
that gives us a contradiction.
Now what was our contradiction? Well we assumed that this
number square root of 5 plus square root of 3 is rational,
therefore it must be irrational.
So that's great, hopefully this gave you a little bit of insight
as to how to solve some problems
where we need to factor, try to find rational roots, or
conjugate roots, or proving that numbers are rational.
It's pretty neat stuff, there's lots of applications of this.
We just talked about a couple in this video,
but hopefully
they give you an idea and you can at least
try to figure out some new problems based on these ideas.
So thank you very much for listening and all the best.