WEBVTT 00:00:00.000 --> 00:00:01.464 align:middle line:90% [SQUEAKING] 00:00:01.464 --> 00:00:02.928 align:middle line:90% [RUSTLING] 00:00:02.928 --> 00:00:05.368 align:middle line:90% [CLICKING] 00:00:05.368 --> 00:00:14.180 align:middle line:90% 00:00:14.180 --> 00:00:16.309 align:middle line:84% CASEY RODRIGUEZ: OK, so let's finish the proof 00:00:16.309 --> 00:00:23.380 align:middle line:84% that we started at the end of the lecture last time. 00:00:23.380 --> 00:00:25.750 align:middle line:84% I usually don't end lectures in the middle of a proof. 00:00:25.750 --> 00:00:30.020 align:middle line:84% But since this lecture and the lecture 00:00:30.020 --> 00:00:33.320 align:middle line:84% before will come to you at the same time, 00:00:33.320 --> 00:00:36.650 align:middle line:90% I didn't feel so bad about it. 00:00:36.650 --> 00:00:40.040 align:middle line:84% All right, so we were trying to prove this theorem 00:00:40.040 --> 00:00:44.690 align:middle line:84% that if there's a rational number so that it's 00:00:44.690 --> 00:00:46.820 align:middle line:84% equal to the supremum of the set, 00:00:46.820 --> 00:00:51.620 align:middle line:84% then x is bigger than or equal to 1 and its square gives me 2. 00:00:51.620 --> 00:00:54.170 align:middle line:90% 00:00:54.170 --> 00:00:58.160 align:middle line:84% And so far, what we've shown is that if I 00:00:58.160 --> 00:01:01.790 align:middle line:84% have an element, a rational number that 00:01:01.790 --> 00:01:05.060 align:middle line:84% is given by the supremum, then it's bigger than or equal to 1 00:01:05.060 --> 00:01:07.970 align:middle line:84% and x squared has to be bigger than or equal to 2. 00:01:07.970 --> 00:01:11.060 align:middle line:84% And so now, we would like to show that, in fact, x squared 00:01:11.060 --> 00:01:13.550 align:middle line:90% equals 2. 00:01:13.550 --> 00:01:17.120 align:middle line:84% Now, since x squared is bigger than or equal to 2, 00:01:17.120 --> 00:01:19.460 align:middle line:84% it's either equal to 2 or bigger than 2. 00:01:19.460 --> 00:01:22.940 align:middle line:84% And we want to show that the second possibility is not 00:01:22.940 --> 00:01:24.770 align:middle line:90% possible. 00:01:24.770 --> 00:01:27.350 align:middle line:84% So we're going to prove that the second possibility is not 00:01:27.350 --> 00:01:30.870 align:middle line:90% possible by contradiction. 00:01:30.870 --> 00:01:35.870 align:middle line:84% So let's assume that the thing that we say cannot happen does 00:01:35.870 --> 00:01:39.140 align:middle line:84% happen and we're going to derive a false statement. 00:01:39.140 --> 00:01:41.840 align:middle line:90% 00:01:41.840 --> 00:01:48.090 align:middle line:90% So define h. 00:01:48.090 --> 00:01:55.020 align:middle line:84% This is going to be x squared minus 2 over 2x 00:01:55.020 --> 00:02:04.170 align:middle line:84% And so note a couple of things that since x squared 00:02:04.170 --> 00:02:12.720 align:middle line:84% is bigger than 2, this implies h is bigger than 0, which implies 00:02:12.720 --> 00:02:17.190 align:middle line:90% x minus h is less than x. 00:02:17.190 --> 00:02:19.860 align:middle line:84% So what is the contradiction going to be? 00:02:19.860 --> 00:02:24.090 align:middle line:84% It's going to be that this element x minus h is, 00:02:24.090 --> 00:02:28.650 align:middle line:84% in fact, an upper bound for the set E. 00:02:28.650 --> 00:02:31.590 align:middle line:84% And this will contradict the fact that x is the supremum. 00:02:31.590 --> 00:02:33.510 align:middle line:84% In other words, it's the least upper bound 00:02:33.510 --> 00:02:36.450 align:middle line:84% for the set E. There's no upper bound for E that 00:02:36.450 --> 00:02:37.740 align:middle line:90% can be less than x. 00:02:37.740 --> 00:02:42.188 align:middle line:90% 00:02:42.188 --> 00:03:02.020 align:middle line:84% All right, so we now prove that x minus h 00:03:02.020 --> 00:03:12.560 align:middle line:84% is an upper bound for this set E. Now, 00:03:12.560 --> 00:03:15.170 align:middle line:84% let's take a look at x minus h squared. 00:03:15.170 --> 00:03:29.390 align:middle line:90% 00:03:29.390 --> 00:03:36.365 align:middle line:84% Then this is equal to x squared minus 2xh plus h squared. 00:03:36.365 --> 00:03:40.890 align:middle line:90% 00:03:40.890 --> 00:03:42.830 align:middle line:84% Now, just given by if we just plug 00:03:42.830 --> 00:03:46.790 align:middle line:84% in what h is, this is equal to x squared 00:03:46.790 --> 00:03:52.481 align:middle line:90% minus x squared minus 2. 00:03:52.481 --> 00:03:54.290 align:middle line:90% So that's putting h in. 00:03:54.290 --> 00:03:57.110 align:middle line:84% And then the 2x here cancels with the 2x there. 00:03:57.110 --> 00:04:01.340 align:middle line:84% So I just get x squared minus 2 plus h squared, which 00:04:01.340 --> 00:04:03.930 align:middle line:90% equals 2 plus h squared. 00:04:03.930 --> 00:04:05.790 align:middle line:84% Now remember, h is a positive number. 00:04:05.790 --> 00:04:09.845 align:middle line:84% So 2 plus a positive number, this is bigger than 2. 00:04:09.845 --> 00:04:14.140 align:middle line:90% 00:04:14.140 --> 00:04:19.600 align:middle line:84% So x minus h squared is bigger than 2. 00:04:19.600 --> 00:04:21.430 align:middle line:90% So now, I want to show-- 00:04:21.430 --> 00:04:23.470 align:middle line:84% I'm going to use this to show that x minus h 00:04:23.470 --> 00:04:34.990 align:middle line:84% is an upper bound for E So let q be in E. q squared 00:04:34.990 --> 00:04:37.210 align:middle line:90% is less than 2. 00:04:37.210 --> 00:04:42.540 align:middle line:90% 00:04:42.540 --> 00:04:47.730 align:middle line:84% Then I have q squared is less than 2, which is less than x 00:04:47.730 --> 00:04:53.880 align:middle line:84% minus h squared by what we just proved here. 00:04:53.880 --> 00:05:00.110 align:middle line:84% So let me write this out or just summarize 00:05:00.110 --> 00:05:02.780 align:middle line:84% that what we proved here was x minus h squared 00:05:02.780 --> 00:05:04.670 align:middle line:90% is bigger than 2. 00:05:04.670 --> 00:05:07.730 align:middle line:84% So we have q squared is less than 2 is less than x 00:05:07.730 --> 00:05:08.970 align:middle line:90% minus h squared. 00:05:08.970 --> 00:05:15.525 align:middle line:84% So that means 0 is bigger than x minus h squared minus q 00:05:15.525 --> 00:05:16.025 align:middle line:90% squared. 00:05:16.025 --> 00:05:19.280 align:middle line:90% 00:05:19.280 --> 00:05:22.160 align:middle line:84% q squared is less than this so than if I just subtract it 00:05:22.160 --> 00:05:23.940 align:middle line:90% I get this inequality. 00:05:23.940 --> 00:05:27.440 align:middle line:84% So that implies that 0 is bigger than-- 00:05:27.440 --> 00:05:29.400 align:middle line:84% now, this is the difference of two squares. 00:05:29.400 --> 00:05:38.290 align:middle line:84% So I can write this as x minus h plus q minus q. 00:05:38.290 --> 00:05:42.810 align:middle line:90% 00:05:42.810 --> 00:05:49.640 align:middle line:84% Now, if I just write out what x minus h 00:05:49.640 --> 00:05:52.550 align:middle line:84% is using the definition of h, this 00:05:52.550 --> 00:06:02.520 align:middle line:84% is equal to x squared plus 2 over 2x plus q over x 00:06:02.520 --> 00:06:07.010 align:middle line:90% minus h minus q. 00:06:07.010 --> 00:06:10.620 align:middle line:84% Now, the product of these two numbers is positive. 00:06:10.620 --> 00:06:13.460 align:middle line:84% This number is positive because q is positive. 00:06:13.460 --> 00:06:21.150 align:middle line:90% 00:06:21.150 --> 00:06:28.770 align:middle line:84% And so that means their sum is positive. 00:06:28.770 --> 00:06:32.220 align:middle line:84% So I have something positive times this. 00:06:32.220 --> 00:06:35.880 align:middle line:84% So I can divide by this positive thing and keep the inequality. 00:06:35.880 --> 00:06:39.230 align:middle line:84% So that implies that 0 is less than x 00:06:39.230 --> 00:06:49.460 align:middle line:84% minus h minus q, which implies that q is less than x minus h. 00:06:49.460 --> 00:06:51.950 align:middle line:84% All right, so we started off with an element of E, 00:06:51.950 --> 00:06:56.630 align:middle line:84% an arbitrary element of E. And we proved that q is less than x 00:06:56.630 --> 00:06:57.245 align:middle line:90% minus h. 00:06:57.245 --> 00:07:00.230 align:middle line:90% 00:07:00.230 --> 00:07:07.370 align:middle line:84% Thus, for all q in E, q is less than x 00:07:07.370 --> 00:07:14.150 align:middle line:84% minus h, which implies that x minus h 00:07:14.150 --> 00:07:32.870 align:middle line:84% is an upper bound for E. So let me keep that board for now 00:07:32.870 --> 00:07:37.170 align:middle line:84% while I write the last sentence or so of this proof. 00:07:37.170 --> 00:07:54.790 align:middle line:90% 00:07:54.790 --> 00:08:02.110 align:middle line:84% So we stated that x minus h is an upper bound for E, which 00:08:02.110 --> 00:08:04.210 align:middle line:84% implies since x is the supremum of E, 00:08:04.210 --> 00:08:23.630 align:middle line:84% it should be less than or equal to all upper bounds, which 00:08:23.630 --> 00:08:25.460 align:middle line:84% implies h is less than or equal to 0. 00:08:25.460 --> 00:08:27.890 align:middle line:84% But this is a direct contradiction 00:08:27.890 --> 00:08:33.780 align:middle line:90% to h being a positive number. 00:08:33.780 --> 00:08:40.100 align:middle line:84% So this is the false statement we ultimately prove. 00:08:40.100 --> 00:08:44.720 align:middle line:84% But the idea is that if x squared is bigger than 2, 00:08:44.720 --> 00:08:46.280 align:middle line:90% you can find a lower bound-- 00:08:46.280 --> 00:08:48.020 align:middle line:84% I mean an upper bound, which is strictly 00:08:48.020 --> 00:08:51.110 align:middle line:84% smaller than x, which contradicts the fact that x 00:08:51.110 --> 00:08:59.400 align:middle line:84% is the greatest lower bound for E. 00:08:59.400 --> 00:09:03.030 align:middle line:84% But remember, what was our original assumption to begin 00:09:03.030 --> 00:09:07.410 align:middle line:90% with in this line of reasoning? 00:09:07.410 --> 00:09:10.710 align:middle line:84% That was that x squared is bigger than 2. 00:09:10.710 --> 00:09:14.160 align:middle line:90% 00:09:14.160 --> 00:09:15.705 align:middle line:90% And that concludes the proof. 00:09:15.705 --> 00:09:35.560 align:middle line:90% 00:09:35.560 --> 00:09:40.150 align:middle line:84% All right, so this was a statement about just 00:09:40.150 --> 00:09:43.420 align:middle line:84% if I have some element that's a rational number 00:09:43.420 --> 00:09:47.290 align:middle line:84% and it's equal to the supremum, then it's square has to be 2. 00:09:47.290 --> 00:09:52.750 align:middle line:84% Now, I'm going to prove that basically no such element 00:09:52.750 --> 00:09:56.040 align:middle line:84% exists, and therefore that the rational numbers do not 00:09:56.040 --> 00:09:57.790 align:middle line:84% have the least upper bound property, which 00:09:57.790 --> 00:10:01.000 align:middle line:84% was the discussion above those two 00:10:01.000 --> 00:10:04.000 align:middle line:84% lines in that circle on that board up there. 00:10:04.000 --> 00:10:08.690 align:middle line:84% So what's the theorem we're going to prove? 00:10:08.690 --> 00:10:11.420 align:middle line:90% 00:10:11.420 --> 00:10:16.350 align:middle line:90% The set E as before. 00:10:16.350 --> 00:10:23.300 align:middle line:84% So this is a rational number such that q is bigger than 0. 00:10:23.300 --> 00:10:25.690 align:middle line:90% q squared is less than 2. 00:10:25.690 --> 00:10:39.804 align:middle line:90% 00:10:39.804 --> 00:10:53.740 align:middle line:84% It is bounded above and has no supremum in Q. So 00:10:53.740 --> 00:10:54.985 align:middle line:90% I'm not just saying that-- 00:10:54.985 --> 00:10:57.870 align:middle line:90% 00:10:57.870 --> 00:11:03.000 align:middle line:84% so this set E sits inside my ordered set, 00:11:03.000 --> 00:11:06.090 align:middle line:84% which is my universe Q. I'm not saying that it doesn't have 00:11:06.090 --> 00:11:07.950 align:middle line:84% a supremum in E. I'm saying it doesn't 00:11:07.950 --> 00:11:13.550 align:middle line:84% have a supremum in the universe where it sits. 00:11:13.550 --> 00:11:15.200 align:middle line:84% And therefore, the rational numbers 00:11:15.200 --> 00:11:19.430 align:middle line:84% do not have the least upper bound property 00:11:19.430 --> 00:11:22.670 align:middle line:84% because this set is a set which is bounded above 00:11:22.670 --> 00:11:24.500 align:middle line:90% and has no supremum in q. 00:11:24.500 --> 00:11:31.020 align:middle line:90% 00:11:31.020 --> 00:11:33.870 align:middle line:84% So the first thing I want to show 00:11:33.870 --> 00:11:35.565 align:middle line:84% is that this set E is bounded above. 00:11:35.565 --> 00:11:43.098 align:middle line:90% 00:11:43.098 --> 00:11:51.740 align:middle line:84% Let q be in E. Then q squared is less than 2, 00:11:51.740 --> 00:11:53.120 align:middle line:90% which is less than 4. 00:11:53.120 --> 00:11:55.700 align:middle line:84% So this tells me 4 minus q squared 00:11:55.700 --> 00:12:00.410 align:middle line:84% is positive, which tells me 2 minus 00:12:00.410 --> 00:12:08.610 align:middle line:90% q times 2 plus q is positive. 00:12:08.610 --> 00:12:12.020 align:middle line:84% Now, since 2 is positive and q is positive, 00:12:12.020 --> 00:12:13.220 align:middle line:90% 2 plus q is positive. 00:12:13.220 --> 00:12:15.560 align:middle line:84% So I can divide both sides of this inequality 00:12:15.560 --> 00:12:18.410 align:middle line:84% by this, 2 plus q, without changing 00:12:18.410 --> 00:12:24.080 align:middle line:90% the inequality, which implies-- 00:12:24.080 --> 00:12:26.450 align:middle line:84% I'm just going to flip this inequality. 00:12:26.450 --> 00:12:30.690 align:middle line:90% 00:12:30.690 --> 00:12:36.840 align:middle line:84% Thus, for all q in E, q is less than 2. 00:12:36.840 --> 00:12:39.430 align:middle line:90% 00:12:39.430 --> 00:12:58.620 align:middle line:84% So 2 is an upper bound for the set E. 00:12:58.620 --> 00:13:03.090 align:middle line:84% OK, so that proves that the set E is bounded above. 00:13:03.090 --> 00:13:15.860 align:middle line:90% 00:13:15.860 --> 00:13:19.420 align:middle line:84% So now, I'm going to prove that no supremum mom exists 00:13:19.420 --> 00:13:20.950 align:middle line:84% by using the previous theorem, which 00:13:20.950 --> 00:13:25.870 align:middle line:84% says if I have a supremum, then its square must be 2. 00:13:25.870 --> 00:13:37.370 align:middle line:90% 00:13:37.370 --> 00:13:51.040 align:middle line:84% OK, so now, we show that the sup of E does not exist. 00:13:51.040 --> 00:13:52.835 align:middle line:90% And we do this by contradiction. 00:13:52.835 --> 00:13:55.210 align:middle line:84% Now, I know it seems like a lot of the proofs we're doing 00:13:55.210 --> 00:13:56.620 align:middle line:90% are by contradiction. 00:13:56.620 --> 00:13:58.270 align:middle line:84% This should not give you the impression 00:13:58.270 --> 00:14:01.060 align:middle line:84% that all proofs should be done by contradiction. 00:14:01.060 --> 00:14:05.050 align:middle line:84% Many proofs that you will do in the homework 00:14:05.050 --> 00:14:11.510 align:middle line:84% can be done directly, meaning not by contradiction. 00:14:11.510 --> 00:14:14.360 align:middle line:84% For example, this first little proof 00:14:14.360 --> 00:14:19.160 align:middle line:84% that E is bounded above, this is a direct proof, meaning I 00:14:19.160 --> 00:14:20.870 align:middle line:84% prove directly that it's bounded above. 00:14:20.870 --> 00:14:23.270 align:middle line:84% I do not assume that it's not bounded above 00:14:23.270 --> 00:14:25.880 align:middle line:90% and arrive at a contradiction. 00:14:25.880 --> 00:14:28.570 align:middle line:90% 00:14:28.570 --> 00:14:34.660 align:middle line:84% But proof by contradiction is very tempting 00:14:34.660 --> 00:14:39.550 align:middle line:84% because it at least gets you going somewhere. 00:14:39.550 --> 00:14:41.230 align:middle line:84% If you can assume not, then you get 00:14:41.230 --> 00:14:45.100 align:middle line:84% to a next step, which this assumption implies 00:14:45.100 --> 00:14:45.830 align:middle line:90% something else. 00:14:45.830 --> 00:14:47.622 align:middle line:84% And hopefully, you can keep going until you 00:14:47.622 --> 00:14:49.840 align:middle line:90% arrive at something false. 00:14:49.840 --> 00:14:54.783 align:middle line:84% But many theorems that you want to prove can be done directly. 00:14:54.783 --> 00:14:55.950 align:middle line:90% All right, but not this one. 00:14:55.950 --> 00:15:01.720 align:middle line:84% Now, we show the supremum of the set does not exist. 00:15:01.720 --> 00:15:12.180 align:middle line:84% So by contradiction, so we will assume that such a supremum 00:15:12.180 --> 00:15:12.690 align:middle line:90% does exist. 00:15:12.690 --> 00:15:25.270 align:middle line:90% 00:15:25.270 --> 00:15:35.680 align:middle line:84% So suppose the supremum exists and call it x. 00:15:35.680 --> 00:15:38.110 align:middle line:84% All right, so x is the supremum arm of the set. 00:15:38.110 --> 00:15:42.880 align:middle line:90% 00:15:42.880 --> 00:15:45.760 align:middle line:84% OK, so by the previous theorem, since x 00:15:45.760 --> 00:15:51.710 align:middle line:84% is an element of the rational numbers, 00:15:51.710 --> 00:15:52.910 align:middle line:90% whose square gives me-- 00:15:52.910 --> 00:15:55.700 align:middle line:90% 00:15:55.700 --> 00:15:57.500 align:middle line:84% which is the supremum of the set, 00:15:57.500 --> 00:16:00.866 align:middle line:84% then x is bigger than or equal to 1 and its square is 2. 00:16:00.866 --> 00:16:04.970 align:middle line:90% 00:16:04.970 --> 00:16:14.090 align:middle line:84% By the previous theorem, x is bigger than or equal to 1 00:16:14.090 --> 00:16:15.320 align:middle line:90% and x squared equals 2. 00:16:15.320 --> 00:16:22.650 align:middle line:90% 00:16:22.650 --> 00:16:25.200 align:middle line:84% In fact, x has to be bigger than 1. 00:16:25.200 --> 00:16:29.970 align:middle line:84% It cannot equal 1 because the square would give me 1 not 2. 00:16:29.970 --> 00:16:32.580 align:middle line:84% All right, so I can make that statement 00:16:32.580 --> 00:16:34.260 align:middle line:84% that x is, in fact, bigger than 1, 00:16:34.260 --> 00:16:35.760 align:middle line:84% not just bigger than or equal to. 00:16:35.760 --> 00:16:41.840 align:middle line:90% 00:16:41.840 --> 00:16:52.890 align:middle line:84% OK, so thus, since x is a rational, 00:16:52.890 --> 00:16:58.020 align:middle line:84% there exists m in natural numbers 00:16:58.020 --> 00:17:11.400 align:middle line:84% such that m is bigger than n and x is equal to m over n. 00:17:11.400 --> 00:17:14.400 align:middle line:84% So that just-- that's what it means 00:17:14.400 --> 00:17:20.339 align:middle line:84% for x to be a rational number and for it to be bigger than 1. 00:17:20.339 --> 00:17:26.450 align:middle line:90% 00:17:26.450 --> 00:17:34.340 align:middle line:84% Thus, there exists an n such that n times 00:17:34.340 --> 00:17:38.670 align:middle line:90% x is a natural number. 00:17:38.670 --> 00:17:40.650 align:middle line:84% All right, just multiplying through by n, 00:17:40.650 --> 00:17:43.137 align:middle line:84% that means that n times x is equal to m, 00:17:43.137 --> 00:17:44.220 align:middle line:90% which is a natural number. 00:17:44.220 --> 00:17:49.210 align:middle line:90% 00:17:49.210 --> 00:17:54.860 align:middle line:84% So let S be the set of natural numbers 00:17:54.860 --> 00:18:01.460 align:middle line:84% so that when I multiply x by k I get a natural number. 00:18:01.460 --> 00:18:04.840 align:middle line:84% And so what we've shown, just based on our assumption, 00:18:04.840 --> 00:18:22.450 align:middle line:84% is that S is non-empty because n is in S 00:18:22.450 --> 00:18:24.590 align:middle line:84% Now, this is a subset of natural numbers. 00:18:24.590 --> 00:18:34.380 align:middle line:84% So by the well-ordering property of N, 00:18:34.380 --> 00:19:01.680 align:middle line:84% this set S has a least element, which we'll call k0 in S. 00:19:01.680 --> 00:19:07.410 align:middle line:84% So what I'm going to show is that, in fact, this little guy 00:19:07.410 --> 00:19:12.360 align:middle line:84% k0 is not, in fact, the least element of this set S, 00:19:12.360 --> 00:19:14.625 align:middle line:84% which would contradict exactly how it's defined. 00:19:14.625 --> 00:19:17.460 align:middle line:90% 00:19:17.460 --> 00:19:32.290 align:middle line:84% So define k1 to be k0 x minus k0. 00:19:32.290 --> 00:19:37.270 align:middle line:90% And note this is an integer. 00:19:37.270 --> 00:19:37.810 align:middle line:90% Why? 00:19:37.810 --> 00:19:43.660 align:middle line:84% Because k0 times x-- so k0 is in the set S, meaning k0 times 00:19:43.660 --> 00:19:44.740 align:middle line:90% x is a natural number. 00:19:44.740 --> 00:19:47.260 align:middle line:90% So it's an integer. 00:19:47.260 --> 00:19:48.820 align:middle line:90% k0 is a natural number. 00:19:48.820 --> 00:19:53.680 align:middle line:84% So the difference of two integers is, again, an integer. 00:19:53.680 --> 00:19:55.800 align:middle line:90% So k0 is a natural number. 00:19:55.800 --> 00:19:58.330 align:middle line:90% k0 times x is a natural number. 00:19:58.330 --> 00:20:00.880 align:middle line:84% So the difference of two natural numbers gives me an integer. 00:20:00.880 --> 00:20:04.050 align:middle line:90% 00:20:04.050 --> 00:20:05.970 align:middle line:84% But in fact, it's a natural number. 00:20:05.970 --> 00:20:19.560 align:middle line:90% 00:20:19.560 --> 00:20:24.295 align:middle line:90% So it's a natural number. 00:20:24.295 --> 00:20:31.608 align:middle line:90% 00:20:31.608 --> 00:20:33.300 align:middle line:90% So let's go over here. 00:20:33.300 --> 00:20:43.910 align:middle line:90% 00:20:43.910 --> 00:20:46.610 align:middle line:84% So far, we have that k1 is a natural number. 00:20:46.610 --> 00:20:49.230 align:middle line:90% 00:20:49.230 --> 00:20:56.510 align:middle line:84% And I'm going to prove that k1 is actually less than k0. 00:20:56.510 --> 00:21:05.690 align:middle line:90% 00:21:05.690 --> 00:21:10.310 align:middle line:90% So since x squared equals 2-- 00:21:10.310 --> 00:21:13.670 align:middle line:90% 00:21:13.670 --> 00:21:14.450 align:middle line:90% I should say-- 00:21:14.450 --> 00:21:20.005 align:middle line:90% 00:21:20.005 --> 00:21:21.760 align:middle line:90% OK, so let's write it this way. 00:21:21.760 --> 00:21:23.880 align:middle line:84% Since x squared is equal to 2, this 00:21:23.880 --> 00:21:37.270 align:middle line:84% implies that 4 minus x squared is positive. 00:21:37.270 --> 00:21:38.680 align:middle line:90% I mean, I just wrote down 2. 00:21:38.680 --> 00:21:41.410 align:middle line:90% 4 minus x squared is just 2. 00:21:41.410 --> 00:21:44.380 align:middle line:84% So this tells me, taking the difference, 2 minus x 00:21:44.380 --> 00:21:47.350 align:middle line:90% times 2 plus x is positive. 00:21:47.350 --> 00:21:49.130 align:middle line:90% 2 and x are positive. 00:21:49.130 --> 00:21:52.070 align:middle line:84% So I can divide this inequality through by this term 00:21:52.070 --> 00:21:53.710 align:middle line:84% and maintain the inequality, which 00:21:53.710 --> 00:21:58.510 align:middle line:90% implies 2 minus x is positive. 00:21:58.510 --> 00:22:04.690 align:middle line:90% And therefore, x is less than 2. 00:22:04.690 --> 00:22:06.130 align:middle line:90% So x is less than 2. 00:22:06.130 --> 00:22:09.700 align:middle line:90% 00:22:09.700 --> 00:22:14.860 align:middle line:84% Then k1, which remember is k0 times x minus 1, 00:22:14.860 --> 00:22:17.380 align:middle line:90% is less than k0. 00:22:17.380 --> 00:22:18.940 align:middle line:90% So k0 is a natural number. 00:22:18.940 --> 00:22:30.970 align:middle line:84% It's positive. x minus 1 is bounded by 2 minus 1 equals k0. 00:22:30.970 --> 00:22:38.020 align:middle line:84% So what we've shown so far is that this number, k1, 00:22:38.020 --> 00:22:39.325 align:middle line:90% is a natural number. 00:22:39.325 --> 00:22:41.890 align:middle line:90% 00:22:41.890 --> 00:22:48.370 align:middle line:90% And it's less than k0. 00:22:48.370 --> 00:22:51.250 align:middle line:84% So I guess I don't have to write this positive part. 00:22:51.250 --> 00:22:58.410 align:middle line:90% 00:22:58.410 --> 00:23:03.060 align:middle line:84% Now, let's remember what k0 was supposed to be. 00:23:03.060 --> 00:23:06.930 align:middle line:84% k0 is supposed to be the smallest element of this set 00:23:06.930 --> 00:23:09.780 align:middle line:84% S. It's the smallest natural number so 00:23:09.780 --> 00:23:13.380 align:middle line:84% that when I multiply it by x I get another natural number. 00:23:13.380 --> 00:23:18.150 align:middle line:84% And from this k0, I constructed a new natural number 00:23:18.150 --> 00:23:27.020 align:middle line:84% called k1, which is less than k0 and it's a natural number. 00:23:27.020 --> 00:23:29.165 align:middle line:90% But here comes the fun part. 00:23:29.165 --> 00:23:35.670 align:middle line:90% 00:23:35.670 --> 00:23:40.640 align:middle line:84% If we compute what x times k1 is, 00:23:40.640 --> 00:23:48.890 align:middle line:84% this is, by definition, x times x times k0 minus k0. 00:23:48.890 --> 00:23:55.280 align:middle line:84% This is equal to x squared k0 minus x times k0, which 00:23:55.280 --> 00:24:05.450 align:middle line:84% equals 2 k0 minus x times k0 because x squared equals 2. 00:24:05.450 --> 00:24:09.890 align:middle line:90% 00:24:09.890 --> 00:24:20.570 align:middle line:84% And this last part, this is equal to k0 plus k0 00:24:20.570 --> 00:24:29.255 align:middle line:90% minus x k0, which equals k1. 00:24:29.255 --> 00:24:34.960 align:middle line:90% 00:24:34.960 --> 00:24:37.180 align:middle line:90% Now, that's a natural number. 00:24:37.180 --> 00:24:38.440 align:middle line:90% That's a natural number. 00:24:38.440 --> 00:24:41.290 align:middle line:84% This natural number is bigger than this natural number. 00:24:41.290 --> 00:24:44.020 align:middle line:84% Therefore, their difference is, again, a natural number. 00:24:44.020 --> 00:24:46.580 align:middle line:90% 00:24:46.580 --> 00:24:50.050 align:middle line:84% So this says that x times k1 is a natural number. 00:24:50.050 --> 00:24:55.000 align:middle line:90% 00:24:55.000 --> 00:25:06.580 align:middle line:84% Thus, k1 is in S and k1 is less than k0, 00:25:06.580 --> 00:25:25.250 align:middle line:84% which implies k0 is not the least element in S, which 00:25:25.250 --> 00:25:30.530 align:middle line:84% is a contradiction to the fact that it 00:25:30.530 --> 00:25:42.710 align:middle line:90% is the least element of S. 00:25:42.710 --> 00:25:51.300 align:middle line:84% So what we've shown is assuming that this set 00:25:51.300 --> 00:25:57.060 align:middle line:84% E has a supremum in the rational numbers, 00:25:57.060 --> 00:26:00.530 align:middle line:84% then we arrive at a contradiction. 00:26:00.530 --> 00:26:02.810 align:middle line:84% So our original assumption must be false. 00:26:02.810 --> 00:26:08.005 align:middle line:90% Thus, sup E does not exist. 00:26:08.005 --> 00:26:26.090 align:middle line:90% 00:26:26.090 --> 00:26:28.700 align:middle line:84% All right, so this proof is maybe a little different than-- 00:26:28.700 --> 00:26:33.170 align:middle line:84% if you've seen a proof of the fact that the square root of 2 00:26:33.170 --> 00:26:36.662 align:middle line:84% is not a rational number, there's 00:26:36.662 --> 00:26:38.870 align:middle line:84% a different proof that maybe is a little bit simpler. 00:26:38.870 --> 00:26:43.620 align:middle line:84% But this one uses ordering instead to prove it, 00:26:43.620 --> 00:26:45.050 align:middle line:90% which I thought was pretty cool. 00:26:45.050 --> 00:26:46.550 align:middle line:84% And it's originally due to Dedekind. 00:26:46.550 --> 00:26:50.240 align:middle line:90% 00:26:50.240 --> 00:27:02.670 align:middle line:84% OK, so we've discussed one aspect of the real numbers that 00:27:02.670 --> 00:27:04.900 align:middle line:84% was in that theorem that I stated earlier, 00:27:04.900 --> 00:27:07.500 align:middle line:84% which I'll restate in a minute, namely 00:27:07.500 --> 00:27:13.530 align:middle line:84% that it is a set that has the least upper bound property. 00:27:13.530 --> 00:27:14.910 align:middle line:90% So I stated that as a theorem. 00:27:14.910 --> 00:27:18.190 align:middle line:84% I'm not going to prove that theorem. 00:27:18.190 --> 00:27:20.670 align:middle line:84% We just want to understand exactly what 00:27:20.670 --> 00:27:22.245 align:middle line:90% sets the real numbers apart. 00:27:22.245 --> 00:27:24.820 align:middle line:90% 00:27:24.820 --> 00:27:27.430 align:middle line:84% And one aspect of that-- there were two things. 00:27:27.430 --> 00:27:29.020 align:middle line:84% One is that it's an ordered field. 00:27:29.020 --> 00:27:31.450 align:middle line:84% And two is that it has the least upper bound property. 00:27:31.450 --> 00:27:34.140 align:middle line:84% So we've now discussed what the least upper bound property 00:27:34.140 --> 00:27:35.970 align:middle line:90% means. 00:27:35.970 --> 00:27:38.640 align:middle line:84% So we just need to fill in one other part 00:27:38.640 --> 00:27:41.550 align:middle line:84% about the real numbers, which is the fact that it's 00:27:41.550 --> 00:27:42.615 align:middle line:90% an ordered field. 00:27:42.615 --> 00:27:45.910 align:middle line:90% 00:27:45.910 --> 00:27:50.500 align:middle line:84% And as I've said before, Q is also an ordered field. 00:27:50.500 --> 00:27:53.770 align:middle line:84% R is not special in that respect. 00:27:53.770 --> 00:27:59.830 align:middle line:84% But as the theorem stated at the middle of the last lecture, 00:27:59.830 --> 00:28:02.590 align:middle line:84% that R is, in fact, the unique ordered field 00:28:02.590 --> 00:28:04.090 align:middle line:84% with the least upper bound property. 00:28:04.090 --> 00:28:06.757 align:middle line:84% Remember, we've just proven that Q does not have the least upper 00:28:06.757 --> 00:28:07.570 align:middle line:90% bound property. 00:28:07.570 --> 00:28:09.775 align:middle line:84% So in some sense, Q is missing stuff. 00:28:09.775 --> 00:28:12.550 align:middle line:90% 00:28:12.550 --> 00:28:13.540 align:middle line:90% It's missing stuff. 00:28:13.540 --> 00:28:16.960 align:middle line:84% It's missing square roots of 2, which 00:28:16.960 --> 00:28:19.930 align:middle line:84% is kind of an algebraic thing, namely 00:28:19.930 --> 00:28:23.080 align:middle line:84% I can't solve the equation x squared minus 2 00:28:23.080 --> 00:28:25.840 align:middle line:84% equals 0 in the rational numbers. 00:28:25.840 --> 00:28:28.330 align:middle line:84% And the fact that it's missing, that you 00:28:28.330 --> 00:28:31.630 align:middle line:84% have this kind of algebraic defect 00:28:31.630 --> 00:28:38.020 align:middle line:84% manifests itself in this fact that Q is also missing things 00:28:38.020 --> 00:28:41.230 align:middle line:90% with respect to an order. 00:28:41.230 --> 00:28:44.260 align:middle line:84% So Q, in short, this is me saying 00:28:44.260 --> 00:28:50.540 align:middle line:90% that has holes and R does not. 00:28:50.540 --> 00:28:59.020 align:middle line:84% I mean, that is probably the most basic statement 00:28:59.020 --> 00:29:03.130 align:middle line:84% one can make about R. And perhaps you 00:29:03.130 --> 00:29:05.200 align:middle line:84% heard in high school calculus and you're 00:29:05.200 --> 00:29:09.610 align:middle line:84% hearing repeated now, but in a nutshell, I mean, Q has holes 00:29:09.610 --> 00:29:10.440 align:middle line:90% and R does not. 00:29:10.440 --> 00:29:12.190 align:middle line:84% But this means something very specifically 00:29:12.190 --> 00:29:16.540 align:middle line:84% that R has the least upper bound property and Q does not. 00:29:16.540 --> 00:29:19.130 align:middle line:90% 00:29:19.130 --> 00:29:24.410 align:middle line:84% All right, so let me talk about what ordered fields are. 00:29:24.410 --> 00:29:31.210 align:middle line:84% So first off, I need to define what a field is. 00:29:31.210 --> 00:29:49.420 align:middle line:84% So a set F is a field if it has two operations-- 00:29:49.420 --> 00:30:00.650 align:middle line:90% 00:30:00.650 --> 00:30:07.670 align:middle line:84% plus-- and I'm going to put a dot here in the middle-- 00:30:07.670 --> 00:30:12.770 align:middle line:84% times basically, such that you have 00:30:12.770 --> 00:30:14.930 align:middle line:84% a list of properties that hold with respect 00:30:14.930 --> 00:30:16.970 align:middle line:90% to these operations. 00:30:16.970 --> 00:30:20.210 align:middle line:90% The first is with respect to-- 00:30:20.210 --> 00:30:21.950 align:middle line:90% so this is plus, this is times. 00:30:21.950 --> 00:30:24.380 align:middle line:84% This is addition, multiplication. 00:30:24.380 --> 00:30:27.950 align:middle line:84% So the first condition is that the set is closed with respect 00:30:27.950 --> 00:30:31.740 align:middle line:90% to taking sums. 00:30:31.740 --> 00:30:43.390 align:middle line:84% So if x y is in F, then x plus y is in F. 00:30:43.390 --> 00:30:49.860 align:middle line:84% This operation of additions should be commutative. 00:30:49.860 --> 00:30:54.110 align:middle line:90% 00:30:54.110 --> 00:30:58.350 align:middle line:84% Commutativity, I hope that's how you spell it. 00:30:58.350 --> 00:31:08.720 align:middle line:84% This means that for all x y in F, x plus y equals y plus x. 00:31:08.720 --> 00:31:20.660 align:middle line:84% Associativity, so this is a condition that for all x, y, 00:31:20.660 --> 00:31:26.690 align:middle line:84% z in F, if I add x and y and then add , 00:31:26.690 --> 00:31:31.790 align:middle line:84% this is equal to adding x to the sum of y and z. 00:31:31.790 --> 00:31:37.230 align:middle line:90% 00:31:37.230 --> 00:31:39.680 align:middle line:84% The fourth is that we have what's 00:31:39.680 --> 00:31:47.170 align:middle line:84% called an identity element, an additive identity element. 00:31:47.170 --> 00:31:52.450 align:middle line:84% There exists an element which I'll label 0 in the set F 00:31:52.450 --> 00:32:02.805 align:middle line:84% such that for all x in F, 0 plus x equals 0. 00:32:02.805 --> 00:32:10.050 align:middle line:90% 00:32:10.050 --> 00:32:20.460 align:middle line:84% And we also have additive inverses, namely for all x in F 00:32:20.460 --> 00:32:23.460 align:middle line:84% there exists an element which I call-- 00:32:23.460 --> 00:32:35.230 align:middle line:84% which I label minus x in F such that x plus minus x equals 0. 00:32:35.230 --> 00:32:40.570 align:middle line:84% So these are the conditions on addition 00:32:40.570 --> 00:32:42.850 align:middle line:84% that should be satisfied for a field. 00:32:42.850 --> 00:32:47.750 align:middle line:90% 00:32:47.750 --> 00:32:49.070 align:middle line:90% So that's about addition. 00:32:49.070 --> 00:32:53.738 align:middle line:84% So the conditions for multiplication are similar. 00:32:53.738 --> 00:32:56.280 align:middle line:84% Namely, it needs to be closed with respect to multiplication. 00:32:56.280 --> 00:33:08.780 align:middle line:84% So if x y is in F, then x times y is in F. 00:33:08.780 --> 00:33:11.360 align:middle line:84% Multiplication should also be commutative. 00:33:11.360 --> 00:33:15.130 align:middle line:90% 00:33:15.130 --> 00:33:21.520 align:middle line:84% So I'm just going to abbreviate that commutative as comm. 00:33:21.520 --> 00:33:28.810 align:middle line:84% For all x y in F, x times y equals y times 00:33:28.810 --> 00:33:31.300 align:middle line:90% x We also have associativity. 00:33:31.300 --> 00:33:34.700 align:middle line:90% 00:33:34.700 --> 00:33:36.380 align:middle line:90% Let's not shorten it by that. 00:33:36.380 --> 00:33:39.380 align:middle line:90% 00:33:39.380 --> 00:33:44.930 align:middle line:84% For all x, y, z in F, if I multiply x times y 00:33:44.930 --> 00:33:49.640 align:middle line:84% and then multiply it by z, it's the same as taking 00:33:49.640 --> 00:33:52.130 align:middle line:84% x and multiplying it by y times z. 00:33:52.130 --> 00:33:57.806 align:middle line:90% 00:33:57.806 --> 00:34:00.770 align:middle line:84% The fourth property is the existence 00:34:00.770 --> 00:34:03.440 align:middle line:90% of multiplicative identity. 00:34:03.440 --> 00:34:13.610 align:middle line:84% There exists an element which I label as 1 in the set F 00:34:13.610 --> 00:34:18.545 align:middle line:84% such that for all x in F, 1 times x equals x. 00:34:18.545 --> 00:34:21.520 align:middle line:90% 00:34:21.520 --> 00:34:26.469 align:middle line:84% And then I also have multiplicative inverses 00:34:26.469 --> 00:34:29.710 align:middle line:90% for non-zero elements. 00:34:29.710 --> 00:34:34.250 align:middle line:84% For all x in F, take away 0-- so for everything in the field 00:34:34.250 --> 00:34:36.400 align:middle line:90% that's non-zero-- 00:34:36.400 --> 00:34:42.699 align:middle line:84% there exists an element which I call x to the minus 1 in F such 00:34:42.699 --> 00:34:48.120 align:middle line:84% that x times x to the minus 1 equals 1. 00:34:48.120 --> 00:34:51.239 align:middle line:84% Now, these are statements about the two operations. 00:34:51.239 --> 00:34:57.420 align:middle line:84% There's one last assumption in the definition 00:34:57.420 --> 00:34:58.470 align:middle line:90% that connects the two. 00:34:58.470 --> 00:35:03.360 align:middle line:84% And that's the distributive law, namely 00:35:03.360 --> 00:35:18.910 align:middle line:84% that for all x, y, z in F, if I take x plus y times z, 00:35:18.910 --> 00:35:25.540 align:middle line:84% this is x times z plus y times z. 00:35:25.540 --> 00:35:29.660 align:middle line:90% So all these conditions-- 00:35:29.660 --> 00:35:34.570 align:middle line:84% so a field is a set with two operations, plus and dot, 00:35:34.570 --> 00:35:35.860 align:middle line:90% meaning multiplication. 00:35:35.860 --> 00:35:39.700 align:middle line:84% And these two operations need to satisfy all of these conditions 00:35:39.700 --> 00:35:43.750 align:middle line:90% for my set to be called a field. 00:35:43.750 --> 00:35:51.380 align:middle line:84% So the clearest example is, of course, 00:35:51.380 --> 00:35:55.190 align:middle line:84% rational numbers with the usual plus and minus, 00:35:55.190 --> 00:35:57.710 align:middle line:84% I mean plus and multiplication defined 00:35:57.710 --> 00:36:02.235 align:middle line:90% as you learned as a child. 00:36:02.235 --> 00:36:05.380 align:middle line:90% 00:36:05.380 --> 00:36:07.540 align:middle line:90% Now, what's a non-example? 00:36:07.540 --> 00:36:13.070 align:middle line:90% 00:36:13.070 --> 00:36:13.790 align:middle line:90% The integers. 00:36:13.790 --> 00:36:18.020 align:middle line:84% So the integers come with plus and multiplication. 00:36:18.020 --> 00:36:24.910 align:middle line:84% However, it doesn't satisfy the existence of inverses, 00:36:24.910 --> 00:36:27.550 align:middle line:84% of multiplicative inverses in 5, but it 00:36:27.550 --> 00:36:30.290 align:middle line:90% does satisfy everything else. 00:36:30.290 --> 00:36:35.020 align:middle line:84% And typically, one would call z what's 00:36:35.020 --> 00:36:47.780 align:middle line:84% called a commutative ring, commutative 00:36:47.780 --> 00:36:51.600 align:middle line:84% because multiplication is also commutative. 00:36:51.600 --> 00:36:53.550 align:middle line:84% But a ring in general does not necessarily 00:36:53.550 --> 00:36:56.700 align:middle line:90% have to satisfy multiplication. 00:36:56.700 --> 00:37:03.690 align:middle line:84% Being commutative, for example, the set of say 2 by 2 matrices 00:37:03.690 --> 00:37:05.340 align:middle line:90% form a ring. 00:37:05.340 --> 00:37:06.960 align:middle line:90% What's an example of another? 00:37:06.960 --> 00:37:11.530 align:middle line:84% What's another example of a field? 00:37:11.530 --> 00:37:16.210 align:middle line:84% Well, we have-- maybe I should have given this one first-- 00:37:16.210 --> 00:37:19.645 align:middle line:84% Z2, which is the simplest field there is. 00:37:19.645 --> 00:37:21.730 align:middle line:84% It's just the element of two-- it's just 00:37:21.730 --> 00:37:26.230 align:middle line:84% a set of two elements, 0 and 1, where 00:37:26.230 --> 00:37:35.395 align:middle line:84% I need to define what 1 plus 1 is, 1 plus 1 defined to be 0. 00:37:35.395 --> 00:37:39.870 align:middle line:90% 00:37:39.870 --> 00:37:42.630 align:middle line:90% And 1 plus 1 is 0. 00:37:42.630 --> 00:37:46.150 align:middle line:90% And yeah, that's it. 00:37:46.150 --> 00:37:48.180 align:middle line:90% I mean, 0 times 0 would be 0. 00:37:48.180 --> 00:37:50.580 align:middle line:90% 0 times 1 would be 0. 00:37:50.580 --> 00:37:52.500 align:middle line:90% 0 plus 0 would be 0. 00:37:52.500 --> 00:37:55.230 align:middle line:84% And that gives you all of the rules 00:37:55.230 --> 00:37:57.960 align:middle line:84% you need to know to be able to define multiplication 00:37:57.960 --> 00:38:02.520 align:middle line:84% and addition on this set of these two elements. 00:38:02.520 --> 00:38:06.570 align:middle line:84% This is a field because what is the inverse of 1? 00:38:06.570 --> 00:38:08.070 align:middle line:90% Well, it's just 1. 00:38:08.070 --> 00:38:10.650 align:middle line:84% And you can check that if I define addition 00:38:10.650 --> 00:38:16.680 align:middle line:84% by this rule along with the other rules that-- 00:38:16.680 --> 00:38:19.450 align:middle line:90% 00:38:19.450 --> 00:38:35.230 align:middle line:84% and along with-- that these defining plus 00:38:35.230 --> 00:38:38.077 align:middle line:84% and multiplication this way satisfies all the conditions 00:38:38.077 --> 00:38:38.785 align:middle line:90% of being a field. 00:38:38.785 --> 00:38:43.494 align:middle line:90% 00:38:43.494 --> 00:38:51.910 align:middle line:84% A more non-trivial example would be, let's say, Z3. 00:38:51.910 --> 00:38:55.900 align:middle line:90% This is a set 0, 1, 2. 00:38:55.900 --> 00:39:00.190 align:middle line:84% Only now, the arithmetic-- so I didn't use such a fancy word 00:39:00.190 --> 00:39:01.030 align:middle line:90% here. 00:39:01.030 --> 00:39:05.410 align:middle line:84% But arithmetic is done mod 3 here. 00:39:05.410 --> 00:39:10.360 align:middle line:84% Here, it was mod 2, meaning if I want to add two elements, 00:39:10.360 --> 00:39:17.140 align:middle line:84% I add them, and then I take the remainder of that sum 00:39:17.140 --> 00:39:18.520 align:middle line:90% after dividing by 3. 00:39:18.520 --> 00:39:28.880 align:middle line:84% So here, addition is defined mod 3. 00:39:28.880 --> 00:39:33.500 align:middle line:84% So if something is a multiple of 3, then I equate it to 0. 00:39:33.500 --> 00:39:38.060 align:middle line:84% So for example, 1 plus 2, which gives me 3, 00:39:38.060 --> 00:39:39.440 align:middle line:90% it's a multiple of 3. 00:39:39.440 --> 00:39:41.015 align:middle line:90% This is defined to be 0. 00:39:41.015 --> 00:39:44.210 align:middle line:90% 00:39:44.210 --> 00:39:49.340 align:middle line:84% 2 times 2, which equals 4, which equals 3 plus 1, multiples of 3 00:39:49.340 --> 00:39:49.850 align:middle line:90% are 0. 00:39:49.850 --> 00:39:50.990 align:middle line:90% So that gives me 1. 00:39:50.990 --> 00:39:54.590 align:middle line:84% In particular, this tells me that in this field, 00:39:54.590 --> 00:39:56.630 align:middle line:90% 2 times 2 equals 1. 00:39:56.630 --> 00:40:01.670 align:middle line:84% So the multiplicative inverse of 2 is given by itself, 2, 00:40:01.670 --> 00:40:03.900 align:middle line:90% all right? 00:40:03.900 --> 00:40:10.750 align:middle line:84% And a times b equals, let's say, d mod 3, all right? 00:40:10.750 --> 00:40:14.200 align:middle line:90% 00:40:14.200 --> 00:40:16.255 align:middle line:84% OK, so that's another example of a field. 00:40:16.255 --> 00:40:27.100 align:middle line:90% 00:40:27.100 --> 00:40:30.363 align:middle line:84% In fact, if you take mod p where p is a prime, 00:40:30.363 --> 00:40:31.280 align:middle line:90% you get another field. 00:40:31.280 --> 00:40:32.250 align:middle line:90% You get a finite field. 00:40:32.250 --> 00:40:34.940 align:middle line:84% So these are examples of what are called finite fields 00:40:34.940 --> 00:40:36.610 align:middle line:84% because exactly that-- they are fields 00:40:36.610 --> 00:40:38.360 align:middle line:84% and they have a finite number of elements. 00:40:38.360 --> 00:40:41.810 align:middle line:90% 00:40:41.810 --> 00:40:48.350 align:middle line:84% Now, OK, based on just these assumptions 00:40:48.350 --> 00:40:51.290 align:middle line:84% that you assume your field to satisfy-- 00:40:51.290 --> 00:40:53.420 align:middle line:84% I mean, for it to be a field, you 00:40:53.420 --> 00:41:01.290 align:middle line:84% can prove all of the simple algebra statements 00:41:01.290 --> 00:41:06.150 align:middle line:84% that you've ever known simply from these axioms of fields. 00:41:06.150 --> 00:41:10.810 align:middle line:90% 00:41:10.810 --> 00:41:13.080 align:middle line:84% So for example, let me just give you 00:41:13.080 --> 00:41:16.290 align:middle line:84% the silliest example of a statement you can prove just 00:41:16.290 --> 00:41:18.450 align:middle line:90% using these elements-- 00:41:18.450 --> 00:41:20.220 align:middle line:90% I mean, these axioms. 00:41:20.220 --> 00:41:26.920 align:middle line:84% You can prove the statement that for all x in F-- 00:41:26.920 --> 00:41:29.690 align:middle line:90% so F is a field here. 00:41:29.690 --> 00:41:31.150 align:middle line:90% So F throughout will be-- 00:41:31.150 --> 00:41:38.394 align:middle line:90% 00:41:38.394 --> 00:41:45.750 align:middle line:84% if x is a field, then for all x in F, 0 times x is 0. 00:41:45.750 --> 00:41:50.500 align:middle line:84% All we know about 0 is that, when you add it to x, 00:41:50.500 --> 00:41:51.360 align:middle line:90% you get x back. 00:41:51.360 --> 00:41:54.150 align:middle line:84% Actually, this should have been-- 00:41:54.150 --> 00:41:55.805 align:middle line:90% yeah, yeah, this is wrong. 00:41:55.805 --> 00:41:56.805 align:middle line:90% This should have been x. 00:41:56.805 --> 00:42:03.904 align:middle line:90% 00:42:03.904 --> 00:42:07.380 align:middle line:84% Let me make sure there's no other errors. 00:42:07.380 --> 00:42:08.530 align:middle line:90% I don't think there are. 00:42:08.530 --> 00:42:12.300 align:middle line:84% That's another danger about not being 00:42:12.300 --> 00:42:19.640 align:middle line:84% able to lecture in person, is that errors on the board 00:42:19.640 --> 00:42:21.150 align:middle line:90% persist. 00:42:21.150 --> 00:42:27.330 align:middle line:84% OK, so you can prove this blockbuster theorem just using 00:42:27.330 --> 00:42:29.890 align:middle line:90% these axioms. 00:42:29.890 --> 00:42:33.520 align:middle line:84% So let's do a quick proof of this. 00:42:33.520 --> 00:42:40.820 align:middle line:84% If x is in F, then 0 is equal to 0 times 00:42:40.820 --> 00:42:44.810 align:middle line:84% x plus the additive inverse of 0 times 00:42:44.810 --> 00:42:46.700 align:middle line:84% x because that's just the definition 00:42:46.700 --> 00:42:47.930 align:middle line:90% of the additive inverse. 00:42:47.930 --> 00:42:51.270 align:middle line:84% Every element has an additive inverse. 00:42:51.270 --> 00:43:00.260 align:middle line:84% And now, 0 is equal to 0 plus 0 times x plus, 00:43:00.260 --> 00:43:03.890 align:middle line:84% again, the additive inverse of x. 00:43:03.890 --> 00:43:05.750 align:middle line:84% And now, I use the distributive law. 00:43:05.750 --> 00:43:12.770 align:middle line:84% This is 0 times x plus 0 times x plus the additive inverse of 0 00:43:12.770 --> 00:43:14.120 align:middle line:90% times x. 00:43:14.120 --> 00:43:16.320 align:middle line:90% And that cancels with that. 00:43:16.320 --> 00:43:17.690 align:middle line:90% And I get 0 times x. 00:43:17.690 --> 00:43:20.300 align:middle line:84% So I started off with 0 and I arrived 00:43:20.300 --> 00:43:23.660 align:middle line:84% at it's equal to 0 times x, all right? 00:43:23.660 --> 00:43:26.860 align:middle line:90% 00:43:26.860 --> 00:43:36.270 align:middle line:84% And you can prove other simple algebraic statements 00:43:36.270 --> 00:43:37.840 align:middle line:90% using these axioms as well. 00:43:37.840 --> 00:43:40.770 align:middle line:84% I think I'll put it in the assignment. 00:43:40.770 --> 00:43:50.650 align:middle line:90% 00:43:50.650 --> 00:43:54.010 align:middle line:84% And I'll just state you can also prove things like-- 00:43:54.010 --> 00:43:59.880 align:middle line:84% if I want to look at minus x, then 00:43:59.880 --> 00:44:04.470 align:middle line:84% this is equal to the additive inverse of 1 times x. 00:44:04.470 --> 00:44:06.000 align:middle line:90% I mean, this is like-- 00:44:06.000 --> 00:44:11.450 align:middle line:84% it's not terribly interesting at the start. 00:44:11.450 --> 00:44:16.700 align:middle line:84% There are essentially some very deep theorems in algebra 00:44:16.700 --> 00:44:19.760 align:middle line:84% that you learn at another point in your life, 00:44:19.760 --> 00:44:21.095 align:middle line:90% but that won't be in this class. 00:44:21.095 --> 00:44:24.220 align:middle line:90% 00:44:24.220 --> 00:44:25.960 align:middle line:84% Actually, today is probably all we're 00:44:25.960 --> 00:44:28.750 align:middle line:84% going to talk about fields, which are algebraic things. 00:44:28.750 --> 00:44:31.810 align:middle line:90% 00:44:31.810 --> 00:44:37.140 align:middle line:84% Algebraic things are, to me, nice 00:44:37.140 --> 00:44:40.980 align:middle line:84% because somehow you always deal with equality. 00:44:40.980 --> 00:44:42.600 align:middle line:84% So how hard could it be to prove two 00:44:42.600 --> 00:44:45.070 align:middle line:90% things are equal to each other? 00:44:45.070 --> 00:44:48.880 align:middle line:84% Yet, analysis deals a lot with inequality, which 00:44:48.880 --> 00:44:52.790 align:middle line:90% somehow is much more subtle. 00:44:52.790 --> 00:44:54.725 align:middle line:90% But that's just a little biased. 00:44:54.725 --> 00:44:57.456 align:middle line:90% 00:44:57.456 --> 00:45:00.860 align:middle line:90% OK, so this is what a field is. 00:45:00.860 --> 00:45:04.250 align:middle line:84% A field is just, again, a set that has these two operations. 00:45:04.250 --> 00:45:05.480 align:middle line:90% What is an ordered field? 00:45:05.480 --> 00:45:26.570 align:middle line:90% 00:45:26.570 --> 00:45:39.330 align:middle line:84% So it's a field, first off, and which is also an ordered set-- 00:45:39.330 --> 00:45:40.845 align:middle line:90% but it can't just-- 00:45:40.845 --> 00:45:44.610 align:middle line:84% you can't have two different structures 00:45:44.610 --> 00:45:46.650 align:middle line:84% on your field and them not interact for that 00:45:46.650 --> 00:45:49.360 align:middle line:90% to be interesting-- 00:45:49.360 --> 00:45:59.270 align:middle line:84% so such that the algebraic structure and the order 00:45:59.270 --> 00:46:03.920 align:middle line:84% on the field are cohabitating nicely, 00:46:03.920 --> 00:46:12.370 align:middle line:84% meaning you have two conditions for all x, y, z in F. 00:46:12.370 --> 00:46:19.390 align:middle line:84% If x is less than y, then x plus z is less than y plus z. 00:46:19.390 --> 00:46:23.460 align:middle line:90% 00:46:23.460 --> 00:46:24.660 align:middle line:90% And one other condition-- 00:46:24.660 --> 00:46:27.840 align:middle line:90% 00:46:27.840 --> 00:46:29.010 align:middle line:90% x is positive. 00:46:29.010 --> 00:46:34.260 align:middle line:84% Or if x is bigger than 0 and y is bigger than 0, 00:46:34.260 --> 00:46:37.665 align:middle line:90% then x times y is bigger than 0. 00:46:37.665 --> 00:46:41.030 align:middle line:90% 00:46:41.030 --> 00:46:46.070 align:middle line:84% And I just use that terminology anyways right now. 00:46:46.070 --> 00:46:54.750 align:middle line:84% But for an ordered field F, if an element is bigger than 0, 00:46:54.750 --> 00:46:56.520 align:middle line:90% then we call it positive. 00:46:56.520 --> 00:47:00.800 align:middle line:90% 00:47:00.800 --> 00:47:02.720 align:middle line:84% Or if it's bigger than or equal to 0, 00:47:02.720 --> 00:47:11.822 align:middle line:84% we say x is positive and respectively non-negative. 00:47:11.822 --> 00:47:14.990 align:middle line:90% 00:47:14.990 --> 00:47:17.780 align:middle line:84% So non-negative if x is bigger than or equal to 0, 00:47:17.780 --> 00:47:24.320 align:middle line:84% positive x is bigger than 0, and then likewise with negative 00:47:24.320 --> 00:47:25.880 align:middle line:90% and non-positive. 00:47:25.880 --> 00:47:28.620 align:middle line:90% 00:47:28.620 --> 00:47:32.658 align:middle line:84% And so the most basic example, again, 00:47:32.658 --> 00:47:37.620 align:middle line:84% is Q. Q is an ordered field with the usual order 00:47:37.620 --> 00:47:43.580 align:middle line:84% and with the usual algebraic structure on Q. 00:47:43.580 --> 00:47:55.210 align:middle line:84% What is not an example is either one of the two fields 00:47:55.210 --> 00:47:57.520 align:middle line:90% I wrote down just a minute ago. 00:47:57.520 --> 00:48:15.810 align:middle line:90% 00:48:15.810 --> 00:48:32.530 align:middle line:84% So a non-example is this field here, 0, 1. 00:48:32.530 --> 00:48:40.170 align:middle line:84% So if I put an order on this, either 0 is less than 1 00:48:40.170 --> 00:48:41.760 align:middle line:90% or 1 is less than 0. 00:48:41.760 --> 00:48:46.650 align:middle line:84% So remember, order does not have to necessarily correspond 00:48:46.650 --> 00:48:50.460 align:middle line:84% to the fact that 0 you connect to 0 in the integers 00:48:50.460 --> 00:48:52.920 align:middle line:84% and that 1 you connect to 1 in the integers. 00:48:52.920 --> 00:48:54.930 align:middle line:84% I mean, these are just two elements of a set. 00:48:54.930 --> 00:48:56.940 align:middle line:84% And an order would be saying either 00:48:56.940 --> 00:48:59.070 align:middle line:84% that element is less than that element 00:48:59.070 --> 00:49:02.230 align:middle line:84% or that element is less than that element. 00:49:02.230 --> 00:49:05.790 align:middle line:84% So let's consider either cases and suppose 00:49:05.790 --> 00:49:11.850 align:middle line:84% we have that order on this set and show that it does not turn 00:49:11.850 --> 00:49:13.335 align:middle line:90% this set into an ordered field. 00:49:13.335 --> 00:49:16.440 align:middle line:90% 00:49:16.440 --> 00:49:35.675 align:middle line:84% So then either 0 is less than 1 or 1 is less than 0. 00:49:35.675 --> 00:49:38.860 align:middle line:90% 00:49:38.860 --> 00:49:45.850 align:middle line:84% So let's do the first case, 0 is less than 1. 00:49:45.850 --> 00:49:51.240 align:middle line:90% 00:49:51.240 --> 00:49:59.580 align:middle line:84% If 0 is less than 1, then what happens 00:49:59.580 --> 00:50:01.050 align:middle line:90% if I add 1 to each side? 00:50:01.050 --> 00:50:11.960 align:middle line:90% 00:50:11.960 --> 00:50:13.940 align:middle line:90% So 1 plus 0 would give me 1. 00:50:13.940 --> 00:50:15.410 align:middle line:90% 1 plus one would be 0. 00:50:15.410 --> 00:50:21.950 align:middle line:84% And therefore, it is not so less than 1 plus 1. 00:50:21.950 --> 00:50:25.190 align:middle line:84% So it does not satisfy the first property. 00:50:25.190 --> 00:50:40.000 align:middle line:90% 00:50:40.000 --> 00:50:45.450 align:middle line:84% So if I have an order on Z2, two possibilities-- either 0 00:50:45.450 --> 00:50:47.310 align:middle line:84% is less than 1 or 1 is less than 0. 00:50:47.310 --> 00:50:52.830 align:middle line:84% If 0 is less than 1, then, by the definition of addition 00:50:52.830 --> 00:50:57.000 align:middle line:84% on this set, if I add 1 to 0, I get 1. 00:50:57.000 --> 00:50:59.040 align:middle line:90% If I add 1 to 1, I get 0. 00:50:59.040 --> 00:51:02.470 align:middle line:84% And therefore, if I were to add 1 to both sides, 00:51:02.470 --> 00:51:04.200 align:middle line:84% I would not get a true inequality 00:51:04.200 --> 00:51:06.780 align:middle line:84% because I'm assuming 0 is less than 1. 00:51:06.780 --> 00:51:12.840 align:middle line:84% So this condition, number 1, does not hold for this order. 00:51:12.840 --> 00:51:15.915 align:middle line:84% And neither does it hold for choosing 1 is less than 0. 00:51:15.915 --> 00:51:20.570 align:middle line:90% 00:51:20.570 --> 00:51:23.375 align:middle line:90% By the same logic, then 1-- 00:51:23.375 --> 00:51:31.750 align:middle line:90% 00:51:31.750 --> 00:51:35.875 align:middle line:90% so 1 plus 1 is not less than-- 00:51:35.875 --> 00:51:41.730 align:middle line:90% 00:51:41.730 --> 00:51:44.190 align:middle line:90% so this set is not-- 00:51:44.190 --> 00:51:47.740 align:middle line:84% this field cannot be turned into an ordered field. 00:51:47.740 --> 00:51:52.830 align:middle line:84% And essentially, the same thing that I've done here shows that 00:51:52.830 --> 00:51:58.510 align:middle line:84% you cannot have any finite ordered field. 00:51:58.510 --> 00:52:17.320 align:middle line:90% 00:52:17.320 --> 00:52:20.185 align:middle line:84% So in general, there are no finite ordered fields. 00:52:20.185 --> 00:52:26.340 align:middle line:90% 00:52:26.340 --> 00:52:36.890 align:middle line:84% Now, just like we proved the blockbuster statement 00:52:36.890 --> 00:52:42.620 align:middle line:84% that 0 times anything in the field equals 0, 00:52:42.620 --> 00:52:48.200 align:middle line:84% we can also prove all of the manipulations of inequalities 00:52:48.200 --> 00:52:59.420 align:middle line:84% that you use without fear simply from, again, 00:52:59.420 --> 00:53:02.490 align:middle line:84% these axioms about an ordered field being a field 00:53:02.490 --> 00:53:06.810 align:middle line:84% and also satisfying these two inequalities-- 00:53:06.810 --> 00:53:10.200 align:middle line:84% I mean, these two conditions here for it 00:53:10.200 --> 00:53:11.590 align:middle line:90% to be an ordered field. 00:53:11.590 --> 00:53:39.000 align:middle line:84% So for example, if F is an ordered field, 00:53:39.000 --> 00:53:44.510 align:middle line:84% then if x is an element of f and x is positive, 00:53:44.510 --> 00:53:52.240 align:middle line:84% this implies its additive inverse is negative and vice 00:53:52.240 --> 00:53:53.950 align:middle line:90% versa. 00:53:53.950 --> 00:54:01.510 align:middle line:84% If x is less than 0, then minus x is positive. 00:54:01.510 --> 00:54:08.460 align:middle line:84% So I know it's tempting to think, well yeah, just 00:54:08.460 --> 00:54:13.080 align:middle line:84% multiply this inequality by minus 1 to get this inequality. 00:54:13.080 --> 00:54:15.120 align:middle line:84% But remember, this is really a statement 00:54:15.120 --> 00:54:16.965 align:middle line:90% about the additive inverse of x. 00:54:16.965 --> 00:54:29.580 align:middle line:90% 00:54:29.580 --> 00:54:30.705 align:middle line:90% So the proof is not hard. 00:54:30.705 --> 00:54:34.320 align:middle line:90% 00:54:34.320 --> 00:54:36.600 align:middle line:90% If x is positive, then-- 00:54:36.600 --> 00:54:39.380 align:middle line:90% 00:54:39.380 --> 00:54:42.346 align:middle line:90% well, I could write it this way. 00:54:42.346 --> 00:54:45.330 align:middle line:84% I can write it as 0 is less than x. 00:54:45.330 --> 00:54:47.930 align:middle line:90% 00:54:47.930 --> 00:54:53.380 align:middle line:84% Then by property number 1, I can add anything to both sides 00:54:53.380 --> 00:54:56.380 align:middle line:84% and get the same inequality-- and preserve the inequality. 00:54:56.380 --> 00:55:01.450 align:middle line:84% Therefore, minus x plus 0 is less than the additive inverse 00:55:01.450 --> 00:55:04.540 align:middle line:90% of x plus x. 00:55:04.540 --> 00:55:13.000 align:middle line:84% Now, by the fact that 0 is the additive identity, a4, 00:55:13.000 --> 00:55:15.910 align:middle line:84% the left side, is going to be minus x. 00:55:15.910 --> 00:55:20.290 align:middle line:84% And by the definition of the additive inverse of x, 00:55:20.290 --> 00:55:23.500 align:middle line:84% The right-hand side is going to be 0. 00:55:23.500 --> 00:55:26.560 align:middle line:90% And that's it. 00:55:26.560 --> 00:55:35.950 align:middle line:84% So what's going in here is this is by a4 and a5. 00:55:35.950 --> 00:55:40.530 align:middle line:90% 00:55:40.530 --> 00:55:46.320 align:middle line:84% And this previous statement is by 1 in this definition before. 00:55:46.320 --> 00:55:49.680 align:middle line:84% And I'm not going to prove this statement as well. 00:55:49.680 --> 00:55:51.270 align:middle line:84% It's the essentially the same proof. 00:55:51.270 --> 00:55:55.650 align:middle line:84% I add minus x to both sides, which I can do by 1. 00:55:55.650 --> 00:55:58.530 align:middle line:84% And then I use a4 and a5 to conclude that x is-- 00:55:58.530 --> 00:56:01.880 align:middle line:90% 00:56:01.880 --> 00:56:03.658 align:middle line:90% that minus x is positive. 00:56:03.658 --> 00:56:25.860 align:middle line:90% 00:56:25.860 --> 00:56:41.370 align:middle line:84% Let me just say See Proposition 1.1.8 in textbook 00:56:41.370 --> 00:56:46.850 align:middle line:90% for the other-- 00:56:46.850 --> 00:57:04.970 align:middle line:84% let's say for the other proofs of standard inequality 00:57:04.970 --> 00:57:06.170 align:middle line:90% manipulations. 00:57:06.170 --> 00:57:16.680 align:middle line:90% 00:57:16.680 --> 00:57:19.690 align:middle line:90% So Q is an ordered field. 00:57:19.690 --> 00:57:26.320 align:middle line:84% And as I stated before about R, R will also-- 00:57:26.320 --> 00:57:29.320 align:middle line:90% R is also an ordered field. 00:57:29.320 --> 00:57:31.480 align:middle line:84% But it has a least upper bound property. 00:57:31.480 --> 00:57:35.350 align:middle line:84% So maybe you are asking yourself, how 00:57:35.350 --> 00:57:40.900 align:middle line:84% about some sort of greatest lower bound property? 00:57:40.900 --> 00:57:43.960 align:middle line:84% The least upper bound property is about sups. 00:57:43.960 --> 00:57:46.840 align:middle line:84% Can we make a similar property about infs? 00:57:46.840 --> 00:57:51.700 align:middle line:84% Are there sets that have the least upper bound property 00:57:51.700 --> 00:57:58.250 align:middle line:84% that don't have, say, a greatest lower bound property, meaning 00:57:58.250 --> 00:58:02.580 align:middle line:84% does there exist something has the greatest lower bound 00:58:02.580 --> 00:58:08.430 align:middle line:84% property if every non-empty set which is bounded below 00:58:08.430 --> 00:58:09.605 align:middle line:90% has an infimum? 00:58:09.605 --> 00:58:11.730 align:middle line:84% That's kind of what I'm referring to, even though I 00:58:11.730 --> 00:58:12.730 align:middle line:90% haven't written it down. 00:58:12.730 --> 00:58:16.600 align:middle line:90% 00:58:16.600 --> 00:58:19.460 align:middle line:90% OK, what is that leading up to? 00:58:19.460 --> 00:58:22.045 align:middle line:84% In the setting of ordered fields, 00:58:22.045 --> 00:58:24.170 align:middle line:84% there really is no difference between a least upper 00:58:24.170 --> 00:58:27.610 align:middle line:84% bound property and a greatest lower bound property. 00:58:27.610 --> 00:58:31.210 align:middle line:84% If I have an ordered field which satisfies the least upper 00:58:31.210 --> 00:58:33.820 align:middle line:84% bound property, then it also satisfies 00:58:33.820 --> 00:58:48.342 align:middle line:84% a greatest lower bound property, which I'll state as a theorem, 00:58:48.342 --> 00:58:51.850 align:middle line:90% and then we'll prove. 00:58:51.850 --> 00:59:11.780 align:middle line:84% Let F be an ordered field with the least upper bound property. 00:59:11.780 --> 00:59:14.170 align:middle line:84% Now, I'm going to prove that it has, if you like, 00:59:14.170 --> 00:59:18.280 align:middle line:90% a greatest lower bound property. 00:59:18.280 --> 00:59:26.120 align:middle line:84% Then if A is a subset of F, which 00:59:26.120 --> 00:59:46.990 align:middle line:84% is non-empty and bounded below, then inf A exists in F, 00:59:46.990 --> 00:59:56.760 align:middle line:84% meaning A has an infimum in the set F. 00:59:56.760 --> 01:00:01.920 align:middle line:84% OK, so the proof of this is basically-- 01:00:01.920 --> 01:00:04.530 align:middle line:84% in some sense, we did something similar 01:00:04.530 --> 01:00:10.350 align:middle line:84% when we proved that one set given by all the minus 1, 01:00:10.350 --> 01:00:13.320 align:middle line:84% minus 2, minus 3 had the greatest upper bound 01:00:13.320 --> 01:00:16.710 align:middle line:84% property or the least upper bound property 01:00:16.710 --> 01:00:20.310 align:middle line:84% by taking its minus and then using, 01:00:20.310 --> 01:00:22.860 align:middle line:84% in a sense, the greatest lower bound property 01:00:22.860 --> 01:00:27.660 align:middle line:84% of the natural numbers, this well-ordering principle, 01:00:27.660 --> 01:00:30.565 align:middle line:84% to conclude that that set had the least upper bound property. 01:00:30.565 --> 01:00:32.190 align:middle line:84% And that's what we're going to do here, 01:00:32.190 --> 01:00:36.040 align:middle line:84% is we're going to take a set which is bounded below, 01:00:36.040 --> 01:00:43.330 align:middle line:84% take its minus, if you like, which is now bounded above, 01:00:43.330 --> 01:00:45.340 align:middle line:84% take the sup of that set, which we can do, 01:00:45.340 --> 01:00:49.780 align:middle line:84% and show that that's the infimum of that set. 01:00:49.780 --> 01:00:52.030 align:middle line:90% So let me write over here. 01:00:52.030 --> 01:00:58.180 align:middle line:84% This should be not part of the proof, but some intuition. 01:00:58.180 --> 01:01:02.910 align:middle line:90% 01:01:02.910 --> 01:01:08.790 align:middle line:84% And I'll draw it like F is a real number line, which 01:01:08.790 --> 01:01:11.370 align:middle line:84% it is because part of the statement 01:01:11.370 --> 01:01:13.020 align:middle line:84% about the real numbers is that it's 01:01:13.020 --> 01:01:15.930 align:middle line:84% the unique ordered field with the least upper bound property. 01:01:15.930 --> 01:01:19.770 align:middle line:84% But don't worry about that for now. 01:01:19.770 --> 01:01:32.790 align:middle line:84% Let's imagine we have a set A. And for now, I'll 01:01:32.790 --> 01:01:37.180 align:middle line:84% draw it like it's an interval, which is bounded below. 01:01:37.180 --> 01:01:39.390 align:middle line:90% So it stops after some point. 01:01:39.390 --> 01:01:41.520 align:middle line:90% And there's nothing there. 01:01:41.520 --> 01:01:45.030 align:middle line:84% So it's bounded below, then if I look at minus A-- 01:01:45.030 --> 01:01:53.190 align:middle line:90% 01:01:53.190 --> 01:01:54.990 align:middle line:90% so here, I'm drawing 0. 01:01:54.990 --> 01:01:58.230 align:middle line:84% If I look at minus A, which is the set of elements-- 01:01:58.230 --> 01:02:00.525 align:middle line:84% the set of the additive inverses of A, 01:02:00.525 --> 01:02:02.520 align:middle line:84% I now have a set which is bounded above. 01:02:02.520 --> 01:02:06.930 align:middle line:84% So if this is a lower bound for A, 01:02:06.930 --> 01:02:10.320 align:middle line:84% then minus B will be an upper bound for minus A. 01:02:10.320 --> 01:02:14.670 align:middle line:84% And therefore, minus A has a least upper bound which, 01:02:14.670 --> 01:02:18.570 align:middle line:84% in this picture I'm drawing, looks like x. 01:02:18.570 --> 01:02:23.085 align:middle line:84% And so then my goal is then to show that-- 01:02:23.085 --> 01:02:29.180 align:middle line:90% 01:02:29.180 --> 01:02:32.620 align:middle line:84% so here, let's go back to B. Here's A. 01:02:32.620 --> 01:02:36.550 align:middle line:84% And here's now minus x to show that minus x 01:02:36.550 --> 01:02:43.870 align:middle line:84% is an infinum of A. So that's the basic intuition on why 01:02:43.870 --> 01:02:44.890 align:middle line:90% this holds. 01:02:44.890 --> 01:02:47.890 align:middle line:84% We're using the ordered field property 01:02:47.890 --> 01:02:51.310 align:middle line:90% to be able to take minuses. 01:02:51.310 --> 01:02:53.230 align:middle line:84% That's where the field property is coming in. 01:02:53.230 --> 01:02:55.670 align:middle line:84% And that minuses-- although we didn't prove this, 01:02:55.670 --> 01:02:56.650 align:middle line:90% it is one of those-- 01:02:56.650 --> 01:03:02.510 align:middle line:90% 01:03:02.510 --> 01:03:04.910 align:middle line:84% essentially, we did prove this, that if I have something 01:03:04.910 --> 01:03:08.450 align:middle line:84% positive and I multiply it through by minus 1 using this 01:03:08.450 --> 01:03:13.100 align:middle line:84% as well, then that reverses the inequality. 01:03:13.100 --> 01:03:14.750 align:middle line:90% But again, these are short-- 01:03:14.750 --> 01:03:18.510 align:middle line:90% 01:03:18.510 --> 01:03:21.840 align:middle line:84% unless I'm telling you to prove a certain inequality, 01:03:21.840 --> 01:03:31.930 align:middle line:84% simple inequality statement like of this type, just 01:03:31.930 --> 01:03:34.660 align:middle line:84% freely use the inequality facts that you 01:03:34.660 --> 01:03:36.970 align:middle line:90% remember from high school. 01:03:36.970 --> 01:03:39.800 align:middle line:90% 01:03:39.800 --> 01:03:42.590 align:middle line:84% And just know that these persist for ordered fields 01:03:42.590 --> 01:03:43.580 align:middle line:90% with the least-- 01:03:43.580 --> 01:03:46.040 align:middle line:90% or ordered fields in general. 01:03:46.040 --> 01:03:55.740 align:middle line:84% OK, so let's turn this intuition into a proof. 01:03:55.740 --> 01:04:01.000 align:middle line:90% 01:04:01.000 --> 01:04:03.910 align:middle line:84% Suppose A-- so I am not stating it here, 01:04:03.910 --> 01:04:07.660 align:middle line:84% but F is an ordered field with the least upper bound property. 01:04:07.660 --> 01:04:09.370 align:middle line:90% I'll go ahead and state it. 01:04:09.370 --> 01:04:12.380 align:middle line:90% 01:04:12.380 --> 01:04:25.030 align:middle line:84% F is an ordered field with the least upper bound property. 01:04:25.030 --> 01:04:27.680 align:middle line:90% 01:04:27.680 --> 01:04:36.060 align:middle line:84% Let A be subset of F, A not equal 01:04:36.060 --> 01:04:44.040 align:middle line:84% to an empty set, A bounded below. 01:04:44.040 --> 01:04:49.430 align:middle line:90% 01:04:49.430 --> 01:04:56.590 align:middle line:84% Then that means there exists an element b in F such 01:04:56.590 --> 01:05:07.170 align:middle line:84% that b is less than or equal to a such 01:05:07.170 --> 01:05:10.950 align:middle line:84% that, for all a in capital A, b is less than or equal to a. 01:05:10.950 --> 01:05:20.370 align:middle line:90% 01:05:20.370 --> 01:05:24.090 align:middle line:84% So I mean, the proof is essentially there 01:05:24.090 --> 01:05:30.630 align:middle line:84% and you just need to turn it into English 01:05:30.630 --> 01:05:32.082 align:middle line:90% because we're in Massachusetts. 01:05:32.082 --> 01:05:33.540 align:middle line:84% If you were in a different country, 01:05:33.540 --> 01:05:36.510 align:middle line:84% you'd turn into the language that's spoken there. 01:05:36.510 --> 01:05:39.960 align:middle line:84% You just need to turn it into written word. 01:05:39.960 --> 01:05:49.570 align:middle line:84% So there exists a lower bound for A. 01:05:49.570 --> 01:05:52.890 align:middle line:84% Let me just note another set called minus-- 01:05:52.890 --> 01:05:57.570 align:middle line:84% which I'll label minus A. This is the set of all elements 01:05:57.570 --> 01:06:02.205 align:middle line:84% in F of the form minus the additive inverses of A, 01:06:02.205 --> 01:06:08.250 align:middle line:84% as a is in capital A. So it's the additive inverse 01:06:08.250 --> 01:06:15.050 align:middle line:84% of all elements in capital A. Then 01:06:15.050 --> 01:06:21.630 align:middle line:84% the fact that I have for all a in A b 01:06:21.630 --> 01:06:30.810 align:middle line:84% is less than or equal to a implies for all a in A, minus a 01:06:30.810 --> 01:06:37.110 align:middle line:84% is less than or equal to minus b because multiplying through 01:06:37.110 --> 01:06:39.510 align:middle line:90% by minus 1 flips the inequality. 01:06:39.510 --> 01:06:44.140 align:middle line:90% 01:06:44.140 --> 01:06:46.720 align:middle line:84% That implies-- so minus b is bigger than 01:06:46.720 --> 01:06:49.540 align:middle line:84% or equal to every element of minus A So 01:06:49.540 --> 01:07:03.230 align:middle line:84% that implies b is an upper bound for the set minus A. 01:07:03.230 --> 01:07:09.620 align:middle line:84% So minus A is a non-empty subset of F, which is bounded above. 01:07:09.620 --> 01:07:13.580 align:middle line:90% Therefore, it has a supremum. 01:07:13.580 --> 01:07:18.320 align:middle line:84% Thus, there exists an x in F such 01:07:18.320 --> 01:07:27.130 align:middle line:84% that x equals sup of the set minus A. 01:07:27.130 --> 01:07:31.130 align:middle line:84% And this is because we are assuming the least upper 01:07:31.130 --> 01:07:33.710 align:middle line:84% bound property, that every non-empty set which is bounded 01:07:33.710 --> 01:07:35.270 align:middle line:90% above has a supremum. 01:07:35.270 --> 01:07:38.750 align:middle line:90% 01:07:38.750 --> 01:07:43.460 align:middle line:84% I'm going to show this guy x is, in fact, the infimum of the set 01:07:43.460 --> 01:07:56.920 align:middle line:84% A, or minus x is the infimum of A. 01:07:56.920 --> 01:07:59.650 align:middle line:84% So the fact that x is the supremum of minus A 01:07:59.650 --> 01:08:05.080 align:middle line:84% implies that for all a in capital A, minus a is less than 01:08:05.080 --> 01:08:07.060 align:middle line:84% or equal to x because x is supposed 01:08:07.060 --> 01:08:09.700 align:middle line:90% to be the least upper bound. 01:08:09.700 --> 01:08:11.260 align:middle line:84% So it's, in fact, an upper bound. 01:08:11.260 --> 01:08:14.180 align:middle line:90% 01:08:14.180 --> 01:08:17.510 align:middle line:84% So that means that for all a in A-- 01:08:17.510 --> 01:08:20.210 align:middle line:84% again, just flipping the inequalities-- minus x is less 01:08:20.210 --> 01:08:23.689 align:middle line:84% than or equal to A, which implies minus x 01:08:23.689 --> 01:08:33.220 align:middle line:90% is a lower bound for capital A. 01:08:33.220 --> 01:08:37.870 align:middle line:84% We now have to show that minus x is the greatest lower bound 01:08:37.870 --> 01:08:41.310 align:middle line:84% of A. If I take any other lower bound of A, 01:08:41.310 --> 01:08:45.885 align:middle line:84% then minus x is bigger than or equal to that lower bound. 01:08:45.885 --> 01:08:53.270 align:middle line:90% 01:08:53.270 --> 01:08:55.710 align:middle line:90% Now show that if-- 01:08:55.710 --> 01:08:58.580 align:middle line:90% let's call it something else-- 01:08:58.580 --> 01:09:11.510 align:middle line:84% y is the lower bound for A, then y is less than 01:09:11.510 --> 01:09:13.955 align:middle line:90% or equal to minus x. 01:09:13.955 --> 01:09:15.330 align:middle line:84% And then that concludes the proof 01:09:15.330 --> 01:09:18.690 align:middle line:84% showing that minus x is, in fact, the infimum of A. 01:09:18.690 --> 01:09:21.899 align:middle line:90% And therefore, A has an infinum. 01:09:21.899 --> 01:09:24.180 align:middle line:84% We've actually identified what the infinum is. 01:09:24.180 --> 01:09:26.830 align:middle line:90% It's the sup of minus A. 01:09:26.830 --> 01:09:41.960 align:middle line:84% OK, so let y be a lower bound for A. 01:09:41.960 --> 01:09:45.649 align:middle line:84% Then exactly how I showed for this single lower bound 01:09:45.649 --> 01:09:49.010 align:middle line:84% that I had for A, you can verify this again. 01:09:49.010 --> 01:09:53.899 align:middle line:84% Just go through the argument and replace b with y. 01:09:53.899 --> 01:09:57.545 align:middle line:90% Then minus y is an upper bound-- 01:09:57.545 --> 01:10:00.930 align:middle line:90% 01:10:00.930 --> 01:10:03.720 align:middle line:84% just look at the picture, replace b with y-- 01:10:03.720 --> 01:10:07.590 align:middle line:90% 01:10:07.590 --> 01:10:13.665 align:middle line:84% for minus A, which implies since x is the supremum of A-- 01:10:13.665 --> 01:10:23.340 align:middle line:90% 01:10:23.340 --> 01:10:25.890 align:middle line:84% so since x is the supremum of minus A, 01:10:25.890 --> 01:10:27.330 align:middle line:90% it's the least upper bound. 01:10:27.330 --> 01:10:29.535 align:middle line:84% So it has to be less than or equal to minus y. 01:10:29.535 --> 01:10:32.460 align:middle line:90% 01:10:32.460 --> 01:10:35.040 align:middle line:84% And flipping the inequality again on that 01:10:35.040 --> 01:10:39.210 align:middle line:84% means that y is less than or equal 01:10:39.210 --> 01:10:48.680 align:middle line:84% to minus x which is what we wanted to prove. 01:10:48.680 --> 01:10:55.650 align:middle line:90% Thus, the inf of A exists. 01:10:55.650 --> 01:10:57.480 align:middle line:84% And we've, in fact, showed that it's 01:10:57.480 --> 01:11:08.770 align:middle line:84% equal to the sup of minus A. So in an ordered field 01:11:08.770 --> 01:11:10.270 align:middle line:84% with the least upper bound property, 01:11:10.270 --> 01:11:16.180 align:middle line:84% not only does every set which is bounded above have a supremum, 01:11:16.180 --> 01:11:18.520 align:middle line:84% every set which is bounded below has an infimum. 01:11:18.520 --> 01:11:22.740 align:middle line:90% 01:11:22.740 --> 01:11:27.240 align:middle line:84% So now, we move on and not talk about generalities, 01:11:27.240 --> 01:11:33.490 align:middle line:84% but we're just going to focus on R, the set of real numbers. 01:11:33.490 --> 01:11:35.610 align:middle line:84% And I'm just going to, again, state 01:11:35.610 --> 01:11:38.220 align:middle line:84% this theorem about the existence of R 01:11:38.220 --> 01:11:40.470 align:middle line:84% and its properties, so just to bring this 01:11:40.470 --> 01:11:54.220 align:middle line:84% all back to our goal of describing exactly what R is 01:11:54.220 --> 01:12:01.190 align:middle line:84% or what separates it from Q is that-- so there 01:12:01.190 --> 01:12:17.490 align:middle line:84% exists a unique ordered field with the least 01:12:17.490 --> 01:12:29.820 align:middle line:84% upper bound property containing Q. 01:12:29.820 --> 01:12:46.030 align:middle line:90% And this field we denote by R. 01:12:46.030 --> 01:12:48.820 align:middle line:84% So Q does-- just to bring this back, 01:12:48.820 --> 01:12:56.980 align:middle line:84% so we started off in ancient times with the natural numbers. 01:12:56.980 --> 01:12:58.458 align:middle line:84% We moved to the integers so that we 01:12:58.458 --> 01:13:00.250 align:middle line:84% could take additive inverses, although they 01:13:00.250 --> 01:13:03.220 align:middle line:90% didn't call them that. 01:13:03.220 --> 01:13:04.752 align:middle line:90% And we wanted 0. 01:13:04.752 --> 01:13:06.460 align:middle line:84% And then we moved to the rational numbers 01:13:06.460 --> 01:13:13.120 align:middle line:84% because we didn't have ways to solve the equation 01:13:13.120 --> 01:13:14.440 align:middle line:90% 2x plus 1 equals 0. 01:13:14.440 --> 01:13:17.440 align:middle line:90% 01:13:17.440 --> 01:13:22.210 align:middle line:84% And so we moved on from Q to R essentially 01:13:22.210 --> 01:13:25.720 align:middle line:84% because we can't solve the equation x squared minus 2 01:13:25.720 --> 01:13:27.100 align:middle line:90% equals 0. 01:13:27.100 --> 01:13:30.940 align:middle line:84% And this inability to solve x squared minus 2 01:13:30.940 --> 01:13:41.190 align:middle line:84% equals 0, although an algebraic fact, in fact 01:13:41.190 --> 01:13:45.150 align:middle line:84% means that Q is incomplete as an ordered set. 01:13:45.150 --> 01:13:47.355 align:middle line:84% It does not have this least upper bound property. 01:13:47.355 --> 01:13:51.220 align:middle line:90% 01:13:51.220 --> 01:13:53.890 align:middle line:84% And what characterizes the real numbers 01:13:53.890 --> 01:13:58.800 align:middle line:84% is that it is an ordered field containing Q. 01:13:58.800 --> 01:14:01.300 align:middle line:84% And it's the unique one with the least upper bound property. 01:14:01.300 --> 01:14:06.510 align:middle line:90% 01:14:06.510 --> 01:14:14.280 align:middle line:84% This unique should be kind of in quotes 01:14:14.280 --> 01:14:19.190 align:middle line:84% because unique up to what's called isomorphism. 01:14:19.190 --> 01:14:27.600 align:middle line:84% Isomorphism is a fancy way of saying what you call apples 01:14:27.600 --> 01:14:33.350 align:middle line:90% I call manzanas, essentially. 01:14:33.350 --> 01:14:34.517 align:middle line:90% OK, so this is what R is. 01:14:34.517 --> 01:14:36.350 align:middle line:84% It's the unique ordered field with the least 01:14:36.350 --> 01:14:42.780 align:middle line:84% upper bound property containing Q. 01:14:42.780 --> 01:14:44.640 align:middle line:84% So I'm not going to prove this theorem. 01:14:44.640 --> 01:14:47.910 align:middle line:84% The way you usually prove this theorem, 01:14:47.910 --> 01:14:53.170 align:middle line:84% you construct R either as what are called Dedekind cuts 01:14:53.170 --> 01:14:57.640 align:middle line:84% or as equivalence classes of Cauchy sequences. 01:14:57.640 --> 01:15:06.620 align:middle line:84% We'll talk about Cauchy sequences soon-ish. 01:15:06.620 --> 01:15:12.560 align:middle line:84% But I'm more interested in proving properties about R, 01:15:12.560 --> 01:15:16.100 align:middle line:84% and then going on to functions on R, limits, 01:15:16.100 --> 01:15:21.560 align:middle line:84% which is what analysis is, is the study of limits, 01:15:21.560 --> 01:15:27.650 align:middle line:84% rather than get tied up in really non-analytic facts, 01:15:27.650 --> 01:15:29.970 align:middle line:84% algebraic facts trying to construct 01:15:29.970 --> 01:15:32.740 align:middle line:90% this R, the actual field. 01:15:32.740 --> 01:15:34.620 align:middle line:84% So we're just going to take this as a given. 01:15:34.620 --> 01:15:36.630 align:middle line:84% And now, we're going to go from here 01:15:36.630 --> 01:15:39.120 align:middle line:84% and start proving facts about real numbers. 01:15:39.120 --> 01:15:44.720 align:middle line:90% 01:15:44.720 --> 01:15:50.645 align:middle line:90% Where Q failed, R succeeds. 01:15:50.645 --> 01:15:54.250 align:middle line:90% 01:15:54.250 --> 01:16:02.130 align:middle line:84% So the first fact is that there exists a unique element 01:16:02.130 --> 01:16:10.440 align:middle line:84% in R such that r is positive and r squared equals 2. 01:16:10.440 --> 01:16:14.070 align:middle line:84% So we saw before that if I replace this 01:16:14.070 --> 01:16:16.830 align:middle line:84% with Q, ta rational number, that's false. 01:16:16.830 --> 01:16:19.770 align:middle line:84% There does not exist a rational number whose square is 2. 01:16:19.770 --> 01:16:21.360 align:middle line:84% But in the real numbers, there does. 01:16:21.360 --> 01:16:25.150 align:middle line:90% 01:16:25.150 --> 01:16:27.400 align:middle line:84% And right now, maybe you're tempted to just say, yeah, 01:16:27.400 --> 01:16:28.960 align:middle line:84% you set R equal to square root of 2. 01:16:28.960 --> 01:16:31.900 align:middle line:90% Well, what is square root of 2? 01:16:31.900 --> 01:16:36.170 align:middle line:84% I mean, how do you come up with that guy? 01:16:36.170 --> 01:16:41.000 align:middle line:84% So we have to come up with some element of R whose square is 2. 01:16:41.000 --> 01:16:42.590 align:middle line:84% Now, we basically did that a minute 01:16:42.590 --> 01:16:44.000 align:middle line:90% ago in the rational numbers. 01:16:44.000 --> 01:16:47.310 align:middle line:84% And kind of the same proof works here. 01:16:47.310 --> 01:17:03.310 align:middle line:84% So let E be the set of all x's in R such that x is positive 01:17:03.310 --> 01:17:04.930 align:middle line:90% and x squared is less than 2. 01:17:04.930 --> 01:17:09.100 align:middle line:84% So earlier in the lecture and at the end of the last lecture, 01:17:09.100 --> 01:17:11.500 align:middle line:90% we had q here, right? 01:17:11.500 --> 01:17:20.850 align:middle line:84% Well, the same proof, basically And what we did earlier-- then 01:17:20.850 --> 01:17:30.800 align:middle line:90% E is bounded above by 2. 01:17:30.800 --> 01:17:32.900 align:middle line:90% So R, which I-- 01:17:32.900 --> 01:17:35.890 align:middle line:90% 01:17:35.890 --> 01:17:46.560 align:middle line:90% so sup E tilde exists in R. 01:17:46.560 --> 01:17:48.330 align:middle line:90% So I have this set. 01:17:48.330 --> 01:17:50.400 align:middle line:90% It's bounded above by 2. 01:17:50.400 --> 01:17:52.510 align:middle line:84% I mean, we did the proof earlier. 01:17:52.510 --> 01:17:55.230 align:middle line:84% So by the least upper bound property of R, 01:17:55.230 --> 01:17:57.300 align:middle line:90% the supremum exists. 01:17:57.300 --> 01:17:58.740 align:middle line:90% Call this element little r. 01:17:58.740 --> 01:18:12.170 align:middle line:90% 01:18:12.170 --> 01:18:13.700 align:middle line:90% Then the same proof-- 01:18:13.700 --> 01:18:18.260 align:middle line:84% I'm not going to do it again because we've already done it. 01:18:18.260 --> 01:18:22.640 align:middle line:84% And all you need to do is replace q's with r's-- 01:18:22.640 --> 01:18:30.040 align:middle line:90% 01:18:30.040 --> 01:18:32.620 align:middle line:84% shows r is bigger than or equal to 1. 01:18:32.620 --> 01:18:35.670 align:middle line:90% In fact, it's bigger than 1. 01:18:35.670 --> 01:18:39.300 align:middle line:90% And r squared equals 2. 01:18:39.300 --> 01:18:46.050 align:middle line:84% So there does exist an element of r which is positive 01:18:46.050 --> 01:18:47.610 align:middle line:90% and whose square is 2. 01:18:47.610 --> 01:18:50.430 align:middle line:84% Now, I will prove that it's unique. 01:18:50.430 --> 01:18:53.470 align:middle line:90% 01:18:53.470 --> 01:18:55.180 align:middle line:84% So now, I want to prove that r is unique. 01:18:55.180 --> 01:19:06.340 align:middle line:90% 01:19:06.340 --> 01:19:08.710 align:middle line:84% That means if I take-- if there's 01:19:08.710 --> 01:19:11.920 align:middle line:84% some other element in capital R that satisfies these two 01:19:11.920 --> 01:19:17.550 align:middle line:84% inequalities, then it must be equal to my original r. 01:19:17.550 --> 01:19:27.420 align:middle line:84% So suppose r tilde is in R. r tilde is bigger than 0. 01:19:27.420 --> 01:19:30.090 align:middle line:90% And r tilde squared equals 2. 01:19:30.090 --> 01:19:35.370 align:middle line:90% 01:19:35.370 --> 01:19:40.250 align:middle line:84% Then since their square is the same number, 01:19:40.250 --> 01:19:47.340 align:middle line:84% namely 2, 0 is equal to r tilde minus r squared, 01:19:47.340 --> 01:19:54.860 align:middle line:84% which equals r tilde plus r, r tilde minus r. 01:19:54.860 --> 01:19:58.790 align:middle line:84% Now, both r and r tilde are positive. 01:19:58.790 --> 01:20:01.160 align:middle line:90% So r plus r tilde is positive. 01:20:01.160 --> 01:20:03.110 align:middle line:90% In particular, it's non-zero. 01:20:03.110 --> 01:20:06.980 align:middle line:84% So since it's a field, I can therefore divide or multiply 01:20:06.980 --> 01:20:10.760 align:middle line:84% both sides by the additive inverse of this thing 01:20:10.760 --> 01:20:17.670 align:middle line:84% and arrive at 0 is equal to r tilde minus r or r 01:20:17.670 --> 01:20:18.870 align:middle line:90% equals r tilde. 01:20:18.870 --> 01:20:22.560 align:middle line:84% So there exist only one element in the real numbers 01:20:22.560 --> 01:20:25.295 align:middle line:84% which is positive and whose square is 2. 01:20:25.295 --> 01:20:30.750 align:middle line:90% 01:20:30.750 --> 01:20:41.830 align:middle line:84% And I'll put it on the assignment, namely that-- 01:20:41.830 --> 01:20:43.470 align:middle line:90% so this shows what? 01:20:43.470 --> 01:20:46.680 align:middle line:90% Square root of 2 exists. 01:20:46.680 --> 01:21:02.250 align:middle line:84% But you can then show a cube root of 2 exists in R. 01:21:02.250 --> 01:21:05.050 align:middle line:90% And we'll not prove this. 01:21:05.050 --> 01:21:13.900 align:middle line:84% But you can show, in general, that if x is in R, then x 01:21:13.900 --> 01:21:18.190 align:middle line:90% to the 1 over n-- 01:21:18.190 --> 01:21:27.800 align:middle line:84% that should say x-- is positive for all natural numbers. 01:21:27.800 --> 01:21:33.620 align:middle line:90% 01:21:33.620 --> 01:21:36.320 align:middle line:90% So where Q failed, R succeeds. 01:21:36.320 --> 01:21:39.380 align:middle line:84% And it's doing this-- or the fact 01:21:39.380 --> 01:21:42.620 align:middle line:84% that it does succeed is not coming 01:21:42.620 --> 01:21:45.320 align:middle line:84% from an algebraic property of R. It's 01:21:45.320 --> 01:21:48.980 align:middle line:84% coming from this property about its order. 01:21:48.980 --> 01:21:52.010 align:middle line:90% 01:21:52.010 --> 01:21:55.210 align:middle line:90% All right, we'll stop there. 01:21:55.210 --> 01:22:04.000 align:middle line:90%