WEBVTT 00:00:00.000 --> 00:00:02.470 align:middle line:90% [SQUEAKING] 00:00:02.470 --> 00:00:04.446 align:middle line:90% [RUSTLING] 00:00:04.446 --> 00:00:07.410 align:middle line:90% [CLICKING] 00:00:07.410 --> 00:00:10.097 align:middle line:90% 00:00:10.097 --> 00:00:12.680 align:middle line:84% TOBIAS COLDING: So this here is just to make sure that you are 00:00:12.680 --> 00:00:28.440 align:middle line:84% in the right place, which is 18.100B or 18.1002. 00:00:28.440 --> 00:00:31.430 align:middle line:84% And my name is Toby, so Toby Colding. 00:00:31.430 --> 00:00:36.560 align:middle line:90% 00:00:36.560 --> 00:00:40.440 align:middle line:84% And by the way, so we will be recorded. 00:00:40.440 --> 00:00:46.160 align:middle line:84% And the video from the lecture will be posted online. 00:00:46.160 --> 00:00:49.700 align:middle line:84% And I will also write lecture notes. 00:00:49.700 --> 00:00:53.140 align:middle line:84% I hope you appreciate it because I wrote the first one. 00:00:53.140 --> 00:00:54.920 align:middle line:84% It actually took me quite a long time. 00:00:54.920 --> 00:00:58.060 align:middle line:84% I don't typically type the lecture notes, 00:00:58.060 --> 00:01:01.340 align:middle line:84% but I will do this for this class. 00:01:01.340 --> 00:01:06.800 align:middle line:84% And I also wanted to just give you an idea of the structure. 00:01:06.800 --> 00:01:09.580 align:middle line:84% First of all, this is one of the most important math 00:01:09.580 --> 00:01:12.620 align:middle line:90% classes at MIT. 00:01:12.620 --> 00:01:16.660 align:middle line:84% And I'll tell you in a minute why it is important. 00:01:16.660 --> 00:01:18.660 align:middle line:84% And of course, I think that, being 00:01:18.660 --> 00:01:20.340 align:middle line:84% one of the most important math classes, 00:01:20.340 --> 00:01:22.600 align:middle line:84% it's one of the most important classes, period. 00:01:22.600 --> 00:01:25.940 align:middle line:90% 00:01:25.940 --> 00:01:30.500 align:middle line:84% But just to give you an idea of what will 00:01:30.500 --> 00:01:32.300 align:middle line:90% be expected from the class-- 00:01:32.300 --> 00:01:34.950 align:middle line:84% so there will be 10 weekly homework. 00:01:34.950 --> 00:01:39.980 align:middle line:90% 00:01:39.980 --> 00:01:47.100 align:middle line:84% And the first one, it won't be due next week. 00:01:47.100 --> 00:01:51.340 align:middle line:84% It will probably be due the week after next week. 00:01:51.340 --> 00:01:56.180 align:middle line:84% And in part, it will just allow you to get settled in a bit. 00:01:56.180 --> 00:01:59.920 align:middle line:84% But also, I'm waiting for somebody else 00:01:59.920 --> 00:02:02.300 align:middle line:90% to assign the TA and the grader. 00:02:02.300 --> 00:02:05.600 align:middle line:84% So I think it will be at least one graduate TA, 00:02:05.600 --> 00:02:10.960 align:middle line:84% and there will be some undergraduate TAs. 00:02:10.960 --> 00:02:14.320 align:middle line:84% And then there will also probably be some graders. 00:02:14.320 --> 00:02:18.020 align:middle line:84% And most of these, including myself, 00:02:18.020 --> 00:02:20.480 align:middle line:90% will have office hours or so. 00:02:20.480 --> 00:02:23.720 align:middle line:90% So it'll be 10 weekly homework. 00:02:23.720 --> 00:02:34.480 align:middle line:84% There will be a midterm, so one midterm and one final. 00:02:34.480 --> 00:02:38.840 align:middle line:90% And the midterm is on March 20. 00:02:38.840 --> 00:02:42.400 align:middle line:90% And the final is-- 00:02:42.400 --> 00:02:43.860 align:middle line:90% I don't know when it is. 00:02:43.860 --> 00:02:48.520 align:middle line:84% I'm not the one that is deciding the date. 00:02:48.520 --> 00:02:53.200 align:middle line:84% And it will be a while before we know what date that will be. 00:02:53.200 --> 00:02:57.940 align:middle line:84% And in terms of the grade, the homework 00:02:57.940 --> 00:03:01.380 align:middle line:84% will count for 50% of your grade, 00:03:01.380 --> 00:03:06.820 align:middle line:84% midterm will count for 20%, and the final for 30% 00:03:06.820 --> 00:03:11.420 align:middle line:84% And you can see all this information about this in-- 00:03:11.420 --> 00:03:13.740 align:middle line:84% there's a syllabus and a course information 00:03:13.740 --> 00:03:16.340 align:middle line:90% posted on the Canvas. 00:03:16.340 --> 00:03:23.260 align:middle line:84% So make sure to check out the Canvas site for the class. 00:03:23.260 --> 00:03:26.300 align:middle line:84% And there's also the information about the book. 00:03:26.300 --> 00:03:30.660 align:middle line:84% And also, the lecture notes will be posted on the Canvas site. 00:03:30.660 --> 00:03:34.320 align:middle line:84% The lecture notes, I think, will probably be posted on the files. 00:03:34.320 --> 00:03:41.310 align:middle line:84% So lecture notes will be posted on the files. 00:03:41.310 --> 00:03:54.200 align:middle line:90% 00:03:54.200 --> 00:03:59.040 align:middle line:84% And one other thing is that I have activated 00:03:59.040 --> 00:04:00.470 align:middle line:90% what's called pset partners. 00:04:00.470 --> 00:04:06.200 align:middle line:90% 00:04:06.200 --> 00:04:12.100 align:middle line:84% So you are encouraged to work with other people on the psets, 00:04:12.100 --> 00:04:15.400 align:middle line:84% but everybody have to hand in their own psets. 00:04:15.400 --> 00:04:21.800 align:middle line:84% And pset partners is a way of finding other people 00:04:21.800 --> 00:04:23.180 align:middle line:90% to work with. 00:04:23.180 --> 00:04:25.260 align:middle line:84% Of course, you can do it another way. 00:04:25.260 --> 00:04:29.130 align:middle line:84% But this is a tool that is used in a lot of classes. 00:04:29.130 --> 00:04:31.460 align:middle line:90% 00:04:31.460 --> 00:04:31.960 align:middle line:90% OK. 00:04:31.960 --> 00:04:36.920 align:middle line:84% I think this covered the basic information. 00:04:36.920 --> 00:04:39.080 align:middle line:90% And so the class here is-- 00:04:39.080 --> 00:04:44.480 align:middle line:84% so the two key topics for this class is-- 00:04:44.480 --> 00:04:54.820 align:middle line:84% so key topics, one is you will learn how 00:04:54.820 --> 00:04:59.150 align:middle line:90% to write a mathematical proof. 00:04:59.150 --> 00:05:05.060 align:middle line:90% 00:05:05.060 --> 00:05:09.080 align:middle line:84% So what does it mean to write something 00:05:09.080 --> 00:05:12.380 align:middle line:84% so that is mathematically rigorous? 00:05:12.380 --> 00:05:14.220 align:middle line:84% And the second thing that you will learn 00:05:14.220 --> 00:05:15.630 align:middle line:90% is how to prove theorems. 00:05:15.630 --> 00:05:25.820 align:middle line:90% 00:05:25.820 --> 00:05:29.940 align:middle line:84% And so here's an example of this. 00:05:29.940 --> 00:05:32.660 align:middle line:84% Let's suppose you have a function. 00:05:32.660 --> 00:05:34.440 align:middle line:84% So suppose you have some interval. 00:05:34.440 --> 00:05:38.380 align:middle line:90% So a and b are real numbers. 00:05:38.380 --> 00:05:44.740 align:middle line:84% And you have a function f defined on this interval. 00:05:44.740 --> 00:05:47.140 align:middle line:90% And you take real values. 00:05:47.140 --> 00:05:49.800 align:middle line:84% And so here, you have the interval here, a. 00:05:49.800 --> 00:05:52.480 align:middle line:90% Here's b. 00:05:52.480 --> 00:05:58.620 align:middle line:84% And let's say that the value at a is negative, 00:05:58.620 --> 00:06:03.600 align:middle line:84% and the value at the other endpoint, b, is positive. 00:06:03.600 --> 00:06:08.280 align:middle line:84% So if you were drawing the graph, here, you have a, maybe. 00:06:08.280 --> 00:06:09.920 align:middle line:90% Here, you have b. 00:06:09.920 --> 00:06:13.160 align:middle line:84% And then the function initially is negative, 00:06:13.160 --> 00:06:15.300 align:middle line:90% and it end up being positive. 00:06:15.300 --> 00:06:17.360 align:middle line:90% So this is f of a. 00:06:17.360 --> 00:06:20.840 align:middle line:90% Here, you have f of b. 00:06:20.840 --> 00:06:23.420 align:middle line:84% And then this function here is continuous. 00:06:23.420 --> 00:06:28.440 align:middle line:90% 00:06:28.440 --> 00:06:31.100 align:middle line:84% So the function looks maybe like this. 00:06:31.100 --> 00:06:34.840 align:middle line:90% 00:06:34.840 --> 00:06:36.500 align:middle line:90% So f is continuous. 00:06:36.500 --> 00:06:39.050 align:middle line:84% So the intermediate value theorem's saying-- 00:06:39.050 --> 00:06:54.850 align:middle line:90% 00:06:54.850 --> 00:06:56.780 align:middle line:84% so the intermediate value theorem. 00:06:56.780 --> 00:07:09.867 align:middle line:90% 00:07:09.867 --> 00:07:11.450 align:middle line:84% It's saying the following, that if you 00:07:11.450 --> 00:07:15.130 align:middle line:84% have a continuous function on an interval, then 00:07:15.130 --> 00:07:18.650 align:middle line:90% it takes all values in between. 00:07:18.650 --> 00:07:21.570 align:middle line:84% So the intermediate value theorem says in particular 00:07:21.570 --> 00:07:27.810 align:middle line:90% that there exists a c. 00:07:27.810 --> 00:07:32.930 align:middle line:84% And the c here is between a and b 00:07:32.930 --> 00:07:38.190 align:middle line:84% so that the value of the function at c is 0. 00:07:38.190 --> 00:07:42.330 align:middle line:84% Because the function starts off negative, and it end positive. 00:07:42.330 --> 00:07:45.810 align:middle line:84% So there must be somewhere in between where it hits 0. 00:07:45.810 --> 00:07:49.150 align:middle line:84% That's the intermediate value theorem. 00:07:49.150 --> 00:07:55.390 align:middle line:84% But now the question is then, how do we make this into-- 00:07:55.390 --> 00:07:57.470 align:middle line:90% and it's kind of obvious. 00:07:57.470 --> 00:08:04.790 align:middle line:84% It's sort of intuitively obvious that if you take a function, 00:08:04.790 --> 00:08:08.590 align:middle line:84% continuous means something like that if you were drawing 00:08:08.590 --> 00:08:13.550 align:middle line:84% the graph, and you'd have this piece of chalk here, 00:08:13.550 --> 00:08:17.110 align:middle line:84% then as you're drawing the graph, 00:08:17.110 --> 00:08:20.690 align:middle line:84% the chalk is not allowed to leave your table. 00:08:20.690 --> 00:08:24.030 align:middle line:84% It can't have a function like this here 00:08:24.030 --> 00:08:28.210 align:middle line:84% that jumps, so go like this, and then jumps like this. 00:08:28.210 --> 00:08:30.330 align:middle line:90% That's not continuous. 00:08:30.330 --> 00:08:34.030 align:middle line:90% This is not continuous. 00:08:34.030 --> 00:08:38.730 align:middle line:84% So continuous function is that you have to, when you draw it, 00:08:38.730 --> 00:08:40.470 align:middle line:90% it's in one motion. 00:08:40.470 --> 00:08:44.130 align:middle line:84% The chalk doesn't leave the blackboard. 00:08:44.130 --> 00:08:47.490 align:middle line:84% And so it seems obvious that there 00:08:47.490 --> 00:08:51.130 align:middle line:84% should be some point in between where 00:08:51.130 --> 00:08:53.530 align:middle line:90% you're crossing this axis here. 00:08:53.530 --> 00:08:56.070 align:middle line:84% But how do we make this into a rigorous proof? 00:08:56.070 --> 00:08:58.510 align:middle line:90% And what is the property? 00:08:58.510 --> 00:09:00.440 align:middle line:84% So how do we make this into a proof? 00:09:00.440 --> 00:09:04.290 align:middle line:90% 00:09:04.290 --> 00:09:20.950 align:middle line:84% So how do we make this intuition into a proof? 00:09:20.950 --> 00:09:25.210 align:middle line:90% 00:09:25.210 --> 00:09:29.350 align:middle line:84% And this is not just a semantic question. 00:09:29.350 --> 00:09:32.450 align:middle line:90% It's not just a crazy question. 00:09:32.450 --> 00:09:35.730 align:middle line:84% Maybe in this case, it seemed clear. 00:09:35.730 --> 00:09:39.850 align:middle line:84% But very quickly, if your foundation is not 00:09:39.850 --> 00:09:43.110 align:middle line:84% totally rigorous, then you get into something 00:09:43.110 --> 00:09:47.510 align:middle line:84% where it seems to be the case, but it might not be the case. 00:09:47.510 --> 00:09:50.910 align:middle line:84% And it's also very difficult to then-- 00:09:50.910 --> 00:09:52.590 align:middle line:84% quickly, that's where you arrive. 00:09:52.590 --> 00:09:54.690 align:middle line:84% And you also quickly arrive to that 00:09:54.690 --> 00:09:59.090 align:middle line:84% you don't even know whether it should be the case or not. 00:09:59.090 --> 00:10:03.610 align:middle line:84% And so it's really important to build things up, step to step, 00:10:03.610 --> 00:10:06.310 align:middle line:84% where each step is totally rigorous. 00:10:06.310 --> 00:10:10.930 align:middle line:84% And you write it because it's not only 00:10:10.930 --> 00:10:13.850 align:middle line:84% you know for sure that things are right, 00:10:13.850 --> 00:10:18.830 align:middle line:84% but it also makes it much easier to predict what should be true. 00:10:18.830 --> 00:10:24.150 align:middle line:84% And so in order to understand why 00:10:24.150 --> 00:10:30.230 align:middle line:84% there should be some c between a and b 00:10:30.230 --> 00:10:36.190 align:middle line:84% where the value is 0, immediately, the first question 00:10:36.190 --> 00:10:37.550 align:middle line:90% that arises-- 00:10:37.550 --> 00:10:38.730 align:middle line:90% what are you using? 00:10:38.730 --> 00:10:43.110 align:middle line:84% What property do we need to make this into a proof? 00:10:43.110 --> 00:10:45.970 align:middle line:84% What property do we need of the real numbers? 00:10:45.970 --> 00:10:49.830 align:middle line:84% So this is the first question that arises. 00:10:49.830 --> 00:11:05.940 align:middle line:84% So what property are we using of the real numbers? 00:11:05.940 --> 00:11:12.810 align:middle line:90% 00:11:12.810 --> 00:11:18.050 align:middle line:84% And so, in fact, what is the real numbers? 00:11:18.050 --> 00:11:19.640 align:middle line:90% So what are the real numbers? 00:11:19.640 --> 00:11:33.770 align:middle line:90% 00:11:33.770 --> 00:11:41.098 align:middle line:84% And so the thing is that, in particular, 00:11:41.098 --> 00:11:43.990 align:middle line:90% what is square root of 2? 00:11:43.990 --> 00:11:45.690 align:middle line:90% So what is square root of 2? 00:11:45.690 --> 00:11:54.040 align:middle line:90% 00:11:54.040 --> 00:11:55.520 align:middle line:90% And is that a real number? 00:11:55.520 --> 00:11:58.270 align:middle line:90% 00:11:58.270 --> 00:12:00.850 align:middle line:84% Is square root of 2 a real number? 00:12:00.850 --> 00:12:13.070 align:middle line:90% 00:12:13.070 --> 00:12:15.870 align:middle line:84% So those are some of the questions 00:12:15.870 --> 00:12:19.670 align:middle line:90% that you immediately get to. 00:12:19.670 --> 00:12:24.910 align:middle line:84% And so now-- but so the answer to, what 00:12:24.910 --> 00:12:27.730 align:middle line:84% are the properties that you're using of the real numbers? 00:12:27.730 --> 00:12:33.810 align:middle line:84% So the answer to this question here is the following. 00:12:33.810 --> 00:12:43.050 align:middle line:90% 00:12:43.050 --> 00:12:45.110 align:middle line:84% So the answer to this question is-- 00:12:45.110 --> 00:12:50.130 align:middle line:84% so what are the property of the real numbers that you're using? 00:12:50.130 --> 00:12:59.850 align:middle line:84% So the answer to that is the following, 00:12:59.850 --> 00:13:17.322 align:middle line:84% that the real numbers-- so R is a complete ordered field 00:13:17.322 --> 00:13:39.430 align:middle line:84% that contains the rational numbers, the rational numbers, 00:13:39.430 --> 00:13:40.430 align:middle line:90% Q. 00:13:40.430 --> 00:13:43.790 align:middle line:84% I denote the rational numbers by Q and the real numbers 00:13:43.790 --> 00:13:54.110 align:middle line:84% by capital R. So in order to make this the intermediate value 00:13:54.110 --> 00:13:58.670 align:middle line:84% theorem, this property that if you have an integral from a 00:13:58.670 --> 00:14:02.290 align:middle line:84% to b, you have a continuous function on that interval. 00:14:02.290 --> 00:14:04.390 align:middle line:84% And it starts off negative, and it 00:14:04.390 --> 00:14:07.830 align:middle line:84% ends positive-- that then there is some value in between, 00:14:07.830 --> 00:14:09.130 align:middle line:90% where 0 is achieved. 00:14:09.130 --> 00:14:12.390 align:middle line:84% In order to make that rigorous, what you're 00:14:12.390 --> 00:14:15.230 align:middle line:84% using about the real numbers is that it's complete 00:14:15.230 --> 00:14:18.430 align:middle line:84% ordered field that contains irrational numbers. 00:14:18.430 --> 00:14:21.430 align:middle line:84% So we need to understand what this here means. 00:14:21.430 --> 00:14:27.070 align:middle line:84% And in this lecture here, I will not explain complete. 00:14:27.070 --> 00:14:29.430 align:middle line:84% I will do that at a later lecture. 00:14:29.430 --> 00:14:35.050 align:middle line:84% But we will understand the rest of what are these properties 00:14:35.050 --> 00:14:36.570 align:middle line:90% that we're using. 00:14:36.570 --> 00:14:38.470 align:middle line:90% So just first, some notation-- 00:14:38.470 --> 00:14:56.570 align:middle line:90% 00:14:56.570 --> 00:15:00.890 align:middle line:84% so the natural numbers here, this here 00:15:00.890 --> 00:15:08.610 align:middle line:90% is 1, 2, 3, 4, 5, et cetera. 00:15:08.610 --> 00:15:10.220 align:middle line:84% So these are the natural numbers. 00:15:10.220 --> 00:15:21.050 align:middle line:90% 00:15:21.050 --> 00:15:27.340 align:middle line:84% The integers-- so 2 set here, these are the integers. 00:15:27.340 --> 00:15:31.590 align:middle line:90% 00:15:31.590 --> 00:15:36.490 align:middle line:84% And the integers are simply the natural numbers. 00:15:36.490 --> 00:15:39.910 align:middle line:84% But then it also allows them to be negative, 00:15:39.910 --> 00:15:41.910 align:middle line:90% and it also includes 0. 00:15:41.910 --> 00:15:47.710 align:middle line:84% So this is 0 plus or minus 1, plus or minus 2, 00:15:47.710 --> 00:15:51.430 align:middle line:90% plus or minus 3, et cetera. 00:15:51.430 --> 00:15:54.110 align:middle line:90% These are the integers. 00:15:54.110 --> 00:16:05.230 align:middle line:84% The rational numbers are then-- so the rational numbers, Q, 00:16:05.230 --> 00:16:10.810 align:middle line:84% so these are numbers of the form m/n, 00:16:10.810 --> 00:16:17.550 align:middle line:84% where m here is an integer and n here-- 00:16:17.550 --> 00:16:18.400 align:middle line:90% sorry. 00:16:18.400 --> 00:16:22.230 align:middle line:84% Yeah, m is an integer, and n here is a natural number. 00:16:22.230 --> 00:16:24.830 align:middle line:90% 00:16:24.830 --> 00:16:27.950 align:middle line:90% That's a rational number. 00:16:27.950 --> 00:16:32.490 align:middle line:84% And so now what are the properties? 00:16:32.490 --> 00:16:36.090 align:middle line:84% So we're building up to trying to understand what property 00:16:36.090 --> 00:16:37.530 align:middle line:90% of the real numbers is. 00:16:37.530 --> 00:16:42.370 align:middle line:84% But let's try to somehow understand it a bit better. 00:16:42.370 --> 00:16:44.610 align:middle line:84% What are the properties that we really have 00:16:44.610 --> 00:16:47.810 align:middle line:90% about the rational numbers? 00:16:47.810 --> 00:16:51.370 align:middle line:84% So for rational numbers, we have-- 00:16:51.370 --> 00:16:56.956 align:middle line:84% so we write a rational number as a quotient, like that. 00:16:56.956 --> 00:16:59.050 align:middle line:84% But of course, there are many ways 00:16:59.050 --> 00:17:01.370 align:middle line:90% that you can write a quotient. 00:17:01.370 --> 00:17:04.040 align:middle line:84% So properties of the rational numbers-- 00:17:04.040 --> 00:17:16.609 align:middle line:90% 00:17:16.609 --> 00:17:21.569 align:middle line:84% so we write it as a quotient, like this. 00:17:21.569 --> 00:17:26.290 align:middle line:84% But this is not uniquely defined. 00:17:26.290 --> 00:17:29.670 align:middle line:84% If you take a quotient, like this, 00:17:29.670 --> 00:17:35.875 align:middle line:84% you think that this here is equal to another quotient, 00:17:35.875 --> 00:17:38.430 align:middle line:90% right? 00:17:38.430 --> 00:17:47.390 align:middle line:84% Again, m1 and m2 are integers, and n1 and n2 00:17:47.390 --> 00:17:50.110 align:middle line:90% are natural numbers. 00:17:50.110 --> 00:17:53.830 align:middle line:84% Two quotients are the same if and only 00:17:53.830 --> 00:18:00.310 align:middle line:84% if that if you multiply this over here and this over there, 00:18:00.310 --> 00:18:02.970 align:middle line:90% then those numbers are the same. 00:18:02.970 --> 00:18:11.670 align:middle line:84% So n2 times N2 is equal to N2 times n1. 00:18:11.670 --> 00:18:15.030 align:middle line:84% Those two rational numbers are the same if 00:18:15.030 --> 00:18:19.230 align:middle line:84% and only if those two natural numbers are the same-- 00:18:19.230 --> 00:18:23.176 align:middle line:84% or integers, because m could be negative. 00:18:23.176 --> 00:18:27.050 align:middle line:90% 00:18:27.050 --> 00:18:33.450 align:middle line:84% So again, a rational number, it has a representative like this. 00:18:33.450 --> 00:18:37.130 align:middle line:84% But you could think about it as always, you 00:18:37.130 --> 00:18:39.270 align:middle line:90% can write this quotient. 00:18:39.270 --> 00:18:41.970 align:middle line:84% And two quotients are the same if and only 00:18:41.970 --> 00:18:45.210 align:middle line:90% if this here is the case. 00:18:45.210 --> 00:18:48.280 align:middle line:84% So this is what it means to be a rational number. 00:18:48.280 --> 00:18:53.205 align:middle line:90% 00:18:53.205 --> 00:18:56.770 align:middle line:84% But then there are some operations. 00:18:56.770 --> 00:18:59.510 align:middle line:84% If you have two rational numbers, 00:18:59.510 --> 00:19:02.200 align:middle line:84% then you can add two rational numbers together. 00:19:02.200 --> 00:19:05.630 align:middle line:90% 00:19:05.630 --> 00:19:09.570 align:middle line:84% And that's really important, that you have some operations 00:19:09.570 --> 00:19:12.370 align:middle line:90% on the rational numbers. 00:19:12.370 --> 00:19:14.030 align:middle line:90% So if you take-- 00:19:14.030 --> 00:19:18.130 align:middle line:90% 00:19:18.130 --> 00:19:21.650 align:middle line:84% this here is one rational number. 00:19:21.650 --> 00:19:25.650 align:middle line:84% And if you now take another rational number, m2/n2-- 00:19:25.650 --> 00:19:29.910 align:middle line:90% 00:19:29.910 --> 00:19:34.196 align:middle line:84% then how do you add two rational numbers together? 00:19:34.196 --> 00:19:36.750 align:middle line:84% Well, of course, you know that you put it 00:19:36.750 --> 00:19:38.630 align:middle line:90% on the common denominator. 00:19:38.630 --> 00:19:49.430 align:middle line:84% So this here is m1 times n2 plus m2 times n1 divided by n1 times 00:19:49.430 --> 00:19:52.270 align:middle line:90% n2. 00:19:52.270 --> 00:19:55.310 align:middle line:84% This here is an integer in the denominator. 00:19:55.310 --> 00:19:58.970 align:middle line:84% And this is a natural number in the denominator. 00:19:58.970 --> 00:20:03.990 align:middle line:84% And so that's representative of a rational number. 00:20:03.990 --> 00:20:06.950 align:middle line:90% So that's an operation. 00:20:06.950 --> 00:20:12.170 align:middle line:84% Now, in order to really know that this is-- 00:20:12.170 --> 00:20:17.350 align:middle line:84% so the idea here is that if you take one rational number, 00:20:17.350 --> 00:20:20.910 align:middle line:84% like this, and you take another rational number, 00:20:20.910 --> 00:20:25.240 align:middle line:84% like that, then you can add them. 00:20:25.240 --> 00:20:30.096 align:middle line:84% And that gives you another rational number. 00:20:30.096 --> 00:20:35.120 align:middle line:84% That's this operation that we know so well and its addition. 00:20:35.120 --> 00:20:38.240 align:middle line:90% 00:20:38.240 --> 00:20:41.440 align:middle line:84% In order for this really to make sense, 00:20:41.440 --> 00:20:48.680 align:middle line:84% you should check that this rational number you get 00:20:48.680 --> 00:20:52.200 align:middle line:84% is independent on the representation. 00:20:52.200 --> 00:20:54.680 align:middle line:84% If you have another representation 00:20:54.680 --> 00:20:59.240 align:middle line:84% for the same rational number and another representation 00:20:59.240 --> 00:21:02.560 align:middle line:84% for the same rational number, then when you add them, 00:21:02.560 --> 00:21:04.920 align:middle line:84% you should get another representation 00:21:04.920 --> 00:21:08.200 align:middle line:90% for the same rational number. 00:21:08.200 --> 00:21:11.200 align:middle line:90% 00:21:11.200 --> 00:21:15.560 align:middle line:84% And we will check one of these things in just a minute. 00:21:15.560 --> 00:21:19.710 align:middle line:90% The other thing is that-- 00:21:19.710 --> 00:21:22.740 align:middle line:90% 00:21:22.740 --> 00:21:24.580 align:middle line:90% so that was one operation. 00:21:24.580 --> 00:21:27.580 align:middle line:84% When you have the rational number, 00:21:27.580 --> 00:21:30.620 align:middle line:90% then addition is one operation. 00:21:30.620 --> 00:21:33.880 align:middle line:84% Another operation is, of course, multiplication. 00:21:33.880 --> 00:21:37.260 align:middle line:90% 00:21:37.260 --> 00:21:39.760 align:middle line:84% So another operation is multiplication. 00:21:39.760 --> 00:21:49.260 align:middle line:90% 00:21:49.260 --> 00:21:53.740 align:middle line:84% And so if you take one rational number, like this, 00:21:53.740 --> 00:21:57.700 align:middle line:84% and you take another rational number, 00:21:57.700 --> 00:22:01.460 align:middle line:84% like that, then if you multiply them 00:22:01.460 --> 00:22:04.340 align:middle line:84% together-- so what does it mean to multiply 00:22:04.340 --> 00:22:06.540 align:middle line:90% two rational number together? 00:22:06.540 --> 00:22:09.320 align:middle line:84% Well, of course, you know this very well. 00:22:09.320 --> 00:22:13.420 align:middle line:84% It's just multiplying the denominator. 00:22:13.420 --> 00:22:15.940 align:middle line:90% So that gives you an integer. 00:22:15.940 --> 00:22:18.760 align:middle line:84% And then multiplying the denominator, 00:22:18.760 --> 00:22:20.260 align:middle line:90% that gives you a natural number. 00:22:20.260 --> 00:22:22.060 align:middle line:84% So this here is a rational number. 00:22:22.060 --> 00:22:25.600 align:middle line:90% 00:22:25.600 --> 00:22:29.180 align:middle line:90% And then there's another thing. 00:22:29.180 --> 00:22:31.760 align:middle line:90% So this is addition. 00:22:31.760 --> 00:22:32.900 align:middle line:90% This was addition. 00:22:32.900 --> 00:22:34.880 align:middle line:90% This is multiplication. 00:22:34.880 --> 00:22:38.600 align:middle line:84% But there's another important property 00:22:38.600 --> 00:22:41.040 align:middle line:84% that you have for the rational number. 00:22:41.040 --> 00:22:42.350 align:middle line:90% And that is an ordering. 00:22:42.350 --> 00:22:46.640 align:middle line:90% 00:22:46.640 --> 00:22:48.920 align:middle line:84% Again, we're trying to understand this question 00:22:48.920 --> 00:22:49.700 align:middle line:90% up there. 00:22:49.700 --> 00:22:52.360 align:middle line:84% But we want to try to understand, how do we 00:22:52.360 --> 00:22:55.660 align:middle line:84% make the intermediate value theorem-- 00:22:55.660 --> 00:22:57.480 align:middle line:90% how do we make this rigorous? 00:22:57.480 --> 00:22:59.800 align:middle line:84% And so we need to understand what property 00:22:59.800 --> 00:23:03.360 align:middle line:90% of the real numbers we need. 00:23:03.360 --> 00:23:05.260 align:middle line:84% And so this is explained in this. 00:23:05.260 --> 00:23:08.080 align:middle line:84% And we were working towards trying to understand 00:23:08.080 --> 00:23:10.060 align:middle line:90% the different concepts in this. 00:23:10.060 --> 00:23:10.940 align:middle line:90% What is a field? 00:23:10.940 --> 00:23:13.960 align:middle line:84% That's what we're trying to understand. 00:23:13.960 --> 00:23:17.960 align:middle line:84% And so we have addition in the rational numbers. 00:23:17.960 --> 00:23:19.000 align:middle line:90% We have addition. 00:23:19.000 --> 00:23:20.420 align:middle line:90% We have multiplication. 00:23:20.420 --> 00:23:24.900 align:middle line:84% But we have another operation on it, which is an ordering. 00:23:24.900 --> 00:23:31.340 align:middle line:84% So if you take a rational number here, 00:23:31.340 --> 00:23:34.380 align:middle line:84% then you can say that this rational number is less 00:23:34.380 --> 00:23:36.460 align:middle line:90% than another rational number. 00:23:36.460 --> 00:23:40.420 align:middle line:84% When is it less than another rational number, 00:23:40.420 --> 00:23:42.620 align:middle line:90% this is equivalent to saying-- 00:23:42.620 --> 00:23:46.680 align:middle line:84% remember that these here are natural numbers. 00:23:46.680 --> 00:23:48.140 align:middle line:90% So they're positive. 00:23:48.140 --> 00:23:51.420 align:middle line:84% So that this rational number is less 00:23:51.420 --> 00:23:55.100 align:middle line:84% than that just means that m1 times 00:23:55.100 --> 00:23:59.120 align:middle line:90% n2 is less than m2 times n1. 00:23:59.120 --> 00:24:01.820 align:middle line:90% 00:24:01.820 --> 00:24:07.160 align:middle line:84% Here, I am just multiplying over here over here. 00:24:07.160 --> 00:24:12.180 align:middle line:84% And I'm using, of course, that these numbers here are positive. 00:24:12.180 --> 00:24:12.860 align:middle line:90% OK. 00:24:12.860 --> 00:24:14.320 align:middle line:90% So this is an ordering. 00:24:14.320 --> 00:24:18.740 align:middle line:84% So in the rational numbers, there is actually ordering. 00:24:18.740 --> 00:24:21.660 align:middle line:90% So that's really important also. 00:24:21.660 --> 00:24:24.860 align:middle line:84% So we have these two operations, and then we have an ordering. 00:24:24.860 --> 00:24:31.460 align:middle line:90% 00:24:31.460 --> 00:24:34.180 align:middle line:84% So here is-- so let me just check. 00:24:34.180 --> 00:24:38.840 align:middle line:84% So I said that when we talked about addition, when 00:24:38.840 --> 00:24:42.000 align:middle line:84% we talk about addition here, then there was something, 00:24:42.000 --> 00:24:46.620 align:middle line:84% really, you had to check, that this even made any sense. 00:24:46.620 --> 00:24:49.480 align:middle line:84% And for this to make any sense means 00:24:49.480 --> 00:24:54.720 align:middle line:84% that this rational number you get from adding these two 00:24:54.720 --> 00:24:58.760 align:middle line:84% is independent on how you write this rational number 00:24:58.760 --> 00:25:01.080 align:middle line:84% and how you write that rational number-- 00:25:01.080 --> 00:25:04.960 align:middle line:90% and the same for multiplication. 00:25:04.960 --> 00:25:07.030 align:middle line:90% So let us check that. 00:25:07.030 --> 00:25:12.100 align:middle line:90% 00:25:12.100 --> 00:25:29.372 align:middle line:84% So this here is checking that multiplication 00:25:29.372 --> 00:25:46.860 align:middle line:84% is a well-defined operation, i.e, it is independent. 00:25:46.860 --> 00:25:57.040 align:middle line:84% It is independent of the representative. 00:25:57.040 --> 00:26:04.220 align:middle line:90% 00:26:04.220 --> 00:26:09.440 align:middle line:84% So we are multiplying-- we take this rational number. 00:26:09.440 --> 00:26:11.920 align:middle line:84% Sorry, let me write a little bit bigger-- 00:26:11.920 --> 00:26:13.840 align:middle line:90% this rational number. 00:26:13.840 --> 00:26:17.480 align:middle line:90% So this is one rational number. 00:26:17.480 --> 00:26:24.280 align:middle line:84% And we multiply this rational number 00:26:24.280 --> 00:26:26.300 align:middle line:90% by another rational number. 00:26:26.300 --> 00:26:29.690 align:middle line:90% Let's call it p1 comma q1. 00:26:29.690 --> 00:26:33.280 align:middle line:90% 00:26:33.280 --> 00:26:35.800 align:middle line:84% And then we take another representative 00:26:35.800 --> 00:26:38.040 align:middle line:90% for the same rational number. 00:26:38.040 --> 00:26:39.500 align:middle line:90% And let's call this m2/n2. 00:26:39.500 --> 00:26:42.840 align:middle line:90% 00:26:42.840 --> 00:26:46.780 align:middle line:84% And for this rational number, we take another representative. 00:26:46.780 --> 00:26:51.040 align:middle line:90% 00:26:51.040 --> 00:26:56.240 align:middle line:84% And we want to prove that this number here is 00:26:56.240 --> 00:27:00.280 align:middle line:90% the same as that number here. 00:27:00.280 --> 00:27:01.970 align:middle line:90% So how do we do that? 00:27:01.970 --> 00:27:08.560 align:middle line:90% 00:27:08.560 --> 00:27:13.920 align:middle line:84% So we have that this here and this here is the same-- 00:27:13.920 --> 00:27:15.460 align:middle line:84% it's two different representation 00:27:15.460 --> 00:27:17.420 align:middle line:90% of the same rational number. 00:27:17.420 --> 00:27:26.460 align:middle line:84% So we have that m1/n1 is equal to m2/n2. 00:27:26.460 --> 00:27:31.900 align:middle line:84% So remember that this just meant that when you multiply over 00:27:31.900 --> 00:27:35.140 align:middle line:84% here, like that, then you get the same integers-- so 00:27:35.140 --> 00:27:40.040 align:middle line:90% this here, n2, plus m2 times n1. 00:27:40.040 --> 00:27:43.500 align:middle line:84% That's what it meant, that these here were the same. 00:27:43.500 --> 00:27:48.620 align:middle line:84% And likewise, you have the two rational numbers here. 00:27:48.620 --> 00:27:49.620 align:middle line:90% This is one. 00:27:49.620 --> 00:27:54.820 align:middle line:84% This is equal to that p2 divided by q2. 00:27:54.820 --> 00:27:57.900 align:middle line:84% This just meant that when you multiply over, 00:27:57.900 --> 00:28:03.750 align:middle line:90% p1 q2 is equal to p2 q1. 00:28:03.750 --> 00:28:07.480 align:middle line:90% 00:28:07.480 --> 00:28:12.000 align:middle line:84% So this just meant that these two represent 00:28:12.000 --> 00:28:13.780 align:middle line:90% the same rational number. 00:28:13.780 --> 00:28:17.560 align:middle line:84% These two represent the same rational number. 00:28:17.560 --> 00:28:21.920 align:middle line:84% And now what we need to prove is that the product here 00:28:21.920 --> 00:28:26.480 align:middle line:84% is independent of these representations. 00:28:26.480 --> 00:28:30.160 align:middle line:90% So the product here is-- 00:28:30.160 --> 00:28:33.420 align:middle line:84% so we need to prove the products of the first-- 00:28:33.420 --> 00:28:36.520 align:middle line:90% 00:28:36.520 --> 00:28:40.320 align:middle line:84% this product here, this product here, 00:28:40.320 --> 00:28:50.040 align:middle line:84% this is just m1 times p1 divided by n1 times q1. 00:28:50.040 --> 00:28:59.960 align:middle line:84% And this is what we want to show, that this thing here 00:28:59.960 --> 00:29:03.720 align:middle line:84% is the same if we took this other representation 00:29:03.720 --> 00:29:07.340 align:middle line:84% and multiply it by the other reputation of this. 00:29:07.340 --> 00:29:15.900 align:middle line:84% So this is m2 times p2, m2 times q2. 00:29:15.900 --> 00:29:19.980 align:middle line:84% That's what we want to prove, that those two are same. 00:29:19.980 --> 00:29:22.660 align:middle line:90% And so we need to prove that. 00:29:22.660 --> 00:29:35.420 align:middle line:84% So we need to show that m1 p1 times 00:29:35.420 --> 00:29:40.940 align:middle line:90% this denominator here into q2-- 00:29:40.940 --> 00:29:50.220 align:middle line:84% we need to prove, need to show that this thing here 00:29:50.220 --> 00:29:58.920 align:middle line:90% is equal to m2 p2 n1 q1. 00:29:58.920 --> 00:30:02.420 align:middle line:90% That's what we need to prove. 00:30:02.420 --> 00:30:04.860 align:middle line:90% But now let's see. 00:30:04.860 --> 00:30:20.640 align:middle line:84% So we know that if we take m1 and together with n2, 00:30:20.640 --> 00:30:25.620 align:middle line:84% m1 together with n2, this is the same as-- 00:30:25.620 --> 00:30:32.340 align:middle line:84% so this here is the same as m2 n1. 00:30:32.340 --> 00:30:35.600 align:middle line:90% This is what this here say. 00:30:35.600 --> 00:30:40.760 align:middle line:84% And it also says that if you take p1 together with q2, then 00:30:40.760 --> 00:30:44.120 align:middle line:84% you may as well write this thing here. 00:30:44.120 --> 00:30:49.400 align:middle line:90% So you may as well write p2 q1. 00:30:49.400 --> 00:30:52.400 align:middle line:84% So this thing here is equal to that. 00:30:52.400 --> 00:30:56.360 align:middle line:84% But now you see that here, you have m2. 00:30:56.360 --> 00:30:57.920 align:middle line:90% Here, you have n1. 00:30:57.920 --> 00:30:59.440 align:middle line:90% Here, you have p2. 00:30:59.440 --> 00:31:01.100 align:middle line:90% Here, you have q1. 00:31:01.100 --> 00:31:03.140 align:middle line:90% So those two are the same. 00:31:03.140 --> 00:31:07.780 align:middle line:84% So this proves that this operation of multiplication 00:31:07.780 --> 00:31:10.340 align:middle line:84% is well defined in the rational number-- 00:31:10.340 --> 00:31:13.940 align:middle line:90% and similarly for addition. 00:31:13.940 --> 00:31:16.460 align:middle line:84% The reason, of course, why I chose multiplication 00:31:16.460 --> 00:31:20.160 align:middle line:84% is that it looks a little simpler on the blackboard. 00:31:20.160 --> 00:31:22.760 align:middle line:84% It's a little more messy if you did addition. 00:31:22.760 --> 00:31:27.212 align:middle line:90% But it's just exactly the same. 00:31:27.212 --> 00:31:27.712 align:middle line:90% OK. 00:31:27.712 --> 00:31:36.660 align:middle line:90% 00:31:36.660 --> 00:31:40.650 align:middle line:84% So this is also supposed to illustrate 00:31:40.650 --> 00:31:42.800 align:middle line:90% how detailed a proof is. 00:31:42.800 --> 00:31:46.260 align:middle line:84% So when we have a concept like this, 00:31:46.260 --> 00:31:50.280 align:middle line:84% we want to make sure that it really makes sense. 00:31:50.280 --> 00:31:53.160 align:middle line:84% And in this case, it makes sense that it's well defined. 00:31:53.160 --> 00:31:56.500 align:middle line:84% It's independent under representation. 00:31:56.500 --> 00:31:59.170 align:middle line:84% And so the proof here, this here would be the proof. 00:31:59.170 --> 00:32:02.793 align:middle line:90% 00:32:02.793 --> 00:32:03.293 align:middle line:90% OK. 00:32:03.293 --> 00:32:05.880 align:middle line:90% 00:32:05.880 --> 00:32:08.320 align:middle line:84% So the takeaway here at the moment 00:32:08.320 --> 00:32:10.860 align:middle line:90% is, we're building towards this. 00:32:10.860 --> 00:32:12.220 align:middle line:90% We're understanding a field. 00:32:12.220 --> 00:32:16.200 align:middle line:84% We want to understand the real numbers. 00:32:16.200 --> 00:32:22.160 align:middle line:84% But the takeaway at the moment is that on the rational numbers, 00:32:22.160 --> 00:32:25.400 align:middle line:84% there is a way of adding two rational numbers 00:32:25.400 --> 00:32:27.540 align:middle line:84% and multiplying two rational numbers. 00:32:27.540 --> 00:32:31.080 align:middle line:90% And there's also an ordering. 00:32:31.080 --> 00:32:36.740 align:middle line:84% And so this leads us to the following definition. 00:32:36.740 --> 00:32:47.680 align:middle line:90% 00:32:47.680 --> 00:32:50.960 align:middle line:84% And so this is the definition of a field. 00:32:50.960 --> 00:33:05.050 align:middle line:84% So a field-- and I will use F here, capital F for a field-- 00:33:05.050 --> 00:33:06.070 align:middle line:90% is the following. 00:33:06.070 --> 00:33:12.290 align:middle line:90% 00:33:12.290 --> 00:33:18.750 align:middle line:84% So it's a set with two operations. 00:33:18.750 --> 00:33:26.450 align:middle line:90% 00:33:26.450 --> 00:33:31.530 align:middle line:84% And the operations we are denoting suggestively by-- 00:33:31.530 --> 00:33:34.410 align:middle line:84% so the two operations are denoted suggestively 00:33:34.410 --> 00:33:49.005 align:middle line:84% by a plus and a multiplication, like a dot. 00:33:49.005 --> 00:33:52.150 align:middle line:90% But this is just abstract. 00:33:52.150 --> 00:33:56.750 align:middle line:84% So we have a set, and we have these two operations. 00:33:56.750 --> 00:34:01.470 align:middle line:84% What property do these two operations need to have? 00:34:01.470 --> 00:34:05.790 align:middle line:84% Each of them have five properties. 00:34:05.790 --> 00:34:08.750 align:middle line:84% So that means that there are 10 properties. 00:34:08.750 --> 00:34:12.350 align:middle line:84% And then there's 11 properties that chain those two operations 00:34:12.350 --> 00:34:13.110 align:middle line:90% together. 00:34:13.110 --> 00:34:15.530 align:middle line:90% So let's start with addition. 00:34:15.530 --> 00:34:21.170 align:middle line:90% So properties of addition-- 00:34:21.170 --> 00:34:25.909 align:middle line:90% 00:34:25.909 --> 00:34:30.449 align:middle line:84% so the first property is that addition make any sense. 00:34:30.449 --> 00:34:35.590 align:middle line:84% So if you take a element in the field 00:34:35.590 --> 00:34:40.270 align:middle line:84% and you take another element in the field, 00:34:40.270 --> 00:34:44.969 align:middle line:84% then addition gives you a third element. 00:34:44.969 --> 00:34:47.230 align:middle line:84% So when you add these together, this 00:34:47.230 --> 00:34:51.790 align:middle line:84% is just a third element in the field. 00:34:51.790 --> 00:34:54.050 align:middle line:84% So this is the first property, that it really 00:34:54.050 --> 00:34:57.730 align:middle line:84% is an operation that when you take two elements of the field, 00:34:57.730 --> 00:35:01.290 align:middle line:84% then you get another element in the field. 00:35:01.290 --> 00:35:03.290 align:middle line:90% So this is one. 00:35:03.290 --> 00:35:10.450 align:middle line:84% The second is that if you add two elements together-- 00:35:10.450 --> 00:35:12.830 align:middle line:90% so x and y are in the field. 00:35:12.830 --> 00:35:15.190 align:middle line:84% If you add two elements together, 00:35:15.190 --> 00:35:20.350 align:middle line:84% then the ordering that you're adding is irrelevant. 00:35:20.350 --> 00:35:23.390 align:middle line:84% So this here is the same as y plus x. 00:35:23.390 --> 00:35:28.530 align:middle line:90% 00:35:28.530 --> 00:35:32.410 align:middle line:84% Then the next one is-- this is sometimes called 00:35:32.410 --> 00:35:35.890 align:middle line:84% abelian or commutative, this property here. 00:35:35.890 --> 00:35:38.890 align:middle line:84% Then there's a third property, which 00:35:38.890 --> 00:35:42.810 align:middle line:84% is that if you add two elements together 00:35:42.810 --> 00:35:48.770 align:middle line:84% and then you add a third element to the sum of these two, 00:35:48.770 --> 00:35:52.830 align:middle line:84% then this here is the same as that you 00:35:52.830 --> 00:35:58.150 align:middle line:84% can add the last two first and then 00:35:58.150 --> 00:36:01.010 align:middle line:84% add the first one to the sum of the other two. 00:36:01.010 --> 00:36:17.790 align:middle line:90% 00:36:17.790 --> 00:36:23.670 align:middle line:84% And so the fourth property is that there 00:36:23.670 --> 00:36:28.470 align:middle line:84% is what is sometimes referred to as a neutral element. 00:36:28.470 --> 00:36:48.190 align:middle line:84% So there exists an element that we denote by 0. 00:36:48.190 --> 00:36:52.290 align:middle line:84% But it's just to make it kind of easier to understand, 00:36:52.290 --> 00:36:54.330 align:middle line:90% that we denote by 0. 00:36:54.330 --> 00:37:07.770 align:middle line:84% And it has the property that if you take this 0 element 00:37:07.770 --> 00:37:12.650 align:middle line:84% and you add it to any other element, then it just 00:37:12.650 --> 00:37:15.250 align:middle line:84% gives you back that other element. 00:37:15.250 --> 00:37:19.770 align:middle line:84% So it's kind of neutral for the operation. 00:37:19.770 --> 00:37:24.050 align:middle line:84% And then there's supposed to be a fifth property. 00:37:24.050 --> 00:37:33.490 align:middle line:84% And the fifth property is that for any element x, 00:37:33.490 --> 00:37:40.200 align:middle line:90% there exist another element. 00:37:40.200 --> 00:37:44.650 align:middle line:90% 00:37:44.650 --> 00:37:49.590 align:middle line:84% And again, we write this suggestively by minus x 00:37:49.590 --> 00:37:52.950 align:middle line:84% with a bracket around, like this. 00:37:52.950 --> 00:38:04.526 align:middle line:84% So there is another element with the property 00:38:04.526 --> 00:38:10.150 align:middle line:84% that if you take x and you're adding this other element, then 00:38:10.150 --> 00:38:12.750 align:middle line:90% you get 0. 00:38:12.750 --> 00:38:18.990 align:middle line:84% So those are the five properties that addition has to have. 00:38:18.990 --> 00:38:23.810 align:middle line:84% And then multiplication have to have five properties also. 00:38:23.810 --> 00:38:29.910 align:middle line:84% And these are similar to this with a little twist. 00:38:29.910 --> 00:38:36.190 align:middle line:84% And so for multiplication, you need 00:38:36.190 --> 00:38:37.850 align:middle line:84% to have the following properties. 00:38:37.850 --> 00:38:41.950 align:middle line:90% 00:38:41.950 --> 00:38:44.470 align:middle line:90% So again, it's five properties. 00:38:44.470 --> 00:38:51.170 align:middle line:84% The first one is the similar property to the property 1 00:38:51.170 --> 00:38:55.370 align:middle line:84% up there, namely that if you take two elements, x and y, 00:38:55.370 --> 00:39:02.890 align:middle line:84% in the field, then when you multiply them together, 00:39:02.890 --> 00:39:06.850 align:middle line:84% that should give you another element in the field. 00:39:06.850 --> 00:39:09.090 align:middle line:90% So that's the first. 00:39:09.090 --> 00:39:16.730 align:middle line:84% The second one is similar to 2 up there, is that x times y 00:39:16.730 --> 00:39:19.350 align:middle line:90% should be the same as y times x. 00:39:19.350 --> 00:39:22.170 align:middle line:90% 00:39:22.170 --> 00:39:25.730 align:middle line:84% So again, that's often referred to that it 00:39:25.730 --> 00:39:30.530 align:middle line:90% is abelian or commutative. 00:39:30.530 --> 00:39:40.010 align:middle line:84% The third property, it's similar to the third property up there-- 00:39:40.010 --> 00:39:43.250 align:middle line:84% is that if you take x times y and then 00:39:43.250 --> 00:39:46.590 align:middle line:84% you multiply it by a third element, 00:39:46.590 --> 00:39:50.030 align:middle line:84% then you could as well have first multiplied 00:39:50.030 --> 00:39:54.990 align:middle line:84% the two last elements and then multiply by the first element, 00:39:54.990 --> 00:39:57.750 align:middle line:90% like that. 00:39:57.750 --> 00:40:05.510 align:middle line:84% And then the fourth property is similar to that property 4 00:40:05.510 --> 00:40:22.710 align:middle line:84% up there, is that there exists an element denoted suggestively 00:40:22.710 --> 00:40:30.390 align:middle line:84% by 1 here, with a property that 1 times any element 00:40:30.390 --> 00:40:35.790 align:middle line:84% x just give you back that element, x. 00:40:35.790 --> 00:40:38.890 align:middle line:84% And then the fifth is similar, except there 00:40:38.890 --> 00:40:40.600 align:middle line:90% is a twist, slight twist. 00:40:40.600 --> 00:40:44.170 align:middle line:90% 00:40:44.170 --> 00:40:49.170 align:middle line:84% And so the fifth property is that-- 00:40:49.170 --> 00:40:52.450 align:middle line:84% but it's very similar to that fifth over there. 00:40:52.450 --> 00:40:54.270 align:middle line:84% The fifth property is the following. 00:40:54.270 --> 00:40:57.410 align:middle line:90% 00:40:57.410 --> 00:41:00.050 align:middle line:84% The fifth property here is that there 00:41:00.050 --> 00:41:12.170 align:middle line:84% exists that for any x in the field except 0-- 00:41:12.170 --> 00:41:14.970 align:middle line:90% you have to disregard 0. 00:41:14.970 --> 00:41:23.940 align:middle line:84% For any other elements, there exist an element-- 00:41:23.940 --> 00:41:27.130 align:middle line:90% 00:41:27.130 --> 00:41:44.270 align:middle line:84% and, again, suggestively denoted by 1/x with the property that x 00:41:44.270 --> 00:41:48.790 align:middle line:90% times 1/x is equal to 1. 00:41:48.790 --> 00:41:49.290 align:middle line:90% OK. 00:41:49.290 --> 00:41:55.910 align:middle line:84% So those are the five properties for addition, five properties 00:41:55.910 --> 00:41:57.290 align:middle line:90% for multiplication. 00:41:57.290 --> 00:42:00.950 align:middle line:84% And you see that the one for multiplication 00:42:00.950 --> 00:42:03.730 align:middle line:84% is like a mirror of the one for addition, 00:42:03.730 --> 00:42:08.150 align:middle line:84% except the fifth one has that you have to look away 00:42:08.150 --> 00:42:09.680 align:middle line:90% from the 0 element. 00:42:09.680 --> 00:42:15.650 align:middle line:90% 00:42:15.650 --> 00:42:18.310 align:middle line:90% And so, again, this is just-- 00:42:18.310 --> 00:42:21.710 align:middle line:84% 1 is just that it is an operation. 00:42:21.710 --> 00:42:25.510 align:middle line:84% 2 is usually called the abelian property or commutative 00:42:25.510 --> 00:42:26.590 align:middle line:90% property. 00:42:26.590 --> 00:42:29.510 align:middle line:84% This is called the associative property. 00:42:29.510 --> 00:42:34.670 align:middle line:84% This, it's often also referred to as a neutral element. 00:42:34.670 --> 00:42:40.110 align:middle line:84% And this here, that exists this element-- 00:42:40.110 --> 00:42:43.090 align:middle line:84% in this case, minus x is sometimes 00:42:43.090 --> 00:42:44.520 align:middle line:90% called the inverse element-- 00:42:44.520 --> 00:42:47.090 align:middle line:90% 00:42:47.090 --> 00:42:50.130 align:middle line:90% and likewise for multiplication. 00:42:50.130 --> 00:42:54.370 align:middle line:84% And then there is, again, an 11th property. 00:42:54.370 --> 00:42:55.890 align:middle line:84% And that's a property because you 00:42:55.890 --> 00:42:58.250 align:middle line:84% see that those-- at the moment, there's 00:42:58.250 --> 00:43:02.050 align:middle line:84% basically no interaction between these two operations. 00:43:02.050 --> 00:43:05.050 align:middle line:84% The only way that you see some interaction, 00:43:05.050 --> 00:43:10.250 align:middle line:84% but this is very little, is that the fifth property over here 00:43:10.250 --> 00:43:13.570 align:middle line:84% refer to the neutral element for addition. 00:43:13.570 --> 00:43:17.130 align:middle line:84% But they don't really chain the two operations together. 00:43:17.130 --> 00:43:20.290 align:middle line:84% And that is the last property that is often referred 00:43:20.290 --> 00:43:22.070 align:middle line:90% to as the distributive law. 00:43:22.070 --> 00:43:32.800 align:middle line:84% So this is an axiom chaining addition-- 00:43:32.800 --> 00:43:42.230 align:middle line:84% write it like this, addition together with multiplication. 00:43:42.230 --> 00:43:53.390 align:middle line:90% 00:43:53.390 --> 00:44:03.950 align:middle line:84% And that's if you take x and you multiply it by y plus z-- 00:44:03.950 --> 00:44:08.030 align:middle line:84% you take the sum here, and then you multiply it by x. 00:44:08.030 --> 00:44:11.190 align:middle line:84% That is supposed to be the same as where 00:44:11.190 --> 00:44:19.830 align:middle line:84% you first multiply x with y and then you multiply x by z. 00:44:19.830 --> 00:44:23.550 align:middle line:84% So this is often called the distributive law. 00:44:23.550 --> 00:44:27.390 align:middle line:84% And that's a law, again, or axiom that is chaining these two 00:44:27.390 --> 00:44:30.303 align:middle line:90% operations together. 00:44:30.303 --> 00:44:30.803 align:middle line:90% OK. 00:44:30.803 --> 00:44:37.610 align:middle line:84% So now let's try to see-- so let me 00:44:37.610 --> 00:44:43.130 align:middle line:84% just establish just one little property of a field. 00:44:43.130 --> 00:44:55.382 align:middle line:84% And you see that when we talk about the axioms for addition, 00:44:55.382 --> 00:44:59.250 align:middle line:84% I just said that there had to exist one element that we 00:44:59.250 --> 00:45:02.410 align:middle line:84% call the 0 element, and it had to have some property. 00:45:02.410 --> 00:45:04.270 align:middle line:84% But maybe there are many elements. 00:45:04.270 --> 00:45:06.080 align:middle line:84% And so this is the first little theorem. 00:45:06.080 --> 00:45:10.090 align:middle line:90% 00:45:10.090 --> 00:45:18.210 align:middle line:84% For any field-- I just write it as capital Field. 00:45:18.210 --> 00:45:20.460 align:middle line:90% But it's just to-- 00:45:20.460 --> 00:45:22.970 align:middle line:90% but you don't really have to. 00:45:22.970 --> 00:45:37.260 align:middle line:84% For any field F, there exist only one zero element. 00:45:37.260 --> 00:45:43.390 align:middle line:90% 00:45:43.390 --> 00:45:44.030 align:middle line:90% OK. 00:45:44.030 --> 00:45:46.750 align:middle line:90% Let's try to prove this. 00:45:46.750 --> 00:45:54.550 align:middle line:84% And again, there's all kind of important concepts 00:45:54.550 --> 00:45:55.610 align:middle line:90% in this class. 00:45:55.610 --> 00:46:02.910 align:middle line:84% But there's also another thing that is important to learn. 00:46:02.910 --> 00:46:04.630 align:middle line:84% And you have to be a little bit patient 00:46:04.630 --> 00:46:09.510 align:middle line:84% because it will take you a little bit of time to learn it-- 00:46:09.510 --> 00:46:13.510 align:middle line:84% is how to write a mathematical proof. 00:46:13.510 --> 00:46:17.710 align:middle line:84% And so this here just gives you just a little bit of idea of it. 00:46:17.710 --> 00:46:23.230 align:middle line:84% So we want to prove that there exists only one zero element. 00:46:23.230 --> 00:46:39.070 align:middle line:84% So assume that O1 and O2 are both zero element. 00:46:39.070 --> 00:46:45.780 align:middle line:90% 00:46:45.780 --> 00:46:54.360 align:middle line:84% So that it's a zero element just means 00:46:54.360 --> 00:46:58.200 align:middle line:84% that if you take O1 and you add anything to it-- 00:46:58.200 --> 00:47:00.440 align:middle line:84% x could be anything in the field-- 00:47:00.440 --> 00:47:02.120 align:middle line:90% then you get x back. 00:47:02.120 --> 00:47:05.080 align:middle line:84% That is that O1 is a zero element. 00:47:05.080 --> 00:47:08.840 align:middle line:84% That O2 is a zero element just means the same thing-- 00:47:08.840 --> 00:47:12.986 align:middle line:84% that if you take anything in x, then you get x back. 00:47:12.986 --> 00:47:17.520 align:middle line:84% And now we want to prove that O1 is equal to O2. 00:47:17.520 --> 00:47:21.400 align:middle line:90% So I just write O1 plus O2. 00:47:21.400 --> 00:47:25.800 align:middle line:84% But now I'm just thinking about this O2 as my x. 00:47:25.800 --> 00:47:30.000 align:middle line:90% So I'm adding 0 O1 to this x. 00:47:30.000 --> 00:47:33.780 align:middle line:84% So I know that I'm just getting this here back. 00:47:33.780 --> 00:47:38.700 align:middle line:90% Is because O1 is a 0 element. 00:47:38.700 --> 00:47:42.540 align:middle line:84% But I could do the same with O2 that I take. 00:47:42.540 --> 00:47:46.380 align:middle line:84% The ordering I know already is irrelevant, 00:47:46.380 --> 00:47:49.160 align:middle line:84% whether it's O1 plus O2 or O2 plus O1. 00:47:49.160 --> 00:47:50.260 align:middle line:90% That's the same. 00:47:50.260 --> 00:47:54.900 align:middle line:84% So I can think about this O1 here as my x. 00:47:54.900 --> 00:47:58.880 align:middle line:90% And so I'm adding the O2 to it. 00:47:58.880 --> 00:48:01.660 align:middle line:84% But then I know that I'm getting this here back. 00:48:01.660 --> 00:48:06.120 align:middle line:84% So this here must also be equal to O1. 00:48:06.120 --> 00:48:08.210 align:middle line:84% And so you see that O1 is equal to O2. 00:48:08.210 --> 00:48:11.940 align:middle line:90% 00:48:11.940 --> 00:48:18.900 align:middle line:84% And of course, in the notes that are post, 00:48:18.900 --> 00:48:21.290 align:middle line:84% things are written slightly more rigorous. 00:48:21.290 --> 00:48:25.540 align:middle line:90% 00:48:25.540 --> 00:48:27.540 align:middle line:84% There's a little bit more meat to it. 00:48:27.540 --> 00:48:31.320 align:middle line:84% Here, I write it a little bit shorter on the board. 00:48:31.320 --> 00:48:33.790 align:middle line:90% So look also at the notes. 00:48:33.790 --> 00:48:36.500 align:middle line:90% 00:48:36.500 --> 00:48:37.000 align:middle line:90% OK. 00:48:37.000 --> 00:48:39.640 align:middle line:90% 00:48:39.640 --> 00:48:42.300 align:middle line:90% So now we know what a field is. 00:48:42.300 --> 00:48:46.360 align:middle line:84% So a field, again, it has these two operations, additions 00:48:46.360 --> 00:48:47.840 align:middle line:90% and multiplication. 00:48:47.840 --> 00:48:50.780 align:middle line:84% Each of those operations have five properties. 00:48:50.780 --> 00:48:53.800 align:middle line:84% And then there's 11 properties that chain those two operations 00:48:53.800 --> 00:48:55.920 align:middle line:90% together. 00:48:55.920 --> 00:48:59.470 align:middle line:84% And so the examples of a fields is-- 00:48:59.470 --> 00:49:04.240 align:middle line:90% 00:49:04.240 --> 00:49:06.390 align:middle line:90% so we already have an example. 00:49:06.390 --> 00:49:11.560 align:middle line:90% 00:49:11.560 --> 00:49:14.290 align:middle line:84% So the rational numbers here is a field. 00:49:14.290 --> 00:49:19.200 align:middle line:90% 00:49:19.200 --> 00:49:26.140 align:middle line:84% It has these two operations and with all the 11 properties. 00:49:26.140 --> 00:49:33.700 align:middle line:84% The natural numbers is not a field, is not a field. 00:49:33.700 --> 00:49:38.480 align:middle line:84% It fails even the most basic thing. 00:49:38.480 --> 00:49:41.240 align:middle line:84% It fails many of the properties it needs, 00:49:41.240 --> 00:49:44.900 align:middle line:84% but it fails that even the serial elements is not in it. 00:49:44.900 --> 00:49:52.260 align:middle line:90% 00:49:52.260 --> 00:49:57.150 align:middle line:84% And the integers here also is not a field. 00:49:57.150 --> 00:50:04.300 align:middle line:90% 00:50:04.300 --> 00:50:11.160 align:middle line:84% That is slightly closer to being a field, but it's still not. 00:50:11.160 --> 00:50:13.180 align:middle line:90% And why is this not a field? 00:50:13.180 --> 00:50:15.320 align:middle line:90% It's for multiplication, right? 00:50:15.320 --> 00:50:19.860 align:middle line:90% If you take 2, then write 2. 00:50:19.860 --> 00:50:23.460 align:middle line:84% If you think about 2, then there should be-- 00:50:23.460 --> 00:50:26.580 align:middle line:84% then the neutral element for multiplication, 00:50:26.580 --> 00:50:29.080 align:middle line:90% that's the usual 1. 00:50:29.080 --> 00:50:31.880 align:middle line:84% But if you take, say, the element 2, 00:50:31.880 --> 00:50:34.620 align:middle line:84% then there should be an element like the inverse. 00:50:34.620 --> 00:50:37.920 align:middle line:84% There should be an element with the property that if you take 2 00:50:37.920 --> 00:50:43.380 align:middle line:84% and you multiply with that element, then you should get 1. 00:50:43.380 --> 00:50:46.060 align:middle line:84% That should be the inverse element, 1 over it. 00:50:46.060 --> 00:50:47.580 align:middle line:90% But this is like a half. 00:50:47.580 --> 00:50:51.860 align:middle line:84% But you're missing a half in the integers. 00:50:51.860 --> 00:50:53.740 align:middle line:84% So that's why this here is not a field. 00:50:53.740 --> 00:50:56.480 align:middle line:90% 00:50:56.480 --> 00:50:57.770 align:middle line:90% OK. 00:50:57.770 --> 00:51:02.200 align:middle line:84% So the next concept is-- so, again, we're 00:51:02.200 --> 00:51:06.200 align:middle line:84% working our way towards trying to understand 00:51:06.200 --> 00:51:08.720 align:middle line:84% what property of the real numbers 00:51:08.720 --> 00:51:13.040 align:middle line:84% that we need in order to make the intermediate value 00:51:13.040 --> 00:51:18.560 align:middle line:90% theorem, the proof regression. 00:51:18.560 --> 00:51:24.600 align:middle line:84% And so we now know what a field is. 00:51:24.600 --> 00:51:29.420 align:middle line:90% 00:51:29.420 --> 00:51:34.620 align:middle line:84% The next few slides, I'll talk about complete, what that means. 00:51:34.620 --> 00:51:40.440 align:middle line:90% But we also have this ordering. 00:51:40.440 --> 00:51:43.420 align:middle line:84% So what does it mean that a field is ordered? 00:51:43.420 --> 00:51:48.680 align:middle line:84% So let's start with an ordered set. 00:51:48.680 --> 00:51:53.940 align:middle line:90% 00:51:53.940 --> 00:51:55.740 align:middle line:90% So an ordered set-- 00:51:55.740 --> 00:52:03.820 align:middle line:84% so this is a set denoted by capital S. 00:52:03.820 --> 00:52:07.300 align:middle line:84% And it has the property that whenever you have two elements 00:52:07.300 --> 00:52:12.780 align:middle line:84% in the set, then it should either be-- and the ordering, 00:52:12.780 --> 00:52:17.280 align:middle line:84% by the way-- so an ordered set with an ordering-- 00:52:17.280 --> 00:52:23.120 align:middle line:90% 00:52:23.120 --> 00:52:27.060 align:middle line:84% and I denote the ordering, again, suggestively by this. 00:52:27.060 --> 00:52:28.660 align:middle line:84% I could have used any other symbol, 00:52:28.660 --> 00:52:31.360 align:middle line:84% but I'm just suggestively using this symbol, 00:52:31.360 --> 00:52:35.380 align:middle line:84% just to make it easier to remember the property. 00:52:35.380 --> 00:52:39.600 align:middle line:84% It's like this property that you know so well. 00:52:39.600 --> 00:52:46.380 align:middle line:84% So an ordered set is a set with an ordering-- 00:52:46.380 --> 00:52:52.760 align:middle line:90% 00:52:52.760 --> 00:52:55.400 align:middle line:90% and, again, denoted by this-- 00:52:55.400 --> 00:53:00.600 align:middle line:84% so that if you take any two elements in the set, 00:53:00.600 --> 00:53:02.830 align:middle line:84% then one of the three properties should hold. 00:53:02.830 --> 00:53:07.040 align:middle line:90% 00:53:07.040 --> 00:53:09.140 align:middle line:84% The following three properties should hold. 00:53:09.140 --> 00:53:26.060 align:middle line:90% 00:53:26.060 --> 00:53:33.380 align:middle line:84% One is that-- so either x should be equal to y. 00:53:33.380 --> 00:53:38.460 align:middle line:84% 2, the other possibility, is that x is smaller than y. 00:53:38.460 --> 00:53:43.740 align:middle line:90% Or 3, y is smaller than x. 00:53:43.740 --> 00:53:46.560 align:middle line:84% So basically, if you have an ordered set, 00:53:46.560 --> 00:53:48.300 align:middle line:84% whenever you take two elements, you 00:53:48.300 --> 00:53:51.180 align:middle line:84% should be able to compare these two elements. 00:53:51.180 --> 00:53:51.680 align:middle line:90% Of course. 00:53:51.680 --> 00:53:54.340 align:middle line:84% These two elements could be the same elements. 00:53:54.340 --> 00:53:56.020 align:middle line:84% But if they're not the same element, 00:53:56.020 --> 00:54:00.340 align:middle line:84% then one should be smaller than the other. 00:54:00.340 --> 00:54:01.310 align:middle line:90% So that's the idea. 00:54:01.310 --> 00:54:03.860 align:middle line:90% 00:54:03.860 --> 00:54:05.000 align:middle line:90% OK. 00:54:05.000 --> 00:54:10.410 align:middle line:84% And so now the next concept is an ordered field. 00:54:10.410 --> 00:54:15.480 align:middle line:90% 00:54:15.480 --> 00:54:22.220 align:middle line:84% So now we have a field, and we have an ordered set. 00:54:22.220 --> 00:54:46.310 align:middle line:84% So an ordered field is a field that is also an ordered set. 00:54:46.310 --> 00:54:51.240 align:middle line:90% 00:54:51.240 --> 00:54:56.720 align:middle line:84% And then the operations from the fields and the ordering, 00:54:56.720 --> 00:54:58.860 align:middle line:90% they should speak to each other. 00:54:58.860 --> 00:55:01.320 align:middle line:84% So there should be an interaction between these two 00:55:01.320 --> 00:55:03.200 align:middle line:90% operations. 00:55:03.200 --> 00:55:17.940 align:middle line:84% So it has to have-- with the following properties, 00:55:17.940 --> 00:55:19.679 align:middle line:90% then if you take-- 00:55:19.679 --> 00:55:23.060 align:middle line:90% 00:55:23.060 --> 00:55:24.900 align:middle line:84% so it's following two properties, 00:55:24.900 --> 00:55:35.000 align:middle line:84% that if x is less than y and you're adding a c to both-- 00:55:35.000 --> 00:55:45.140 align:middle line:84% so this here implies that for any z in the field, 00:55:45.140 --> 00:55:47.800 align:middle line:84% if you add this z to both sides-- 00:55:47.800 --> 00:55:53.620 align:middle line:84% so x plus z is less than y plus z. 00:55:53.620 --> 00:55:58.600 align:middle line:84% So you should be able to add something to both sides 00:55:58.600 --> 00:56:03.340 align:middle line:84% and it's still less than, like that. 00:56:03.340 --> 00:56:06.260 align:middle line:90% 00:56:06.260 --> 00:56:09.620 align:middle line:84% So this here is, how do the ordering 00:56:09.620 --> 00:56:11.780 align:middle line:90% interacts with addition? 00:56:11.780 --> 00:56:15.160 align:middle line:84% And the other thing is, how do it act interacts 00:56:15.160 --> 00:56:17.800 align:middle line:90% with multiplication? 00:56:17.800 --> 00:56:22.440 align:middle line:84% And there, the property is that if you take a x that 00:56:22.440 --> 00:56:29.440 align:middle line:90% is positive and y is positive-- 00:56:29.440 --> 00:56:30.480 align:middle line:90% and the y. 00:56:30.480 --> 00:56:33.920 align:middle line:84% So you have both x and y that are positive. 00:56:33.920 --> 00:56:39.480 align:middle line:84% Then the product of these two should also be positive. 00:56:39.480 --> 00:56:42.960 align:middle line:84% So then this here, the product of the two, 00:56:42.960 --> 00:56:44.560 align:middle line:90% should also be positive. 00:56:44.560 --> 00:56:51.060 align:middle line:84% And so this is how the ordering interact with multiplication. 00:56:51.060 --> 00:56:53.720 align:middle line:84% This is how it interacts with addition. 00:56:53.720 --> 00:56:56.100 align:middle line:84% This is how it interacts with multiplication. 00:56:56.100 --> 00:57:00.590 align:middle line:90% 00:57:00.590 --> 00:57:07.360 align:middle line:84% And so now-- well, the first thing you observe 00:57:07.360 --> 00:57:11.840 align:middle line:84% is that, again, you already know an example of this. 00:57:11.840 --> 00:57:15.540 align:middle line:90% 00:57:15.540 --> 00:57:24.520 align:middle line:90% So the rational numbers-- 00:57:24.520 --> 00:57:28.230 align:middle line:90% so Q here is an ordered field. 00:57:28.230 --> 00:57:36.300 align:middle line:90% 00:57:36.300 --> 00:57:39.580 align:middle line:84% We have already seen that it is a field, 00:57:39.580 --> 00:57:41.120 align:middle line:90% has those 11 properties. 00:57:41.120 --> 00:57:43.860 align:middle line:90% It has addition, multiplication. 00:57:43.860 --> 00:57:45.480 align:middle line:90% Each of those have 5 property. 00:57:45.480 --> 00:57:49.380 align:middle line:84% It has the 11 that chain the two together. 00:57:49.380 --> 00:57:51.940 align:middle line:84% And we also discussed the ordering. 00:57:51.940 --> 00:57:53.960 align:middle line:84% If you take two rational numbers, 00:57:53.960 --> 00:57:56.240 align:middle line:84% what does it mean that one is less than the other? 00:57:56.240 --> 00:57:58.620 align:middle line:84% It has the property of an ordering, 00:57:58.620 --> 00:58:02.340 align:middle line:84% that you can compare any two elements, any two 00:58:02.340 --> 00:58:05.240 align:middle line:84% rational numbers, that either they are the same 00:58:05.240 --> 00:58:08.360 align:middle line:84% or, if you have two, one is less than the other. 00:58:08.360 --> 00:58:14.040 align:middle line:90% 00:58:14.040 --> 00:58:18.480 align:middle line:84% And it also has this, the additive property. 00:58:18.480 --> 00:58:23.400 align:middle line:84% And also, the ordering interact with addition in this way. 00:58:23.400 --> 00:58:26.600 align:middle line:84% And it interacts with multiplication in this way. 00:58:26.600 --> 00:58:33.280 align:middle line:84% So the rational number is indeed an ordered field. 00:58:33.280 --> 00:58:39.360 align:middle line:84% Now let me just establish one last property. 00:58:39.360 --> 00:58:42.670 align:middle line:90% And that is that if I take-- 00:58:42.670 --> 00:58:47.880 align:middle line:90% 00:58:47.880 --> 00:58:50.180 align:middle line:84% well, let me try to prove a little theorem here. 00:58:50.180 --> 00:58:57.840 align:middle line:90% 00:58:57.840 --> 00:59:03.110 align:middle line:90% So here's the theorem. 00:59:03.110 --> 00:59:09.420 align:middle line:90% 00:59:09.420 --> 00:59:15.420 align:middle line:84% So F here is supposed to be an ordered field. 00:59:15.420 --> 00:59:20.500 align:middle line:90% 00:59:20.500 --> 00:59:25.940 align:middle line:84% And then we have that x here is less than y. 00:59:25.940 --> 00:59:34.260 align:middle line:90% And z here, this is positive. 00:59:34.260 --> 00:59:36.820 align:middle line:90% So I assume I have this. 00:59:36.820 --> 00:59:40.060 align:middle line:84% And now I want to show that-- then I claim that 00:59:40.060 --> 00:59:43.500 align:middle line:84% from these properties that our ordered field have, 00:59:43.500 --> 00:59:51.580 align:middle line:84% I get automatically then x times c is less than y times c. 00:59:51.580 --> 00:59:55.700 align:middle line:84% So I claim I get this property here 00:59:55.700 --> 00:59:59.580 align:middle line:84% that you know these properties over here. 00:59:59.580 --> 01:00:01.720 align:middle line:90% So let's try to prove that. 01:00:01.720 --> 01:00:19.310 align:middle line:90% 01:00:19.310 --> 01:00:23.550 align:middle line:90% So what I want to do is-- 01:00:23.550 --> 01:00:35.890 align:middle line:84% so we want to show that x times z is less than y times z. 01:00:35.890 --> 01:00:38.350 align:middle line:90% This is what we want to show. 01:00:38.350 --> 01:00:42.090 align:middle line:90% Now, to say this here-- 01:00:42.090 --> 01:00:58.050 align:middle line:84% this here is equivalent to showing that-- 01:00:58.050 --> 01:01:01.030 align:middle line:90% so if you take, this you could-- 01:01:01.030 --> 01:01:09.890 align:middle line:84% because this here, if you add minus y times z-- sorry. 01:01:09.890 --> 01:01:12.570 align:middle line:90% 01:01:12.570 --> 01:01:17.470 align:middle line:84% If you take this thing here and to both sides-- 01:01:17.470 --> 01:01:26.890 align:middle line:84% so you take x times z and then you plus minus x times z, 01:01:26.890 --> 01:01:31.830 align:middle line:90% then this here is the same as-- 01:01:31.830 --> 01:01:35.490 align:middle line:84% to say that this is less than that is equivalent to saying 01:01:35.490 --> 01:01:36.430 align:middle line:90% this thing here. 01:01:36.430 --> 01:01:41.530 align:middle line:90% 01:01:41.530 --> 01:01:43.810 align:middle line:90% Why is that equivalent? 01:01:43.810 --> 01:01:48.530 align:middle line:84% This is this law here that is chaining the ordering together 01:01:48.530 --> 01:01:50.970 align:middle line:90% with addition. 01:01:50.970 --> 01:01:56.310 align:middle line:84% And this here, this element here, 01:01:56.310 --> 01:02:00.510 align:middle line:84% this is just the inverse element of this product here. 01:02:00.510 --> 01:02:04.679 align:middle line:90% 01:02:04.679 --> 01:02:08.710 align:middle line:84% But now you see that these two things together, 01:02:08.710 --> 01:02:10.082 align:middle line:90% that just gives you 0. 01:02:10.082 --> 01:02:19.030 align:middle line:90% 01:02:19.030 --> 01:02:26.950 align:middle line:84% This thing here, this here is the same as x. 01:02:26.950 --> 01:02:30.790 align:middle line:90% So maybe we need to show this. 01:02:30.790 --> 01:02:32.790 align:middle line:84% But this here, it's not hard to believe 01:02:32.790 --> 01:02:43.990 align:middle line:84% that this here is the same as y times z minus x times z. 01:02:43.990 --> 01:02:46.850 align:middle line:84% So this here is now-- you can factor out. 01:02:46.850 --> 01:02:51.910 align:middle line:84% So this is this law, the 11th law that allow you to factor 01:02:51.910 --> 01:02:53.070 align:middle line:90% this one out. 01:02:53.070 --> 01:02:58.350 align:middle line:84% So this is the same as y minus x times z. 01:02:58.350 --> 01:03:04.450 align:middle line:84% This was the distributive law and was the 11th law. 01:03:04.450 --> 01:03:10.170 align:middle line:84% But now you see that because y was bigger than x-- 01:03:10.170 --> 01:03:20.210 align:middle line:84% since y here was bigger than x, this 01:03:20.210 --> 01:03:23.850 align:middle line:84% is equivalent to saying that y is bigger than x-- 01:03:23.850 --> 01:03:28.450 align:middle line:84% if you subtract x on both sides, this 01:03:28.450 --> 01:03:34.090 align:middle line:84% is equivalent to saying that y minus x is bigger than 0. 01:03:34.090 --> 01:03:40.520 align:middle line:84% This is, again, using the first axiom over here. 01:03:40.520 --> 01:03:44.150 align:middle line:84% And so you see that now we have that. 01:03:44.150 --> 01:03:49.810 align:middle line:84% So we want to prove that this thing here, this 01:03:49.810 --> 01:03:53.410 align:middle line:84% here is all equivalent to proving that this thing here 01:03:53.410 --> 01:03:55.890 align:middle line:90% is bigger than 0. 01:03:55.890 --> 01:03:58.750 align:middle line:84% So the claim up there is equivalent to proving 01:03:58.750 --> 01:04:02.390 align:middle line:84% that this thing here is bigger than 0. 01:04:02.390 --> 01:04:07.070 align:middle line:84% But the difference here, this difference here is positive. 01:04:07.070 --> 01:04:09.590 align:middle line:90% This is assumed to be positive. 01:04:09.590 --> 01:04:13.590 align:middle line:84% The second law that is chaining the ordering together 01:04:13.590 --> 01:04:17.390 align:middle line:84% with multiplication say that if you have something positive 01:04:17.390 --> 01:04:19.330 align:middle line:84% and you multiply it by something positive, 01:04:19.330 --> 01:04:21.230 align:middle line:84% you're getting something positive. 01:04:21.230 --> 01:04:24.550 align:middle line:84% And so in order to prove this here, 01:04:24.550 --> 01:04:27.870 align:middle line:84% it was the same as proving that this thing here 01:04:27.870 --> 01:04:29.670 align:middle line:90% is actually positive. 01:04:29.670 --> 01:04:35.450 align:middle line:84% And we have now just reduced it to this second axiom. 01:04:35.450 --> 01:04:35.950 align:middle line:90% OK. 01:04:35.950 --> 01:04:38.630 align:middle line:90% 01:04:38.630 --> 01:04:41.710 align:middle line:90% Is that clear? 01:04:41.710 --> 01:04:43.010 align:middle line:90% OK. 01:04:43.010 --> 01:04:44.910 align:middle line:84% So we go just a little bit easier 01:04:44.910 --> 01:04:46.130 align:middle line:90% here in the first lecture. 01:04:46.130 --> 01:04:49.990 align:middle line:84% This is what I have for the first lecture. 01:04:49.990 --> 01:04:53.750 align:middle line:84% But there's a bunch of different concepts. 01:04:53.750 --> 01:05:00.290 align:middle line:84% And again, feel free, of course, to ask any questions. 01:05:00.290 --> 01:05:03.770 align:middle line:84% On the class, I typically just linger around for a few minutes 01:05:03.770 --> 01:05:07.290 align:middle line:84% afterwards if anybody has any questions after the class. 01:05:07.290 --> 01:05:09.970 align:middle line:84% And we will also have office hours. 01:05:09.970 --> 01:05:14.290 align:middle line:84% I'm just waiting for the assignment of the TAs. 01:05:14.290 --> 01:05:17.743 align:middle line:84% So that will be put all on online. 01:05:17.743 --> 01:05:20.890 align:middle line:90% 01:05:20.890 --> 01:05:21.470 align:middle line:90% Yeah. 01:05:21.470 --> 01:05:22.210 align:middle line:90% OK. 01:05:22.210 --> 01:05:23.510 align:middle line:90% Let me know how it goes. 01:05:23.510 --> 01:05:26.330 align:middle line:84% And one more thing is that this-- again, I 01:05:26.330 --> 01:05:28.370 align:middle line:90% mentioned before. 01:05:28.370 --> 01:05:31.370 align:middle line:84% So my experience is that it takes people a little bit 01:05:31.370 --> 01:05:35.130 align:middle line:84% of time to get a hang of this with writing proofs. 01:05:35.130 --> 01:05:39.146 align:middle line:90% So don't be discouraged. 01:05:39.146 --> 01:05:43.530 align:middle line:84% A few weeks or months into the class, things 01:05:43.530 --> 01:05:46.830 align:middle line:84% will really turn around, and it will all make sense. 01:05:46.830 --> 01:05:50.130 align:middle line:84% And then it will be more fun also. 01:05:50.130 --> 01:05:51.680 align:middle line:90% OK. 01:05:51.680 --> 01:05:55.000 align:middle line:90%