1 00:00:00,000 --> 00:00:03,800 Rationality of numbers, so 
this is the last topic of this 2 00:00:03,800 --> 00:00:08,200 video. Prove that the square root of 5 
plus the square root of 3 is irrational. 3 00:00:08,200 --> 00:00:10,733 Now I often get students very 
confused about this because it, 4 00:00:10,733 --> 00:00:13,566 you know, you just think, “Oh well I 
know that square root of 5 is irrational 5 00:00:13,566 --> 00:00:16,200 and square root of 3 is irrational, so if 
I add them it's going to be irrational.” 6 00:00:16,200 --> 00:00:20,000 That logic’s wrong, right? I mean square 
root of 2 is irrational and then negative 7 00:00:20,000 --> 00:00:21,733 square root of 2 is irrational, 8 00:00:21,733 --> 00:00:23,700 so if I add those together I get 0, 9 00:00:23,700 --> 00:00:26,100 which is definitely a rational 
number, it's an integer. 10 00:00:26,100 --> 00:00:28,133 11 00:00:28,133 --> 00:00:31,566 So you have to be a little bit careful, it's not 
just enough to say that each of these terms 12 00:00:31,566 --> 00:00:33,666 is individually irrational, 13 00:00:33,666 --> 00:00:35,800 you need to say something about the sum 14 00:00:35,800 --> 00:00:36,266 15 00:00:36,266 --> 00:00:39,566 of these two terms. So you can add two 
irrational numbers and get something rational, 16 00:00:39,566 --> 00:00:42,866 that is possible. Not likely, but it's possible. 17 00:00:42,866 --> 00:00:44,600 18 00:00:44,600 --> 00:00:47,333 Give this a shot, pause the video, 
see if you can determine this. 19 00:00:47,333 --> 00:00:51,466 If you can't figure this out, you should at least 
know what kind of proof technique to use. 20 00:00:51,466 --> 00:00:53,733 That at least counts for something. 21 00:00:53,733 --> 00:00:54,900 22 00:00:54,900 --> 00:00:57,066 Now that you're back, hopefully 
you've at least approached this by 23 00:00:57,066 --> 00:00:59,866 using a contradiction argument. 24 00:00:59,866 --> 00:01:02,900 Proving that something 
is irrational is difficult 25 00:01:02,900 --> 00:01:06,833 because it's a “not” condition, right? Being 
irrational means you're not rational, 26 00:01:06,833 --> 00:01:09,933 because of that “not” condition, 
you usually want to 27 00:01:09,933 --> 00:01:12,433 proceed by contradiction or contrapositive something like this. 28 00:01:12,433 --> 00:01:16,300 Here we don't have an implication 
so contrapositive isn’t a possibility, 29 00:01:16,300 --> 00:01:18,366 and to be honest with you, contradiction 
is just always stronger anyway, 30 00:01:18,366 --> 00:01:20,400 so I always like to use contradiction. 31 00:01:20,400 --> 00:01:21,333 32 00:01:21,333 --> 00:01:24,833 So I'm going to do a proof by contradiction, we're 
going to assume that this number is rational, 33 00:01:24,833 --> 00:01:26,800 and see where we can go from there. 34 00:01:26,800 --> 00:01:27,833 35 00:01:27,833 --> 00:01:30,300 So I'm going to assume that this number, 
square root of 5 plus square root of 3 36 00:01:30,300 --> 00:01:32,633 is equal to some x, and that x is rational. 37 00:01:32,633 --> 00:01:33,566 38 00:01:33,566 --> 00:01:35,900 Now what am I going to do? Well it 
seems natural to want to square, 39 00:01:35,900 --> 00:01:40,000 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. 40 00:01:40,000 --> 00:01:41,366 41 00:01:41,366 --> 00:01:44,000 Now it's a little bit ugly and 
I still have a square root, but 42 00:01:44,000 --> 00:01:46,466 instead of having two square roots I only have one. 43 00:01:46,466 --> 00:01:48,800 So it seems…it stands 
to reason that if I 44 00:01:48,800 --> 00:01:51,633 isolate for that square 
root, like I've done here, 45 00:01:51,633 --> 00:01:52,433 46 00:01:52,433 --> 00:01:54,800 and I square again, I'm going 
to get rid of the square root. 47 00:01:54,800 --> 00:01:56,900 I'm going to be left with something to deal with. 48 00:01:56,900 --> 00:01:57,533 49 00:01:57,533 --> 00:02:01,033 So I square again, and I 
get this polynomial here. 50 00:02:01,033 --> 00:02:03,366 I'm going to isolate by bringing 
it over and now I'm left with 51 00:02:03,366 --> 00:02:08,266 0 is equal to x to the power 4, 
minus 16x squared plus 4x. 52 00:02:08,266 --> 00:02:12,533 Now what can I do? Well if I think about this - 
so again x was a given rational number… 53 00:02:12,533 --> 00:02:14,066 54 00:02:14,066 --> 00:02:17,300 oh there should be no x, 
here. It should just be plus 4, 55 00:02:17,300 --> 00:02:18,733 56 00:02:18,733 --> 00:02:21,300 I apologize for that typo. 57 00:02:21,300 --> 00:02:24,000 So now I have this plus 4, okay? 58 00:02:24,000 --> 00:02:27,866 So by the Rational Roots 
Theorem, I know that the 59 00:02:27,866 --> 00:02:30,933 only possible roots have 
numerator that's a factor of 4, 60 00:02:30,933 --> 00:02:33,100 and the denominator that's a factor of x, 61 00:02:33,100 --> 00:02:36,333 or a factor the coefficient 
of x to the 4 which is 1. 62 00:02:36,333 --> 00:02:37,966 This only leaves me with 63 00:02:37,966 --> 00:02:41,600 6 roots: plus minus 1, plus 
minus 2, plus minus 4. 64 00:02:41,600 --> 00:02:43,733 Now what you have to do - so 
I didn't do this I just said it, 65 00:02:43,733 --> 00:02:46,233 a quick check shows that none of these work. You have to actually plug these in 66 00:02:46,233 --> 00:02:49,633 and make sure that none of these roots are actually roots of this thing, right? 67 00:02:49,633 --> 00:02:53,033 It's possible that the square root of 5 plus the square root of 3 is equal to 2, 68 00:02:53,033 --> 00:02:57,333 or 4, or minus 1, it's 
possible, a priori, right? 69 00:02:57,333 --> 00:03:00,866 So make sure that none of these roots 
actually work, and then we have a contradiction. 70 00:03:00,866 --> 00:03:03,400 Well why? Well we assumed that x, 71 00:03:03,400 --> 00:03:07,266 right, which is a rational number, is a 
root of the polynomial that I'm given by 72 00:03:07,266 --> 00:03:11,000 x to the 4 minus 16x squared plus 4. Get rid of this x. 73 00:03:11,000 --> 00:03:12,566 74 00:03:12,566 --> 00:03:15,000 But of course it's not, 
right, none of these work 75 00:03:15,000 --> 00:03:18,166 and so this polynomial - this rational number, 76 00:03:18,166 --> 00:03:21,600 or this number that we assumed was rational, 
doesn't actually satisfy this polynomial, 77 00:03:21,600 --> 00:03:24,000 that gives us a contradiction. 78 00:03:24,000 --> 00:03:26,000 Now what was our contradiction? 
Well we assumed that this 79 00:03:26,000 --> 00:03:28,666 number square root of 5 plus 
square root of 3 is rational, 80 00:03:28,666 --> 00:03:30,633 therefore it must be irrational. 81 00:03:30,633 --> 00:03:32,100 82 00:03:32,100 --> 00:03:35,000 So that's great, hopefully this 
gave you a little bit of insight 83 00:03:35,000 --> 00:03:36,766 as to how to solve some problems 84 00:03:36,766 --> 00:03:39,133 where we need to factor, 
try to find rational roots, or 85 00:03:39,133 --> 00:03:42,000 conjugate roots, or proving 
that numbers are rational. 86 00:03:42,000 --> 00:03:44,100 It's pretty neat stuff, there's 
lots of applications of this. 87 00:03:44,100 --> 00:03:46,666 We just talked about 
a couple in this video, 88 00:03:46,666 --> 00:03:48,066 but hopefully 89 00:03:48,066 --> 00:03:50,666 they give you an idea 
and you can at least 90 00:03:50,666 --> 00:03:53,533 try to figure out some new 
problems based on these ideas. 91 00:03:53,533 --> 00:03:54,200 92 00:03:54,200 --> 00:03:57,633 So thank you very much for 
listening and all the best.