WEBVTT 00:00:00.000 --> 00:00:02.000 align:middle line:90% [SQUEAKING] 00:00:02.000 --> 00:00:04.500 align:middle line:90% [RUSTLING] 00:00:04.500 --> 00:00:07.000 align:middle line:90% [CLICKING] 00:00:07.000 --> 00:00:11.017 align:middle line:90% 00:00:11.017 --> 00:00:11.600 align:middle line:90% PROFESSOR: OK. 00:00:11.600 --> 00:00:16.580 align:middle line:90% So just to remind you-- 00:00:16.580 --> 00:00:22.160 align:middle line:90% so last time, we looked at-- 00:00:22.160 --> 00:00:26.020 align:middle line:84% we had the motivating example that we had a graph. 00:00:26.020 --> 00:00:31.840 align:middle line:84% We had a function f defined on some interval from a to b. 00:00:31.840 --> 00:00:35.960 align:middle line:84% And it was going inside the real numbers. 00:00:35.960 --> 00:00:38.660 align:middle line:84% And it started off negative here. 00:00:38.660 --> 00:00:40.520 align:middle line:90% This is f of a. 00:00:40.520 --> 00:00:42.400 align:middle line:90% This is negative. 00:00:42.400 --> 00:00:47.000 align:middle line:84% And it ended up positive, f of b. 00:00:47.000 --> 00:00:48.880 align:middle line:90% And we wanted to say-- 00:00:48.880 --> 00:00:50.510 align:middle line:84% and this was a continuous function. 00:00:50.510 --> 00:01:00.950 align:middle line:90% 00:01:00.950 --> 00:01:04.450 align:middle line:84% And we wanted to say that there was some c. 00:01:04.450 --> 00:01:16.510 align:middle line:84% So we wanted to show that there's a c between a and b 00:01:16.510 --> 00:01:21.170 align:middle line:84% so that the value at c is equal to 0. 00:01:21.170 --> 00:01:23.220 align:middle line:84% So this was the intermediate value theorem. 00:01:23.220 --> 00:01:30.150 align:middle line:90% 00:01:30.150 --> 00:01:38.790 align:middle line:84% And so this here was a motivating example 00:01:38.790 --> 00:01:43.810 align:middle line:84% for trying to understand, what property of R do we need? 00:01:43.810 --> 00:02:10.650 align:middle line:84% So what property of R do we need to make this rigorous? 00:02:10.650 --> 00:02:13.790 align:middle line:84% So that was the motivating example. 00:02:13.790 --> 00:02:19.530 align:middle line:84% And so in order to make this rigorous, 00:02:19.530 --> 00:02:27.050 align:middle line:84% well, then we need to concept of that R-- 00:02:27.050 --> 00:02:31.370 align:middle line:84% we need to understand the following statement, 00:02:31.370 --> 00:02:42.600 align:middle line:84% that R is a complete ordered field. 00:02:42.600 --> 00:02:46.660 align:middle line:90% 00:02:46.660 --> 00:02:50.610 align:middle line:84% And a field here-- so just briefly, 00:02:50.610 --> 00:03:07.560 align:middle line:84% fields here was a set with two operations, a plus 00:03:07.560 --> 00:03:09.820 align:middle line:84% and what we called multiplication. 00:03:09.820 --> 00:03:16.360 align:middle line:90% 00:03:16.360 --> 00:03:30.130 align:middle line:84% So each of these two operations have a 5 property. 00:03:30.130 --> 00:03:36.120 align:middle line:90% 00:03:36.120 --> 00:03:39.040 align:middle line:90% And then there's an 11 property. 00:03:39.040 --> 00:03:47.230 align:middle line:84% Plus, there's an 11th property that chain them together. 00:03:47.230 --> 00:03:55.950 align:middle line:90% 00:03:55.950 --> 00:03:58.590 align:middle line:84% And the last one is usually referred to 00:03:58.590 --> 00:04:00.230 align:middle line:90% as the distributive law. 00:04:00.230 --> 00:04:05.790 align:middle line:84% So that was one of the two important concepts 00:04:05.790 --> 00:04:08.790 align:middle line:90% that we talked about last time. 00:04:08.790 --> 00:04:11.030 align:middle line:84% The other concept that we talked about 00:04:11.030 --> 00:04:13.060 align:middle line:84% was what's called an ordered set. 00:04:13.060 --> 00:04:18.990 align:middle line:90% 00:04:18.990 --> 00:04:27.030 align:middle line:84% So that's a set with an ordering. 00:04:27.030 --> 00:04:30.490 align:middle line:84% And the ordering has to have the property that-- 00:04:30.490 --> 00:04:36.990 align:middle line:84% so this here, the ordering has the property-- 00:04:36.990 --> 00:04:38.770 align:middle line:84% so it has actually two properties. 00:04:38.770 --> 00:04:47.750 align:middle line:90% 00:04:47.750 --> 00:04:49.870 align:middle line:84% The first one is that whenever you 00:04:49.870 --> 00:04:57.460 align:middle line:84% take an element x and an element y in the set, then 00:04:57.460 --> 00:05:01.060 align:middle line:84% you should be able to relate these two by the ordering. 00:05:01.060 --> 00:05:09.740 align:middle line:84% So either x is equal to y or x is smaller 00:05:09.740 --> 00:05:17.040 align:middle line:90% than y or y is smaller than x. 00:05:17.040 --> 00:05:21.580 align:middle line:90% 00:05:21.580 --> 00:05:23.940 align:middle line:84% And the second property is what's 00:05:23.940 --> 00:05:31.420 align:middle line:84% called transitive, that if x and y and c are elements in the set 00:05:31.420 --> 00:05:40.580 align:middle line:84% and x is less than y and y is less than c, then 00:05:40.580 --> 00:05:42.780 align:middle line:90% x should be less than c. 00:05:42.780 --> 00:05:45.740 align:middle line:84% So this is the transitive property. 00:05:45.740 --> 00:05:48.300 align:middle line:84% So the first property, again, is that you 00:05:48.300 --> 00:05:50.840 align:middle line:90% can compare any two elements. 00:05:50.840 --> 00:05:53.880 align:middle line:84% And the second one is this transitive law. 00:05:53.880 --> 00:05:57.800 align:middle line:90% 00:05:57.800 --> 00:06:01.600 align:middle line:84% So that was the two concepts we talked about last time. 00:06:01.600 --> 00:06:07.620 align:middle line:84% And then we talked about what is meant to be an ordered field. 00:06:07.620 --> 00:06:17.560 align:middle line:84% So then we also talked about ordered fields. 00:06:17.560 --> 00:06:23.160 align:middle line:84% And then ordered fields is, you have a field that 00:06:23.160 --> 00:06:25.440 align:middle line:90% is also an ordered set. 00:06:25.440 --> 00:06:33.560 align:middle line:84% And then the relationship-- so that one being less than 00:06:33.560 --> 00:06:34.880 align:middle line:90% the other one-- 00:06:34.880 --> 00:06:37.720 align:middle line:84% to interact. have So this is these two things 00:06:37.720 --> 00:06:41.680 align:middle line:84% that, if you take x is less than y 00:06:41.680 --> 00:06:46.080 align:middle line:84% and you're adding any elements on both sides, 00:06:46.080 --> 00:06:50.350 align:middle line:84% then it still preserve the order. 00:06:50.350 --> 00:06:52.110 align:middle line:90% So this is one thing. 00:06:52.110 --> 00:06:57.310 align:middle line:84% And the second is that if you take x that's bigger than 0 00:06:57.310 --> 00:07:04.630 align:middle line:84% and y that's bigger than 0, then the product of x and y 00:07:04.630 --> 00:07:06.990 align:middle line:90% should be bigger than 0. 00:07:06.990 --> 00:07:10.690 align:middle line:90% And so this is how the-- 00:07:10.690 --> 00:07:15.670 align:middle line:84% so an ordered field, again, is a field that has a set, 00:07:15.670 --> 00:07:17.830 align:middle line:90% also has an ordering. 00:07:17.830 --> 00:07:22.750 align:middle line:84% And then the ordering together with the operation of the field 00:07:22.750 --> 00:07:25.030 align:middle line:84% should be chained together by these two rules. 00:07:25.030 --> 00:07:27.550 align:middle line:90% 00:07:27.550 --> 00:07:32.770 align:middle line:84% So we're getting close to understanding this. 00:07:32.770 --> 00:07:35.370 align:middle line:90% So this question is-- 00:07:35.370 --> 00:07:39.750 align:middle line:84% so in order for all of this to make sense, 00:07:39.750 --> 00:07:45.780 align:middle line:84% we need to understand what it means that the real numbers is 00:07:45.780 --> 00:07:48.060 align:middle line:90% a complete ordered field. 00:07:48.060 --> 00:07:51.720 align:middle line:84% And so we have now this part of it, that it's an ordered field. 00:07:51.720 --> 00:07:53.060 align:middle line:90% We understand that. 00:07:53.060 --> 00:07:58.200 align:middle line:84% This here, we need to understand what this means. 00:07:58.200 --> 00:07:58.700 align:middle line:90% OK. 00:07:58.700 --> 00:08:04.380 align:middle line:84% So now before I get into that, let's 00:08:04.380 --> 00:08:15.760 align:middle line:84% just talk about-- so what is the difference 00:08:15.760 --> 00:08:24.460 align:middle line:84% between the real numbers and the rational numbers? 00:08:24.460 --> 00:08:28.420 align:middle line:84% Remember, the rational numbers are numbers of this form 00:08:28.420 --> 00:08:33.085 align:middle line:84% here, where this is an integer and this is a natural number. 00:08:33.085 --> 00:08:37.460 align:middle line:84% So what is the difference between these two? 00:08:37.460 --> 00:08:48.200 align:middle line:84% Well, one difference is-- so one difference 00:08:48.200 --> 00:09:00.120 align:middle line:84% is that that square root of 2 is a real number but not 00:09:00.120 --> 00:09:00.970 align:middle line:90% a rational number. 00:09:00.970 --> 00:09:13.384 align:middle line:90% 00:09:13.384 --> 00:09:18.650 align:middle line:84% Now let me try to explain first why it's not a rational number. 00:09:18.650 --> 00:09:21.240 align:middle line:90% 00:09:21.240 --> 00:09:23.560 align:middle line:90% So let's try to see-- 00:09:23.560 --> 00:09:25.260 align:middle line:90% so what is square root of 2? 00:09:25.260 --> 00:09:30.600 align:middle line:90% 00:09:30.600 --> 00:09:37.080 align:middle line:84% The square root of 2, you can think about it as a number, x. 00:09:37.080 --> 00:09:40.520 align:middle line:84% And it has to have the property that if you multiply it 00:09:40.520 --> 00:09:42.510 align:middle line:90% by itself-- 00:09:42.510 --> 00:09:46.550 align:middle line:84% so x times x or, if you will, x squared-- 00:09:46.550 --> 00:09:49.910 align:middle line:90% this here should be equal to 2. 00:09:49.910 --> 00:09:51.710 align:middle line:90% It also should be positive. 00:09:51.710 --> 00:09:57.950 align:middle line:90% 00:09:57.950 --> 00:10:04.110 align:middle line:84% Now suppose-- so I claim that square root of 2 00:10:04.110 --> 00:10:07.010 align:middle line:90% is not a rational number. 00:10:07.010 --> 00:10:09.050 align:middle line:90% So by that, I mean that-- 00:10:09.050 --> 00:10:12.170 align:middle line:84% so I want to claim the following. 00:10:12.170 --> 00:10:17.110 align:middle line:90% 00:10:17.110 --> 00:10:40.600 align:middle line:84% There does not exist a rational number x equal to m/n, where-- 00:10:40.600 --> 00:10:42.740 align:middle line:90% again, this is an integer. 00:10:42.740 --> 00:10:45.240 align:middle line:90% This is a natural number-- 00:10:45.240 --> 00:10:53.540 align:middle line:90% so that x squared is equal to 2. 00:10:53.540 --> 00:10:58.780 align:middle line:84% So I claim that you don't have a rational number that 00:10:58.780 --> 00:11:00.840 align:middle line:90% is square root of 2. 00:11:00.840 --> 00:11:01.340 align:middle line:90% OK. 00:11:01.340 --> 00:11:03.120 align:middle line:90% Now let's try to prove that. 00:11:03.120 --> 00:11:18.500 align:middle line:90% 00:11:18.500 --> 00:11:23.140 align:middle line:84% So suppose there was such a number. 00:11:23.140 --> 00:11:24.540 align:middle line:90% So we want to prove it. 00:11:24.540 --> 00:11:26.240 align:middle line:90% So we prove by contradiction. 00:11:26.240 --> 00:11:36.220 align:middle line:90% 00:11:36.220 --> 00:11:49.180 align:middle line:84% So assume that there exists x that is a rational number. 00:11:49.180 --> 00:11:50.680 align:middle line:84% So it can be written in this form 00:11:50.680 --> 00:11:59.200 align:middle line:84% here, where m is an integer, and n is a natural number. 00:11:59.200 --> 00:12:03.620 align:middle line:84% So assume that there exists an x that is a rational number 00:12:03.620 --> 00:12:07.080 align:middle line:84% and so that the squared is equal to 2. 00:12:07.080 --> 00:12:10.320 align:middle line:84% Well, if that's the case, then m/n-- 00:12:10.320 --> 00:12:14.140 align:middle line:90% 00:12:14.140 --> 00:12:18.280 align:middle line:84% so this here squared is supposed to be equal to 2. 00:12:18.280 --> 00:12:21.740 align:middle line:90% 00:12:21.740 --> 00:12:29.720 align:middle line:84% So this implies that m squared over n squared is equal to 2. 00:12:29.720 --> 00:12:36.550 align:middle line:84% And this, just multiplying over, implies that m squared is equal 00:12:36.550 --> 00:12:40.750 align:middle line:90% to 2n squared. 00:12:40.750 --> 00:12:46.050 align:middle line:84% Now, when we write x as a rational number-- of course, 00:12:46.050 --> 00:12:48.150 align:middle line:84% we know that there are many ways of writing it 00:12:48.150 --> 00:12:49.530 align:middle line:90% as rational number. 00:12:49.530 --> 00:12:51.990 align:middle line:84% Because if you have one, then you 00:12:51.990 --> 00:12:56.810 align:middle line:84% can just multiply the numerator and denominator 00:12:56.810 --> 00:13:01.142 align:middle line:84% by a natural number, and then you get another representation. 00:13:01.142 --> 00:13:03.190 align:middle line:90% But we can assume-- 00:13:03.190 --> 00:13:05.510 align:middle line:90% so we may assume. 00:13:05.510 --> 00:13:07.270 align:middle line:90% So when we have written-- 00:13:07.270 --> 00:13:23.700 align:middle line:84% so we may assume that m and n does not have a common factor. 00:13:23.700 --> 00:13:27.950 align:middle line:90% 00:13:27.950 --> 00:13:30.130 align:middle line:84% Because otherwise, we could just reduce it. 00:13:30.130 --> 00:13:34.230 align:middle line:84% So we could reduce it until they don't have a common factor. 00:13:34.230 --> 00:13:44.760 align:middle line:90% 00:13:44.760 --> 00:13:45.260 align:middle line:90% OK. 00:13:45.260 --> 00:13:49.780 align:middle line:84% So we will assume that m and n do not have a common factor. 00:13:49.780 --> 00:13:53.820 align:middle line:84% We have now that m squared is equal to this. 00:13:53.820 --> 00:13:59.300 align:middle line:84% But this means that m here must be even. 00:13:59.300 --> 00:14:08.720 align:middle line:84% So we have that since m squared is equal to 2n squared-- 00:14:08.720 --> 00:14:13.040 align:middle line:84% so it means that 2 is a factor in m squared. 00:14:13.040 --> 00:14:15.340 align:middle line:84% So it must actually be a factor in m. 00:14:15.340 --> 00:14:24.580 align:middle line:84% So it means that 2 here is a factor in m. 00:14:24.580 --> 00:14:27.980 align:middle line:90% In other words, m is even. 00:14:27.980 --> 00:14:34.946 align:middle line:84% So you can write m here as 2 times another integer here. 00:14:34.946 --> 00:14:40.730 align:middle line:90% 00:14:40.730 --> 00:14:48.050 align:middle line:84% But this means that m squared is equal to 2 m1 squared, which 00:14:48.050 --> 00:14:51.640 align:middle line:90% is equal to 4 m1 squared. 00:14:51.640 --> 00:14:56.050 align:middle line:84% And we already know that that is equal to 2n squared. 00:14:56.050 --> 00:14:59.170 align:middle line:90% 00:14:59.170 --> 00:15:03.030 align:middle line:84% So this means dividing by 2 on both sides. 00:15:03.030 --> 00:15:08.730 align:middle line:84% It means that 2 m1 squared is equal to n. 00:15:08.730 --> 00:15:12.090 align:middle line:84% But this means that-- sorry, is equal to n squared. 00:15:12.090 --> 00:15:17.650 align:middle line:84% So this means that 2 here must be a factor of n. 00:15:17.650 --> 00:15:29.290 align:middle line:84% So this means here that 2 is also a factor in n. 00:15:29.290 --> 00:15:33.550 align:middle line:84% But now you see that this is a contradiction. 00:15:33.550 --> 00:15:35.390 align:middle line:90% So this is a contradiction. 00:15:35.390 --> 00:15:41.190 align:middle line:84% Because now what we have is that 2 is a factor in m 00:15:41.190 --> 00:15:43.170 align:middle line:90% and 2 is a factor on n. 00:15:43.170 --> 00:15:44.890 align:middle line:90% So they have a common factor. 00:15:44.890 --> 00:15:46.950 align:middle line:84% But we assumed they didn't have a common factor. 00:15:46.950 --> 00:15:50.090 align:middle line:90% 00:15:50.090 --> 00:15:52.430 align:middle line:84% So we argue by contradiction-- we assumed 00:15:52.430 --> 00:15:55.750 align:middle line:84% there was a rational number so that the square of it 00:15:55.750 --> 00:15:57.230 align:middle line:90% was equal to 2. 00:15:57.230 --> 00:15:58.990 align:middle line:90% And we get a contradiction. 00:15:58.990 --> 00:16:03.390 align:middle line:84% So there could not be a rational number 00:16:03.390 --> 00:16:06.850 align:middle line:84% that has a property that the square is equal to 2. 00:16:06.850 --> 00:16:10.030 align:middle line:84% So square root of 2 is definitely 00:16:10.030 --> 00:16:11.100 align:middle line:90% not a rational number. 00:16:11.100 --> 00:16:18.150 align:middle line:90% 00:16:18.150 --> 00:16:25.230 align:middle line:84% So now, how do we think about square root of 2? 00:16:25.230 --> 00:16:34.910 align:middle line:84% So how do we fit square root of 2 into the number system? 00:16:34.910 --> 00:16:47.060 align:middle line:90% 00:16:47.060 --> 00:16:50.380 align:middle line:84% Well, square root of 2 is-- if you 00:16:50.380 --> 00:16:55.100 align:middle line:84% think about it as a decimal number, 00:16:55.100 --> 00:17:09.859 align:middle line:90% then it starts with 1.4142136. 00:17:09.859 --> 00:17:12.460 align:middle line:90% And then it continue like this. 00:17:12.460 --> 00:17:18.020 align:middle line:84% So one way of thinking about square root of 2 00:17:18.020 --> 00:17:20.220 align:middle line:84% is that it is-- and we'll come back 00:17:20.220 --> 00:17:22.460 align:middle line:90% to this at some later lecture. 00:17:22.460 --> 00:17:24.660 align:middle line:90% So we think about-- 00:17:24.660 --> 00:17:40.330 align:middle line:84% so one way of thinking about square root of 2 00:17:40.330 --> 00:17:47.960 align:middle line:84% is as a limit of a sequence of number. 00:17:47.960 --> 00:17:52.530 align:middle line:90% 00:17:52.530 --> 00:17:58.530 align:middle line:84% And that brings us to another really important concept 00:17:58.530 --> 00:18:02.130 align:middle line:84% that we will talk much more about later, 00:18:02.130 --> 00:18:06.050 align:middle line:90% is sequences and limits. 00:18:06.050 --> 00:18:10.610 align:middle line:84% And so if we want to think about it as a limit of a sequence, 00:18:10.610 --> 00:18:14.310 align:middle line:84% we think about maybe the number 1 first. 00:18:14.310 --> 00:18:16.690 align:middle line:90% And then we think about 1.4. 00:18:16.690 --> 00:18:30.830 align:middle line:84% And then we think 1.4.1, and then 1.414, and then 1.4142, 00:18:30.830 --> 00:18:31.710 align:middle line:90% et cetera. 00:18:31.710 --> 00:18:37.230 align:middle line:84% And so you see that this here is a sequence of numbers. 00:18:37.230 --> 00:18:44.340 align:middle line:84% And so this is an increasing sequence of numbers. 00:18:44.340 --> 00:18:53.950 align:middle line:90% 00:18:53.950 --> 00:19:02.880 align:middle line:84% And square root of 2 is the limit of the sequence. 00:19:02.880 --> 00:19:08.270 align:middle line:90% 00:19:08.270 --> 00:19:13.830 align:middle line:84% So again, we'll talk much more about sequences and limits 00:19:13.830 --> 00:19:14.330 align:middle line:90% later. 00:19:14.330 --> 00:19:16.710 align:middle line:84% But this is just something to keep in mind 00:19:16.710 --> 00:19:23.222 align:middle line:84% as also motivating why we need to understand sequences 00:19:23.222 --> 00:19:23.930 align:middle line:90% and their limits. 00:19:23.930 --> 00:19:30.620 align:middle line:90% 00:19:30.620 --> 00:19:33.940 align:middle line:90% So we now know-- 00:19:33.940 --> 00:19:37.980 align:middle line:84% maybe not super surprisingly, but we know and see 00:19:37.980 --> 00:19:43.980 align:middle line:84% there was a very simple proof why the real numbers is larger 00:19:43.980 --> 00:19:45.420 align:middle line:90% than the rational numbers. 00:19:45.420 --> 00:19:49.100 align:middle line:84% It contain a particular square root of 2 00:19:49.100 --> 00:19:53.180 align:middle line:84% that the rational numbers does not contain. 00:19:53.180 --> 00:19:54.480 align:middle line:90% We still need to-- 00:19:54.480 --> 00:19:58.500 align:middle line:84% and this is related to the completeness of R 00:19:58.500 --> 00:20:02.520 align:middle line:84% and why the rational numbers is not complete. 00:20:02.520 --> 00:20:07.220 align:middle line:84% And this is, again, what we're now working towards. 00:20:07.220 --> 00:20:10.380 align:middle line:84% So we're working towards what it means 00:20:10.380 --> 00:20:11.970 align:middle line:90% for a field to be complete. 00:20:11.970 --> 00:20:31.570 align:middle line:90% 00:20:31.570 --> 00:20:36.330 align:middle line:84% So let me just write that, and I'll explain it more. 00:20:36.330 --> 00:20:46.290 align:middle line:84% But this here is what we are aiming towards, completeness, 00:20:46.290 --> 00:20:48.290 align:middle line:84% which is-- and I'll explain that-- 00:20:48.290 --> 00:20:49.800 align:middle line:84% at the least upper bound property. 00:20:49.800 --> 00:21:02.708 align:middle line:90% 00:21:02.708 --> 00:21:03.990 align:middle line:90% OK. 00:21:03.990 --> 00:21:07.110 align:middle line:84% So suppose you take an ordered set. 00:21:07.110 --> 00:21:15.080 align:middle line:90% So suppose S is an ordered set. 00:21:15.080 --> 00:21:20.160 align:middle line:90% 00:21:20.160 --> 00:21:29.280 align:middle line:84% And the ordering-- as usual, we're ordering, 00:21:29.280 --> 00:21:33.000 align:middle line:90% we denote by this. 00:21:33.000 --> 00:21:35.460 align:middle line:90% So suppose S is an ordered set. 00:21:35.460 --> 00:21:40.880 align:middle line:84% And suppose that A here is a subset of S. 00:21:40.880 --> 00:21:59.328 align:middle line:84% Then we say that M is an upper bound for the subset for A 00:21:59.328 --> 00:22:15.280 align:middle line:84% if, for all elements in A, M here is bigger or equal to A. 00:22:15.280 --> 00:22:19.780 align:middle line:84% So an upper bound for a subset is 00:22:19.780 --> 00:22:22.540 align:middle line:84% something that has to be bigger than all 00:22:22.540 --> 00:22:25.240 align:middle line:84% the elements in the subset, bigger or equal to. 00:22:25.240 --> 00:22:27.410 align:middle line:84% It could be equal to one of the elements. 00:22:27.410 --> 00:22:30.580 align:middle line:90% 00:22:30.580 --> 00:22:32.960 align:middle line:90% So that's a upper bound. 00:22:32.960 --> 00:22:36.150 align:middle line:84% And so, of course, if you look at-- 00:22:36.150 --> 00:22:41.140 align:middle line:90% 00:22:41.140 --> 00:22:44.820 align:middle line:90% so a couple of examples here. 00:22:44.820 --> 00:22:57.580 align:middle line:84% So if the set here is-- let's say that the set is-- 00:22:57.580 --> 00:23:01.660 align:middle line:84% so S is the natural numbers here. 00:23:01.660 --> 00:23:10.240 align:middle line:90% And the subset here is 1, 2, 3. 00:23:10.240 --> 00:23:20.770 align:middle line:84% So then the element 4 here is an upper bound 00:23:20.770 --> 00:23:38.690 align:middle line:84% for A. So 4 is an upper bound for A, whereas 2 is not. 00:23:38.690 --> 00:23:41.090 align:middle line:90% Obviously, 3 is bigger than 2. 00:23:41.090 --> 00:23:43.240 align:middle line:90% So 2 is not an upper bound. 00:23:43.240 --> 00:23:48.810 align:middle line:90% 00:23:48.810 --> 00:23:54.170 align:middle line:90% Another example is if you take-- 00:23:54.170 --> 00:23:58.010 align:middle line:90% 00:23:58.010 --> 00:24:01.290 align:middle line:84% let's say S is the rational numbers, 00:24:01.290 --> 00:24:07.520 align:middle line:84% and A here is the natural numbers. 00:24:07.520 --> 00:24:14.280 align:middle line:90% 00:24:14.280 --> 00:24:18.070 align:middle line:84% Then A here does not have an upper bound. 00:24:18.070 --> 00:24:30.240 align:middle line:90% 00:24:30.240 --> 00:24:37.850 align:middle line:84% And I'll return to this second example in just a minute. 00:24:37.850 --> 00:24:44.532 align:middle line:90% 00:24:44.532 --> 00:24:49.080 align:middle line:84% s that is what it means to be an upper bound. 00:24:49.080 --> 00:24:54.840 align:middle line:90% So now there's another concept. 00:24:54.840 --> 00:24:56.930 align:middle line:90% And that is a least upper bound. 00:24:56.930 --> 00:25:16.060 align:middle line:90% 00:25:16.060 --> 00:25:19.330 align:middle line:84% So there's another concept of the least upper bound. 00:25:19.330 --> 00:25:26.140 align:middle line:90% 00:25:26.140 --> 00:25:30.430 align:middle line:84% So again, we have S is an ordered set. 00:25:30.430 --> 00:25:35.700 align:middle line:90% 00:25:35.700 --> 00:25:43.860 align:middle line:90% We have A here is a subset of S. 00:25:43.860 --> 00:25:53.340 align:middle line:84% Then we say that it's a subset of S. Maybe let me just 00:25:53.340 --> 00:25:55.430 align:middle line:90% point out that it's bounded. 00:25:55.430 --> 00:26:08.890 align:middle line:90% 00:26:08.890 --> 00:26:11.420 align:middle line:84% So And is a subset of S that is bounded. 00:26:11.420 --> 00:26:14.850 align:middle line:90% 00:26:14.850 --> 00:26:36.770 align:middle line:84% Then we say that capital M is a least upper bound 00:26:36.770 --> 00:26:47.570 align:middle line:90% for a if M is an upper bound-- 00:26:47.570 --> 00:26:50.730 align:middle line:90% so it has to be an upper bound-- 00:26:50.730 --> 00:27:02.190 align:middle line:84% and if, for any other upper bound-- 00:27:02.190 --> 00:27:11.440 align:middle line:90% 00:27:11.440 --> 00:27:14.740 align:middle line:84% we have to have that for any other upper bound, 00:27:14.740 --> 00:27:25.760 align:middle line:84% say, M1, we have that M1 has to be bigger or equal to M. 00:27:25.760 --> 00:27:32.080 align:middle line:84% So a bounded set has a least upper 00:27:32.080 --> 00:27:38.000 align:middle line:84% bound if there is an element that is the smallest, so 00:27:38.000 --> 00:27:39.450 align:middle line:90% to speak, of the upper bound. 00:27:39.450 --> 00:27:42.760 align:middle line:90% 00:27:42.760 --> 00:27:45.035 align:middle line:90% OK. 00:27:45.035 --> 00:27:53.360 align:middle line:84% And so now we are ready to state what it means 00:27:53.360 --> 00:27:55.490 align:middle line:90% to be a complete ordered field. 00:27:55.490 --> 00:28:03.960 align:middle line:90% 00:28:03.960 --> 00:28:35.340 align:middle line:84% So a complete ordered field is an ordered field 00:28:35.340 --> 00:28:53.150 align:middle line:84% so that any bounded subset has a least upper bound. 00:28:53.150 --> 00:29:03.010 align:middle line:90% 00:29:03.010 --> 00:29:05.782 align:middle line:90% OK. 00:29:05.782 --> 00:29:08.610 align:middle line:84% So that's a complete ordered field. 00:29:08.610 --> 00:29:11.610 align:middle line:84% And so we have the following theorem 00:29:11.610 --> 00:29:15.970 align:middle line:84% that we're not going to prove because that really takes us 00:29:15.970 --> 00:29:16.720 align:middle line:90% off field. 00:29:16.720 --> 00:29:21.022 align:middle line:90% 00:29:21.022 --> 00:29:22.900 align:middle line:84% So we'll just take it for granted. 00:29:22.900 --> 00:29:27.810 align:middle line:90% 00:29:27.810 --> 00:29:53.030 align:middle line:84% So theorem-- so there exists a smallest complete ordered field 00:29:53.030 --> 00:29:55.020 align:middle line:84% that contains the rational numbers. 00:29:55.020 --> 00:30:11.840 align:middle line:90% 00:30:11.840 --> 00:30:12.340 align:middle line:90% OK. 00:30:12.340 --> 00:30:13.500 align:middle line:90% So this is a theorem. 00:30:13.500 --> 00:30:17.000 align:middle line:84% Again, we won't prove it because it's really 00:30:17.000 --> 00:30:19.300 align:middle line:90% outside the scope of this class. 00:30:19.300 --> 00:30:22.520 align:middle line:84% It's not really what we want to focus on here. 00:30:22.520 --> 00:30:27.600 align:middle line:90% But you can prove this. 00:30:27.600 --> 00:30:41.720 align:middle line:84% And so that field is what we call R, 00:30:41.720 --> 00:30:43.780 align:middle line:90% is R. It's the real numbers. 00:30:43.780 --> 00:30:48.800 align:middle line:90% 00:30:48.800 --> 00:30:49.600 align:middle line:90% OK. 00:30:49.600 --> 00:30:58.640 align:middle line:84% So now I want to see that if we work with that field instead 00:30:58.640 --> 00:31:03.630 align:middle line:84% of the rational numbers, then actually, we have in particular 00:31:03.630 --> 00:31:07.580 align:middle line:84% that the square root of 2 is in that field. 00:31:07.580 --> 00:31:25.930 align:middle line:90% 00:31:25.930 --> 00:31:41.640 align:middle line:84% So we want to show that a square root of 2 is in the reals. 00:31:41.640 --> 00:31:45.246 align:middle line:90% 00:31:45.246 --> 00:31:51.350 align:middle line:84% So now I'm going to look at the set A. 00:31:51.350 --> 00:31:58.450 align:middle line:84% So I'm going to look at the set A, which 00:31:58.450 --> 00:32:06.570 align:middle line:84% is the element in R such that x here is bigger than 0 00:32:06.570 --> 00:32:13.930 align:middle line:90% and x squared is less than 2. 00:32:13.930 --> 00:32:17.950 align:middle line:84% So I'm going to look at this set. 00:32:17.950 --> 00:32:23.410 align:middle line:84% Now, of course, this set here-- so A here is bounded. 00:32:23.410 --> 00:32:24.270 align:middle line:90% It's not empty. 00:32:24.270 --> 00:32:25.960 align:middle line:90% We'll say first, it's not empty. 00:32:25.960 --> 00:32:29.370 align:middle line:90% 00:32:29.370 --> 00:32:38.090 align:middle line:84% It's not empty since 1 here is in A. 1 is positive. 00:32:38.090 --> 00:32:44.690 align:middle line:84% And 1 squared here is, of course, 1, which is less than 2. 00:32:44.690 --> 00:32:46.450 align:middle line:90% So it's non-empty. 00:32:46.450 --> 00:32:50.460 align:middle line:90% And A here is also A is bounded. 00:32:50.460 --> 00:32:53.970 align:middle line:90% 00:32:53.970 --> 00:33:09.480 align:middle line:84% Because any element-- so for all x in A, you have that x squared, 00:33:09.480 --> 00:33:12.360 align:middle line:84% x is positive and is less than 2. 00:33:12.360 --> 00:33:17.320 align:middle line:84% So this means that x here must be itself less than 2. 00:33:17.320 --> 00:33:20.980 align:middle line:84% Because 2, of course, when you square it, you're getting 4. 00:33:20.980 --> 00:33:25.760 align:middle line:90% 00:33:25.760 --> 00:33:30.010 align:middle line:84% So A here is a bounded subset of the reals. 00:33:30.010 --> 00:33:40.600 align:middle line:90% 00:33:40.600 --> 00:33:45.130 align:middle line:84% So now the real has the least upper bound property. 00:33:45.130 --> 00:33:47.960 align:middle line:90% 00:33:47.960 --> 00:33:51.720 align:middle line:84% So this means that I can now define-- 00:33:51.720 --> 00:34:03.015 align:middle line:84% so define square root of 2 to be the least upper bound. 00:34:03.015 --> 00:34:20.030 align:middle line:84% So define it to be the least upper bound of A. A 00:34:20.030 --> 00:34:23.429 align:middle line:90% was a nonempty bounded subset. 00:34:23.429 --> 00:34:29.350 align:middle line:84% So because we are in the reals, if you grant this theorem, 00:34:29.350 --> 00:34:30.970 align:middle line:84% then there is a least upper bound. 00:34:30.970 --> 00:34:34.790 align:middle line:90% 00:34:34.790 --> 00:34:38.550 align:middle line:90% So now let's denote-- 00:34:38.550 --> 00:34:56.010 align:middle line:84% so let me denote this number by x. 00:34:56.010 --> 00:34:58.910 align:middle line:84% I'm going to denoted by x instead of square root of 2. 00:34:58.910 --> 00:35:02.090 align:middle line:84% Because if I denote it by square root of 2, 00:35:02.090 --> 00:35:03.810 align:middle line:90% then it's going to be confusing. 00:35:03.810 --> 00:35:07.010 align:middle line:84% And it's like you will automatically 00:35:07.010 --> 00:35:11.290 align:middle line:84% use what you think square root 2 should satisfy. 00:35:11.290 --> 00:35:17.175 align:middle line:90% So let's call it x. 00:35:17.175 --> 00:35:20.570 align:middle line:90% 00:35:20.570 --> 00:35:23.870 align:middle line:90% So I want to prove-- 00:35:23.870 --> 00:35:41.528 align:middle line:84% so it's clear that since x here is at least upper bound for A 00:35:41.528 --> 00:35:45.890 align:middle line:84% and, in particular, 1 is in A, then 00:35:45.890 --> 00:35:51.120 align:middle line:84% it means that 1 here has to be less than or equal to x. 00:35:51.120 --> 00:35:54.760 align:middle line:90% And 1, of course, is positive. 00:35:54.760 --> 00:36:00.160 align:middle line:84% So it means that x here is definitely positive. 00:36:00.160 --> 00:36:04.610 align:middle line:84% Now we need to show that the square-- 00:36:04.610 --> 00:36:13.840 align:middle line:90% 00:36:13.840 --> 00:36:31.480 align:middle line:84% so we need to show that x squared is equal to 2. 00:36:31.480 --> 00:36:33.320 align:middle line:84% That's what we need to prove in order 00:36:33.320 --> 00:36:36.580 align:middle line:84% to prove that x is really the square root of 2. 00:36:36.580 --> 00:36:39.406 align:middle line:90% 00:36:39.406 --> 00:36:41.380 align:middle line:90% So I'm going to divide it up. 00:36:41.380 --> 00:36:51.520 align:middle line:84% So we divide the proof into two steps. 00:36:51.520 --> 00:37:02.390 align:middle line:90% 00:37:02.390 --> 00:37:06.810 align:middle line:84% The first one is that we want to-- 00:37:06.810 --> 00:37:14.830 align:middle line:84% so step 1, we're going to show that x squared is less than 00:37:14.830 --> 00:37:16.790 align:middle line:90% or equal to 2. 00:37:16.790 --> 00:37:19.470 align:middle line:84% So this is what we would show in the first step. 00:37:19.470 --> 00:37:22.750 align:middle line:84% And then the second step would be the reverse inequality. 00:37:22.750 --> 00:37:24.930 align:middle line:84% And of course, together, this would be the proof. 00:37:24.930 --> 00:37:28.190 align:middle line:90% 00:37:28.190 --> 00:37:34.210 align:middle line:84% So again, just like we argued before, in this case, 00:37:34.210 --> 00:37:36.450 align:middle line:84% we will also argue by contradiction. 00:37:36.450 --> 00:37:39.390 align:middle line:90% 00:37:39.390 --> 00:37:41.590 align:middle line:84% You may get the feeling that we're always 00:37:41.590 --> 00:37:43.930 align:middle line:84% going to argue by contradiction in this class. 00:37:43.930 --> 00:37:45.110 align:middle line:90% And we won't. 00:37:45.110 --> 00:37:49.290 align:middle line:84% And it's generally thought of that it's better style 00:37:49.290 --> 00:37:51.290 align:middle line:90% to prove things directly. 00:37:51.290 --> 00:37:55.050 align:middle line:84% But sometimes it's a much more logical 00:37:55.050 --> 00:37:58.730 align:middle line:84% and easier in a way-- at least, easier the first time-- 00:37:58.730 --> 00:38:00.270 align:middle line:90% to prove it by contradiction. 00:38:00.270 --> 00:38:03.970 align:middle line:84% And there's nothing wrong with proving it by contradiction. 00:38:03.970 --> 00:38:06.390 align:middle line:84% So we will show this by contradiction. 00:38:06.390 --> 00:38:20.150 align:middle line:90% 00:38:20.150 --> 00:38:20.650 align:middle line:90% OK. 00:38:20.650 --> 00:38:50.000 align:middle line:90% 00:38:50.000 --> 00:38:51.820 align:middle line:84% So we show this by contradiction. 00:38:51.820 --> 00:38:53.500 align:middle line:90% And so we assume first-- 00:38:53.500 --> 00:38:56.300 align:middle line:90% so assume not. 00:38:56.300 --> 00:38:59.760 align:middle line:90% 00:38:59.760 --> 00:39:05.260 align:middle line:84% So assume that x squared is not less than or equal to 2. 00:39:05.260 --> 00:39:06.360 align:middle line:90% So we assume not. 00:39:06.360 --> 00:39:09.160 align:middle line:90% 00:39:09.160 --> 00:39:17.980 align:middle line:84% So assume that x squared is strictly bigger than 2. 00:39:17.980 --> 00:39:20.720 align:middle line:90% 00:39:20.720 --> 00:39:26.440 align:middle line:84% Now I will look at x minus some small number. 00:39:26.440 --> 00:39:29.200 align:middle line:84% And I'm going to eventually prove 00:39:29.200 --> 00:39:34.100 align:middle line:84% that if I'm looking at x minus some small number-- 00:39:34.100 --> 00:39:39.400 align:middle line:90% 00:39:39.400 --> 00:39:42.510 align:middle line:84% I'm going to prove now that if this here was strictly bigger 00:39:42.510 --> 00:39:46.910 align:middle line:84% than 2, then I can find some small number so that this here 00:39:46.910 --> 00:39:48.770 align:middle line:90% would also be an upper bound. 00:39:48.770 --> 00:39:53.710 align:middle line:90% So h would be positive. 00:39:53.710 --> 00:39:56.870 align:middle line:84% This here would be a number strictly less than x. 00:39:56.870 --> 00:39:59.890 align:middle line:84% And I would show that this here is also an upper bound. 00:39:59.890 --> 00:40:02.142 align:middle line:90% That's what I want to do. 00:40:02.142 --> 00:40:09.750 align:middle line:84% So I take x minus h, and I square it. 00:40:09.750 --> 00:40:21.790 align:middle line:84% If I do that, then I get that this is x squared minus 2xh plus 00:40:21.790 --> 00:40:24.630 align:middle line:90% h squared. 00:40:24.630 --> 00:40:27.050 align:middle line:90% The squared is non-negative. 00:40:27.050 --> 00:40:31.590 align:middle line:84% So this here is definitely bigger or equal to x squared 00:40:31.590 --> 00:40:33.394 align:middle line:90% minus 2xh. 00:40:33.394 --> 00:40:37.030 align:middle line:90% 00:40:37.030 --> 00:40:38.770 align:middle line:90% And so you see that-- 00:40:38.770 --> 00:40:39.270 align:middle line:90% right. 00:40:39.270 --> 00:40:52.280 align:middle line:90% 00:40:52.280 --> 00:40:56.980 align:middle line:84% So we have that x squared is bigger than 2. 00:40:56.980 --> 00:41:04.280 align:middle line:84% So this means that x squared minus 2 is positive. 00:41:04.280 --> 00:41:06.520 align:middle line:84% I'm just subtracting a 2 from both sides. 00:41:06.520 --> 00:41:10.260 align:middle line:84% So we're assuming that this here is positive. 00:41:10.260 --> 00:41:13.340 align:middle line:90% 00:41:13.340 --> 00:41:16.870 align:middle line:84% And so now if I just make sure that-- 00:41:16.870 --> 00:41:23.780 align:middle line:90% 00:41:23.780 --> 00:41:27.380 align:middle line:90% so now I'm going to choose. 00:41:27.380 --> 00:41:28.760 align:middle line:90% So I want to choose. 00:41:28.760 --> 00:41:31.600 align:middle line:84% So I want to prove that x minus-- 00:41:31.600 --> 00:41:33.540 align:middle line:90% so I want to show-- 00:41:33.540 --> 00:41:46.920 align:middle line:84% so I want to get the contradiction 00:41:46.920 --> 00:42:04.960 align:middle line:84% by finding a h bigger than 0 so that minus h is also 00:42:04.960 --> 00:42:11.720 align:middle line:90% an upper bound for A. 00:42:11.720 --> 00:42:16.060 align:middle line:84% Because this here is clearly smaller than x, right? 00:42:16.060 --> 00:42:18.280 align:middle line:84% And this would then contradict that x 00:42:18.280 --> 00:42:20.840 align:middle line:90% was the smallest upper bound. 00:42:20.840 --> 00:42:23.840 align:middle line:90% So that's what I want to do. 00:42:23.840 --> 00:42:29.120 align:middle line:90% And so now, how do I do that? 00:42:29.120 --> 00:42:31.190 align:middle line:84% So using this little calculation here-- 00:42:31.190 --> 00:42:34.680 align:middle line:90% 00:42:34.680 --> 00:42:36.950 align:middle line:90% so what I want to do is-- 00:42:36.950 --> 00:42:50.390 align:middle line:84% so I want to find h positive so that when 00:42:50.390 --> 00:42:55.170 align:middle line:84% I square, when I subtract h from x and I square it, 00:42:55.170 --> 00:42:59.030 align:middle line:84% then I still get something bigger than 2. 00:42:59.030 --> 00:43:05.070 align:middle line:84% So I want to find h positive so that this thing here 00:43:05.070 --> 00:43:08.390 align:middle line:90% is bigger than 2. 00:43:08.390 --> 00:43:13.310 align:middle line:84% And all I need to do is, I need to show that I can find h 00:43:13.310 --> 00:43:28.390 align:middle line:84% so that this thing here-- so enough to show that I can find h 00:43:28.390 --> 00:43:36.940 align:middle line:84% such that x squared minus 2xh, that this thing here is bigger 00:43:36.940 --> 00:43:37.520 align:middle line:90% than 2. 00:43:37.520 --> 00:43:41.060 align:middle line:90% 00:43:41.060 --> 00:43:44.580 align:middle line:84% And so you see, now I just move over 00:43:44.580 --> 00:43:48.820 align:middle line:84% on the other side to say that this here is bigger than 2. 00:43:48.820 --> 00:43:51.700 align:middle line:84% It's the same as saying I can just move this 2 over 00:43:51.700 --> 00:43:54.540 align:middle line:90% on that side. 00:43:54.540 --> 00:43:57.180 align:middle line:84% And I can move this here over on the other side. 00:43:57.180 --> 00:44:03.860 align:middle line:84% So I just need to find h with this property. 00:44:03.860 --> 00:44:10.180 align:middle line:84% But that, of course, is equivalent to saying that h has 00:44:10.180 --> 00:44:17.060 align:middle line:84% to be less than x squared minus 2 over 2x. 00:44:17.060 --> 00:44:18.980 align:middle line:90% x is positive. 00:44:18.980 --> 00:44:21.500 align:middle line:84% This thing here is strictly positive. 00:44:21.500 --> 00:44:24.120 align:middle line:84% So I can certainly find a positive number. 00:44:24.120 --> 00:44:26.460 align:middle line:84% This here is something strictly positive. 00:44:26.460 --> 00:44:30.140 align:middle line:84% So I can certainly find some small h 00:44:30.140 --> 00:44:32.910 align:middle line:84% so that it's less than this strictly positive number. 00:44:32.910 --> 00:44:35.960 align:middle line:90% 00:44:35.960 --> 00:44:39.200 align:middle line:90% So this is step one. 00:44:39.200 --> 00:44:42.660 align:middle line:84% This is just showing that x squared was less than 00:44:42.660 --> 00:44:44.360 align:middle line:90% or equal to 2. 00:44:44.360 --> 00:44:52.320 align:middle line:84% Now I want to argue similarly to prove the other direction. 00:44:52.320 --> 00:45:15.900 align:middle line:90% 00:45:15.900 --> 00:45:18.800 align:middle line:84% In this case, there's not really any difference. 00:45:18.800 --> 00:45:23.600 align:middle line:84% You see in a minute that the one inequality is not 00:45:23.600 --> 00:45:25.460 align:middle line:90% any easier than the other. 00:45:25.460 --> 00:45:28.180 align:middle line:90% 00:45:28.180 --> 00:45:29.760 align:middle line:90% I have a colleague here. 00:45:29.760 --> 00:45:31.550 align:middle line:90% He's is retired now. 00:45:31.550 --> 00:45:34.430 align:middle line:84% And he collaborated with some other guy 00:45:34.430 --> 00:45:37.250 align:middle line:90% and made a lot of famous work. 00:45:37.250 --> 00:45:40.270 align:middle line:84% And the other guy maybe become even more famous. 00:45:40.270 --> 00:45:42.853 align:middle line:84% And he used to say that when you do mathematics-- 00:45:42.853 --> 00:45:45.270 align:middle line:84% so it doesn't really apply here because the two inequality 00:45:45.270 --> 00:45:46.590 align:middle line:90% is just as easy. 00:45:46.590 --> 00:45:50.470 align:middle line:84% But he used to say that when you do mathematics, 00:45:50.470 --> 00:45:52.230 align:middle line:90% there's three things you do. 00:45:52.230 --> 00:45:54.150 align:middle line:90% You locate a problem. 00:45:54.150 --> 00:45:58.110 align:middle line:84% You prove one of the two inequality, whichever 00:45:58.110 --> 00:45:59.590 align:middle line:90% one is easier. 00:45:59.590 --> 00:46:00.810 align:middle line:90% And you ask Raghu-- 00:46:00.810 --> 00:46:03.460 align:middle line:84% Raghu was his collaborator-- to prove the other inequality. 00:46:03.460 --> 00:46:05.778 align:middle line:90% [LAUGHTER] 00:46:05.778 --> 00:46:07.510 align:middle line:90% 00:46:07.510 --> 00:46:08.010 align:middle line:90% OK. 00:46:08.010 --> 00:46:15.430 align:middle line:90% 00:46:15.430 --> 00:46:22.950 align:middle line:84% So step two is that x squared is bigger or equal to 2. 00:46:22.950 --> 00:46:26.010 align:middle line:90% And we will again prove this. 00:46:26.010 --> 00:46:36.401 align:middle line:84% So again, we'll show this by contradiction. 00:46:36.401 --> 00:46:43.926 align:middle line:90% 00:46:43.926 --> 00:46:50.340 align:middle line:90% So I assume not. 00:46:50.340 --> 00:47:01.640 align:middle line:84% So assume that x squared here is strictly smaller than 2. 00:47:01.640 --> 00:47:04.460 align:middle line:90% 00:47:04.460 --> 00:47:06.900 align:middle line:90% Now what we want to prove is-- 00:47:06.900 --> 00:47:12.140 align:middle line:84% so then we want to get the contradiction. 00:47:12.140 --> 00:47:36.330 align:middle line:84% We want to get the contradiction by showing that x is not 00:47:36.330 --> 00:47:46.930 align:middle line:84% an upper bound, an upper bound for A. x 00:47:46.930 --> 00:47:49.850 align:middle line:84% was supposed to be the least upper bound, so in particular 00:47:49.850 --> 00:47:52.718 align:middle line:90% an upper bound. 00:47:52.718 --> 00:47:54.730 align:middle line:90% So how do we do that? 00:47:54.730 --> 00:47:59.650 align:middle line:84% Well, we're going to add just a little bit to x. 00:47:59.650 --> 00:48:02.380 align:middle line:84% So this is similar to how we argued before. 00:48:02.380 --> 00:48:14.570 align:middle line:90% 00:48:14.570 --> 00:48:19.930 align:middle line:90% So I'm going to add-- 00:48:19.930 --> 00:48:24.070 align:middle line:90% so h, again, is being positive. 00:48:24.070 --> 00:48:26.550 align:middle line:90% So h is positive. 00:48:26.550 --> 00:48:30.590 align:middle line:84% And I'm going to add just a little bit to x. 00:48:30.590 --> 00:48:38.030 align:middle line:84% And I'm going to see that if I choose h small, then 00:48:38.030 --> 00:48:46.890 align:middle line:84% I can find something in the set if I choose it so that x-- 00:48:46.890 --> 00:48:50.850 align:middle line:84% so I want to prove that if h is small enough, 00:48:50.850 --> 00:48:57.390 align:middle line:84% then x plus h is in A, contradicting 00:48:57.390 --> 00:49:00.710 align:middle line:90% that x was an upper bound. 00:49:00.710 --> 00:49:03.690 align:middle line:84% And so h is going to be chosen small. 00:49:03.690 --> 00:49:08.710 align:middle line:84% So in particular, it's going to be between 0 and 1. 00:49:08.710 --> 00:49:13.070 align:middle line:90% And I'm looking at x plus h. 00:49:13.070 --> 00:49:14.750 align:middle line:90% And I'm squaring it. 00:49:14.750 --> 00:49:19.815 align:middle line:84% So this thing here is x squared plus h squared plus 2hx. 00:49:19.815 --> 00:49:23.100 align:middle line:90% 00:49:23.100 --> 00:49:27.620 align:middle line:90% Now h is less than 1. 00:49:27.620 --> 00:49:34.100 align:middle line:84% So this means that h squared-- since h is positive 00:49:34.100 --> 00:49:40.780 align:middle line:84% and is less than 1, then you have that h squared 00:49:40.780 --> 00:49:46.420 align:middle line:90% is less than actual h. 00:49:46.420 --> 00:50:00.460 align:middle line:84% And so this thing here is less than x squared plus h plus 2hx 00:50:00.460 --> 00:50:02.350 align:middle line:90% And I want to show-- 00:50:02.350 --> 00:50:05.920 align:middle line:90% 00:50:05.920 --> 00:50:12.100 align:middle line:84% so I want to show that this x plus h, this is still in A. 00:50:12.100 --> 00:50:33.650 align:middle line:84% So I want to show that we can find such a h so that this here 00:50:33.650 --> 00:50:37.070 align:middle line:90% squared is still less than 2. 00:50:37.070 --> 00:50:39.650 align:middle line:90% 00:50:39.650 --> 00:50:43.410 align:middle line:84% So all I need to do is, because of this inequality, 00:50:43.410 --> 00:50:47.650 align:middle line:84% I just need to find an h so that this thing here is less than 2. 00:50:47.650 --> 00:51:06.550 align:middle line:84% So I want to find h such that this thing here is less than 2. 00:51:06.550 --> 00:51:22.670 align:middle line:90% 00:51:22.670 --> 00:51:27.090 align:middle line:84% Right now, again, we're just moving over on the other side. 00:51:27.090 --> 00:51:29.870 align:middle line:84% In this case, we're moving this x squared over here. 00:51:29.870 --> 00:51:35.250 align:middle line:84% So we have h plus 2hx is less than-- 00:51:35.250 --> 00:51:38.790 align:middle line:90% this is what we want to find-- 00:51:38.790 --> 00:51:48.110 align:middle line:84% want to find an h such that this thing here is less than 2 00:51:48.110 --> 00:51:50.310 align:middle line:90% minus a x squared. 00:51:50.310 --> 00:51:53.270 align:middle line:84% This thing here is now-- because x squared 00:51:53.270 --> 00:51:54.570 align:middle line:90% is strictly less than 2. 00:51:54.570 --> 00:51:57.070 align:middle line:90% This is positive. 00:51:57.070 --> 00:52:00.990 align:middle line:84% Here, you can just factor out an h. 00:52:00.990 --> 00:52:03.030 align:middle line:84% So if you factor out an h, then you 00:52:03.030 --> 00:52:09.670 align:middle line:84% have this thing here is less than 2 minus x squared. 00:52:09.670 --> 00:52:13.190 align:middle line:84% And this is just saying that you want to find an h so that this 00:52:13.190 --> 00:52:20.940 align:middle line:84% thing here is strictly less than 2 minus x squared over 1 plus 00:52:20.940 --> 00:52:22.320 align:middle line:90% 2x. 00:52:22.320 --> 00:52:24.460 align:middle line:90% x is positive. 00:52:24.460 --> 00:52:27.740 align:middle line:90% So this thing here is positive. 00:52:27.740 --> 00:52:29.960 align:middle line:84% So this and this here is also positive. 00:52:29.960 --> 00:52:31.520 align:middle line:84% So this here is a positive number. 00:52:31.520 --> 00:52:35.060 align:middle line:84% So you can definitely find some x with this property. 00:52:35.060 --> 00:52:36.840 align:middle line:90% And then you see that-- 00:52:36.840 --> 00:52:41.980 align:middle line:84% so this means that this x plus h would also be in the set a, 00:52:41.980 --> 00:52:44.440 align:middle line:84% contradicting that x was an upper bound. 00:52:44.440 --> 00:52:52.140 align:middle line:90% 00:52:52.140 --> 00:52:52.640 align:middle line:90% OK. 00:52:52.640 --> 00:52:54.900 align:middle line:84% So now, accordingly, we immediately 00:52:54.900 --> 00:53:07.200 align:middle line:84% get of this is that the rationals is not complete. 00:53:07.200 --> 00:53:10.560 align:middle line:84% So we get the following corollarys. 00:53:10.560 --> 00:53:16.450 align:middle line:90% 00:53:16.450 --> 00:53:18.300 align:middle line:84% The rational number is not complete. 00:53:18.300 --> 00:53:26.010 align:middle line:90% 00:53:26.010 --> 00:53:27.550 align:middle line:90% And this is just because-- 00:53:27.550 --> 00:53:31.350 align:middle line:84% so the proof of that is that we have already-- 00:53:31.350 --> 00:53:43.150 align:middle line:84% so if we're looking at a field that is complete and contains 00:53:43.150 --> 00:53:46.170 align:middle line:84% a rational, then we know that square root of 2 00:53:46.170 --> 00:53:47.970 align:middle line:90% is in that field. 00:53:47.970 --> 00:54:08.250 align:middle line:84% So we just proven that if f is a complete field containing 00:54:08.250 --> 00:54:23.630 align:middle line:84% the rationals, then square root of 2 is in that field. 00:54:23.630 --> 00:54:40.030 align:middle line:84% But we have also shown that square root of 2 00:54:40.030 --> 00:54:42.290 align:middle line:90% is not a rational number. 00:54:42.290 --> 00:54:46.930 align:middle line:90% 00:54:46.930 --> 00:54:49.510 align:middle line:84% So the rational in itself is not complete. 00:54:49.510 --> 00:54:54.710 align:middle line:90% 00:54:54.710 --> 00:54:56.420 align:middle line:90% OK. 00:54:56.420 --> 00:55:00.150 align:middle line:84% Now the next thing I want to do is 00:55:00.150 --> 00:55:03.670 align:middle line:84% prove what's called the Archimedean property, 00:55:03.670 --> 00:55:05.990 align:middle line:90% the Archimedean property. 00:55:05.990 --> 00:55:07.750 align:middle line:84% And so this is actually something 00:55:07.750 --> 00:55:15.280 align:middle line:84% that was sort of used here, sort of used it a little bit already. 00:55:15.280 --> 00:55:20.100 align:middle line:84% But let me just prove it rigorously. 00:55:20.100 --> 00:55:28.740 align:middle line:90% 00:55:28.740 --> 00:55:30.920 align:middle line:90% So the Archimedean property-- 00:55:30.920 --> 00:55:41.140 align:middle line:90% 00:55:41.140 --> 00:55:55.260 align:middle line:84% so this is the following, that for all real numbers, every time 00:55:55.260 --> 00:56:07.050 align:middle line:84% you take a real number, then there exists an integer-- 00:56:07.050 --> 00:56:11.450 align:middle line:90% 00:56:11.450 --> 00:56:25.370 align:middle line:84% actually, even a natural number n such that n here 00:56:25.370 --> 00:56:28.130 align:middle line:90% is bigger than x. 00:56:28.130 --> 00:56:28.630 align:middle line:90% OK. 00:56:28.630 --> 00:56:31.490 align:middle line:84% This seems like a kind of trivial fact. 00:56:31.490 --> 00:56:35.917 align:middle line:84% But let me try to actually prove it in a rigorous way. 00:56:35.917 --> 00:56:38.000 align:middle line:84% So this is what's called the Archimedean property. 00:56:38.000 --> 00:56:43.610 align:middle line:90% 00:56:43.610 --> 00:56:48.590 align:middle line:84% And so, again, we prove this by contradiction. 00:56:48.590 --> 00:56:56.770 align:middle line:90% 00:56:56.770 --> 00:57:09.960 align:middle line:84% So suppose not-- so if there existed an x so that you did not 00:57:09.960 --> 00:57:17.800 align:middle line:84% have this, well then-- so suppose not, then-- 00:57:17.800 --> 00:57:19.560 align:middle line:84% so the negation of that statement 00:57:19.560 --> 00:57:29.120 align:middle line:84% is that then there exists an x such that 00:57:29.120 --> 00:57:36.520 align:middle line:84% for all natural numbers, for all, 00:57:36.520 --> 00:57:40.520 align:middle line:84% you have that n here is less than or equal to x. 00:57:40.520 --> 00:57:43.880 align:middle line:90% 00:57:43.880 --> 00:57:45.720 align:middle line:90% Yeah, it's actually equal to x. 00:57:45.720 --> 00:57:50.320 align:middle line:84% Because this is the negation of that. 00:57:50.320 --> 00:58:02.350 align:middle line:84% But this means that x here is an upper bound. 00:58:02.350 --> 00:58:09.030 align:middle line:90% 00:58:09.030 --> 00:58:16.780 align:middle line:90% And so then you could take-- 00:58:16.780 --> 00:58:21.390 align:middle line:84% so you could replace by the least upper bound property. 00:58:21.390 --> 00:58:23.900 align:middle line:90% 00:58:23.900 --> 00:58:25.200 align:middle line:90% So it's upper bound. 00:58:25.200 --> 00:58:34.340 align:middle line:84% So in particular, the natural number 00:58:34.340 --> 00:58:45.340 align:middle line:84% is a bounded subset of the reals. 00:58:45.340 --> 00:58:50.340 align:middle line:84% And then you could take-- so then we let, 00:58:50.340 --> 00:59:10.030 align:middle line:84% let's say, y be the least upper bound of the natural numbers. 00:59:10.030 --> 00:59:12.690 align:middle line:84% If the natural numbers was really bounded, 00:59:12.690 --> 00:59:14.630 align:middle line:84% because it's a subset of the reals, 00:59:14.630 --> 00:59:17.510 align:middle line:84% we could now take the least upper bound. 00:59:17.510 --> 00:59:18.010 align:middle line:90% OK. 00:59:18.010 --> 00:59:21.370 align:middle line:84% So now what we have is that we have 00:59:21.370 --> 00:59:26.630 align:middle line:90% that for any natural number-- 00:59:26.630 --> 00:59:38.070 align:middle line:84% so for all, we have that n here is less than or equal to y. 00:59:38.070 --> 00:59:40.410 align:middle line:84% But whenever you take a natural number, 00:59:40.410 --> 00:59:44.250 align:middle line:84% then add 1 to that natural number-- 00:59:44.250 --> 00:59:48.110 align:middle line:90% so we would also have that-- 00:59:48.110 --> 00:59:52.730 align:middle line:90% 00:59:52.730 --> 00:59:57.770 align:middle line:84% so y is not just an upper bound for n, but it actually-- 00:59:57.770 --> 01:00:00.950 align:middle line:84% because n plus 1 is also not a natural number. 01:00:00.950 --> 01:00:05.680 align:middle line:84% So y here would have to be bigger or equal to n plus 1 01:00:05.680 --> 01:00:07.880 align:middle line:90% also. 01:00:07.880 --> 01:00:10.120 align:middle line:90% And this now holds for all n. 01:00:10.120 --> 01:00:14.420 align:middle line:84% So this means that-- just moving the 1 over on the other side-- 01:00:14.420 --> 01:00:22.240 align:middle line:84% so hence, n here is less than y minus 1. 01:00:22.240 --> 01:00:34.080 align:middle line:84% But this here say that-- so this means that y minus 1 01:00:34.080 --> 01:00:49.980 align:middle line:84% is also an upper bound and that this is a contradiction. 01:00:49.980 --> 01:00:55.910 align:middle line:84% So this contradicts that y was the least upper bound. 01:00:55.910 --> 01:01:04.500 align:middle line:90% 01:01:04.500 --> 01:01:06.150 align:middle line:84% And that's the Archimedean property. 01:01:06.150 --> 01:01:08.700 align:middle line:90% 01:01:08.700 --> 01:01:10.364 align:middle line:90% OK. 01:01:10.364 --> 01:01:17.100 align:middle line:84% Now I want to give one application of this. 01:01:17.100 --> 01:01:18.840 align:middle line:90% And so this is the following. 01:01:18.840 --> 01:01:41.700 align:middle line:90% 01:01:41.700 --> 01:01:47.660 align:middle line:84% So I now want to prove the following corollary 01:01:47.660 --> 01:01:48.790 align:middle line:90% of the Archimedean problem. 01:01:48.790 --> 01:01:53.540 align:middle line:90% 01:01:53.540 --> 01:01:56.620 align:middle line:84% And the corollary is that whenever 01:01:56.620 --> 01:02:01.730 align:middle line:84% you have two real numbers, x and y-- 01:02:01.730 --> 01:02:04.090 align:middle line:90% so these are real numbers. 01:02:04.090 --> 01:02:07.010 align:middle line:90% And x here is less than y. 01:02:07.010 --> 01:02:26.130 align:middle line:84% Then the claim is that then there exists a rational number 01:02:26.130 --> 01:02:34.130 align:middle line:84% m/n with the property that is between these two 01:02:34.130 --> 01:02:42.770 align:middle line:84% so that x is less than m/n and less than 1. 01:02:42.770 --> 01:02:45.900 align:middle line:84% that's what I want to prove using the Archimedean property. 01:02:45.900 --> 01:02:49.690 align:middle line:90% 01:02:49.690 --> 01:03:04.480 align:middle line:84% And so to prove that, I look at beta. 01:03:04.480 --> 01:03:07.980 align:middle line:90% Say beta is y minus x. 01:03:07.980 --> 01:03:11.240 align:middle line:90% 01:03:11.240 --> 01:03:13.400 align:middle line:90% And I'm looking at-- 01:03:13.400 --> 01:03:14.720 align:middle line:90% so y minus x. 01:03:14.720 --> 01:03:19.640 align:middle line:84% Because y is bigger than x, this here is positive. 01:03:19.640 --> 01:03:23.120 align:middle line:84% And I can now look at 1 over beta. 01:03:23.120 --> 01:03:28.600 align:middle line:84% So 1 over beta is some positive real number. 01:03:28.600 --> 01:03:31.940 align:middle line:84% And now the Archimedean property say that-- 01:03:31.940 --> 01:03:50.520 align:middle line:84% so by the Archimedean property, there 01:03:50.520 --> 01:04:05.350 align:middle line:84% exists a n, a natural number such that n is 01:04:05.350 --> 01:04:08.290 align:middle line:90% bigger than 1 over beta. 01:04:08.290 --> 01:04:10.990 align:middle line:90% 01:04:10.990 --> 01:04:12.710 align:middle line:90% That's the Archimedean property. 01:04:12.710 --> 01:04:15.390 align:middle line:90% It gives us this n. 01:04:15.390 --> 01:04:18.510 align:middle line:84% But now, of course, if that's the case, 01:04:18.510 --> 01:04:23.110 align:middle line:84% this implies you can just take 1 over it on both sides, 01:04:23.110 --> 01:04:25.470 align:middle line:84% and that reverses the inequality. 01:04:25.470 --> 01:04:31.810 align:middle line:84% This implies that 1/n is smaller than beta. 01:04:31.810 --> 01:04:32.730 align:middle line:90% So we get that. 01:04:32.730 --> 01:04:35.950 align:middle line:90% 01:04:35.950 --> 01:04:41.300 align:middle line:90% And so now we look at-- 01:04:41.300 --> 01:04:49.070 align:middle line:90% 01:04:49.070 --> 01:04:54.430 align:middle line:84% so now like define the step size to some extent. 01:04:54.430 --> 01:05:00.570 align:middle line:84% So we have here x, and we have here y. 01:05:00.570 --> 01:05:05.790 align:middle line:84% And beta is really this distance here. 01:05:05.790 --> 01:05:10.970 align:middle line:84% This here is beta, the distance between the two. 01:05:10.970 --> 01:05:13.970 align:middle line:90% And n here is-- 01:05:13.970 --> 01:05:17.970 align:middle line:84% we've now seen that there's an n so that when you're looking 01:05:17.970 --> 01:05:21.570 align:middle line:90% at 1/n, then-- 01:05:21.570 --> 01:05:24.950 align:middle line:84% so 1/n here is smaller than beta. 01:05:24.950 --> 01:05:29.930 align:middle line:84% It's something positive and smaller than beta. 01:05:29.930 --> 01:05:33.590 align:middle line:90% So now you're looking at the m. 01:05:33.590 --> 01:05:36.370 align:middle line:90% 01:05:36.370 --> 01:05:52.200 align:middle line:84% So now look at the integers so that m minus 1 01:05:52.200 --> 01:05:59.680 align:middle line:90% is less than x times m. 01:05:59.680 --> 01:06:03.660 align:middle line:84% So suppose you're looking at this thing here. 01:06:03.660 --> 01:06:07.620 align:middle line:84% If you take x times n, this is an integer. 01:06:07.620 --> 01:06:11.760 align:middle line:84% So there's definition m with this property. 01:06:11.760 --> 01:06:14.220 align:middle line:90% And you can find the biggest-- 01:06:14.220 --> 01:06:19.160 align:middle line:90% so look at m with this property. 01:06:19.160 --> 01:06:31.240 align:middle line:84% And let's denote n and choose the largest one, the largest 01:06:31.240 --> 01:06:32.060 align:middle line:90% such m. 01:06:32.060 --> 01:06:38.960 align:middle line:90% 01:06:38.960 --> 01:06:45.980 align:middle line:84% So if you choose the largest such m, then you see that-- 01:06:45.980 --> 01:06:48.630 align:middle line:84% so this inequality here, this will just 01:06:48.630 --> 01:06:58.690 align:middle line:84% tell you that m minus 1 over n, that this is smaller than x. 01:06:58.690 --> 01:07:02.830 align:middle line:90% 01:07:02.830 --> 01:07:05.950 align:middle line:84% But you would also have that because it 01:07:05.950 --> 01:07:07.670 align:middle line:90% was the largest such m-- 01:07:07.670 --> 01:07:11.150 align:middle line:90% 01:07:11.150 --> 01:07:26.510 align:middle line:84% since m was the largest such m, then 01:07:26.510 --> 01:07:32.590 align:middle line:84% you would have that if you had m itself in the denominator, 01:07:32.590 --> 01:07:35.210 align:middle line:84% then it wouldn't be smaller than x. 01:07:35.210 --> 01:07:37.830 align:middle line:84% It would actually be larger or equal to x. 01:07:37.830 --> 01:07:40.710 align:middle line:90% 01:07:40.710 --> 01:07:44.790 align:middle line:84% And I just want to now prove that this-- so this 01:07:44.790 --> 01:07:46.590 align:middle line:90% here is bigger than x. 01:07:46.590 --> 01:07:48.970 align:middle line:84% That's part of what I needed to prove. 01:07:48.970 --> 01:07:51.870 align:middle line:84% I also wanted to show that it's smaller than y. 01:07:51.870 --> 01:07:58.170 align:middle line:90% 01:07:58.170 --> 01:08:01.530 align:middle line:90% But if you now add-- 01:08:01.530 --> 01:08:02.910 align:middle line:90% but now the step size. 01:08:02.910 --> 01:08:05.330 align:middle line:90% So when you have here-- 01:08:05.330 --> 01:08:07.330 align:middle line:90% here, you have x. 01:08:07.330 --> 01:08:08.386 align:middle line:90% Here, you have y. 01:08:08.386 --> 01:08:11.130 align:middle line:90% 01:08:11.130 --> 01:08:14.390 align:middle line:84% 1/n was smaller than this distance. 01:08:14.390 --> 01:08:16.649 align:middle line:84% So if you are at some point here-- 01:08:16.649 --> 01:08:21.330 align:middle line:84% this is where you are when you are at n minus 1/n. 01:08:21.330 --> 01:08:24.270 align:middle line:84% Then you can't go further to the right. 01:08:24.270 --> 01:08:28.750 align:middle line:90% You go 1/n to the right. 01:08:28.750 --> 01:08:32.729 align:middle line:84% But that can't take you all the way past y. 01:08:32.729 --> 01:08:37.609 align:middle line:84% So that's why, if you're looking at this number here, 01:08:37.609 --> 01:08:42.450 align:middle line:90% this m must be smaller than y. 01:08:42.450 --> 01:08:45.800 align:middle line:84% And if I had just chosen m so that this here 01:08:45.800 --> 01:08:48.000 align:middle line:84% was smaller or equal to that, then I 01:08:48.000 --> 01:08:50.800 align:middle line:90% would get this [INAUDIBLE]. 01:08:50.800 --> 01:08:51.652 align:middle line:90% OK. 01:08:51.652 --> 01:08:56.899 align:middle line:84% So that's a corollary of the Archimedean property. 01:08:56.899 --> 01:08:59.880 align:middle line:84% I think there would be something along these lines 01:08:59.880 --> 01:09:01.200 align:middle line:90% on the homework. 01:09:01.200 --> 01:09:05.080 align:middle line:84% And again, I'm just waiting for the office hour. 01:09:05.080 --> 01:09:07.080 align:middle line:84% I know that the office hour, there will at least 01:09:07.080 --> 01:09:11.000 align:middle line:90% be four office hours next week. 01:09:11.000 --> 01:09:15.399 align:middle line:84% On Mondays and Wednesday, there would be office hours from 5:00 01:09:15.399 --> 01:09:20.319 align:middle line:90% to 6:00 on Monday and Wednesday. 01:09:20.319 --> 01:09:25.340 align:middle line:84% And the other days, it would be 1:15 to 2:15. 01:09:25.340 --> 01:09:30.560 align:middle line:84% But you can check-- the time is actually already posted. 01:09:30.560 --> 01:09:33.000 align:middle line:84% The room number, I hope to have that today. 01:09:33.000 --> 01:09:35.680 align:middle line:84% Once I have that, then I will also 01:09:35.680 --> 01:09:39.319 align:middle line:84% assign the homework, probably today, or maybe more likely, 01:09:39.319 --> 01:09:40.859 align:middle line:90% tomorrow, the homework. 01:09:40.859 --> 01:09:46.550 align:middle line:84% But it won't be due next week, just the following week. 01:09:46.550 --> 01:09:47.729 align:middle line:90% Any questions? 01:09:47.729 --> 01:09:50.310 align:middle line:90% 01:09:50.310 --> 01:09:52.109 align:middle line:84% And, of course, I very much encourage 01:09:52.109 --> 01:09:56.070 align:middle line:90% you to go to the office hours. 01:09:56.070 --> 01:09:59.490 align:middle line:90% And yeah. 01:09:59.490 --> 01:09:59.990 align:middle line:90% Yeah? 01:09:59.990 --> 01:10:01.350 align:middle line:84% STUDENT: Are you upper bounding-- 01:10:01.350 --> 01:10:01.890 align:middle line:90% PROFESSOR: Sorry. 01:10:01.890 --> 01:10:02.682 align:middle line:90% STUDENT: Oh, sorry. 01:10:02.682 --> 01:10:03.310 align:middle line:90% PROFESSOR: Oh. 01:10:03.310 --> 01:10:06.950 align:middle line:84% STUDENT: So we defined plus in the solution. 01:10:06.950 --> 01:10:11.110 align:middle line:84% So how do we define minus 1, for example? 01:10:11.110 --> 01:10:12.990 align:middle line:84% PROFESSOR: So remember that the minus 1 01:10:12.990 --> 01:10:18.750 align:middle line:84% was this element that you had when you had-- 01:10:18.750 --> 01:10:22.310 align:middle line:84% remember that a field had the two operation, plus 01:10:22.310 --> 01:10:23.650 align:middle line:90% and multiplication. 01:10:23.650 --> 01:10:27.610 align:middle line:84% And for each of these that had 5 property, 01:10:27.610 --> 01:10:32.670 align:middle line:84% they were almost the same property. 01:10:32.670 --> 01:10:35.390 align:middle line:84% There was one exception, that 0 didn't have 01:10:35.390 --> 01:10:37.830 align:middle line:90% a inverse for multiplication. 01:10:37.830 --> 01:10:40.510 align:middle line:90% But minus 1 is the-- 01:10:40.510 --> 01:10:43.570 align:middle line:84% so you have 1, but 1 is the neutral element 01:10:43.570 --> 01:10:45.410 align:middle line:90% for multiplication. 01:10:45.410 --> 01:10:45.910 align:middle line:90% Why? 01:10:45.910 --> 01:10:48.290 align:middle line:84% That was how 1 was defined, right? 01:10:48.290 --> 01:10:52.290 align:middle line:84% And we proved that there was a unique 0. 01:10:52.290 --> 01:10:56.450 align:middle line:84% That same argument would prove that there's a unique 1. 01:10:56.450 --> 01:10:58.890 align:middle line:90% So there's just one 1. 01:10:58.890 --> 01:11:01.410 align:middle line:90% So it's a well-defined notion. 01:11:01.410 --> 01:11:04.730 align:middle line:84% And then you can look at the inverse element of that 01:11:04.730 --> 01:11:06.390 align:middle line:90% with respect to addition. 01:11:06.390 --> 01:11:09.710 align:middle line:90% That's minus 1. 01:11:09.710 --> 01:11:10.210 align:middle line:90% Yeah. 01:11:10.210 --> 01:11:12.568 align:middle line:90% 01:11:12.568 --> 01:11:14.610 align:middle line:84% STUDENT: Why did we define an upper bound and not 01:11:14.610 --> 01:11:15.570 align:middle line:90% a lower bound? 01:11:15.570 --> 01:11:17.670 align:middle line:84% PROFESSOR: Yeah, and that's a good question. 01:11:17.670 --> 01:11:20.510 align:middle line:84% And I will come back to that next time. 01:11:20.510 --> 01:11:25.210 align:middle line:84% And it would seem more logical to define both upper 01:11:25.210 --> 01:11:26.390 align:middle line:90% and lower bound. 01:11:26.390 --> 01:11:30.290 align:middle line:84% But you can look at the complement. 01:11:30.290 --> 01:11:33.130 align:middle line:84% And so if you take a set and you're 01:11:33.130 --> 01:11:36.330 align:middle line:84% looking at the complement, then the upper bound for that 01:11:36.330 --> 01:11:38.080 align:middle line:90% is the same as the lower bound. 01:11:38.080 --> 01:11:40.420 align:middle line:90% So that's a kind of sneaky way-- 01:11:40.420 --> 01:11:44.270 align:middle line:90% that's the reason why we did it. 01:11:44.270 --> 01:11:46.500 align:middle line:84% I might have done it the other way around, 01:11:46.500 --> 01:11:49.180 align:middle line:84% but then it was a little bit overkill. 01:11:49.180 --> 01:11:52.780 align:middle line:84% But I agree, it seemed more logical to define both things. 01:11:52.780 --> 01:11:54.700 align:middle line:84% But it actually is contained in this. 01:11:54.700 --> 01:11:57.280 align:middle line:90% 01:11:57.280 --> 01:11:59.640 align:middle line:90% Any other questions? 01:11:59.640 --> 01:12:00.180 align:middle line:90% Yeah? 01:12:00.180 --> 01:12:02.555 align:middle line:84% STUDENT: I was a little confused because I was thinking-- 01:12:02.555 --> 01:12:05.040 align:middle line:84% I wonder if you could please speak about the completeness 01:12:05.040 --> 01:12:05.940 align:middle line:90% of [INAUDIBLE]? 01:12:05.940 --> 01:12:16.520 align:middle line:84% Like you make the statement [INAUDIBLE] 01:12:16.520 --> 01:12:21.000 align:middle line:84% say a bounded subset of rational numbers 01:12:21.000 --> 01:12:22.610 align:middle line:90% would not have this upper bound. 01:12:22.610 --> 01:12:25.400 align:middle line:90% 01:12:25.400 --> 01:12:30.840 align:middle line:84% PROFESSOR: So again, we saw that square root of 2 01:12:30.840 --> 01:12:32.560 align:middle line:90% is not a rational number. 01:12:32.560 --> 01:12:36.910 align:middle line:84% And if we're looking at all rational numbers, 01:12:36.910 --> 01:12:40.190 align:middle line:84% all rational numbers so that the square is 01:12:40.190 --> 01:12:45.950 align:middle line:84% less than or equal to 2, that set, 01:12:45.950 --> 01:12:48.530 align:middle line:84% it has an upper bound in the rational number, 01:12:48.530 --> 01:12:51.710 align:middle line:84% but it doesn't have a least upper bound. 01:12:51.710 --> 01:12:53.990 align:middle line:84% And that's because we first prove 01:12:53.990 --> 01:12:59.310 align:middle line:84% that there wasn't a rational number so that it multiplied 01:12:59.310 --> 01:13:01.350 align:middle line:90% itself was equal to 2. 01:13:01.350 --> 01:13:02.690 align:middle line:90% We proved that-- 01:13:02.690 --> 01:13:05.790 align:middle line:84% STUDENT: But does the least [INAUDIBLE] also have to belong 01:13:05.790 --> 01:13:06.410 align:middle line:90% to the-- 01:13:06.410 --> 01:13:07.130 align:middle line:90% PROFESSOR: Yeah, that's right. 01:13:07.130 --> 01:13:07.530 align:middle line:90% That's right. 01:13:07.530 --> 01:13:07.830 align:middle line:90% That's right. 01:13:07.830 --> 01:13:08.030 align:middle line:90% STUDENT: --[INAUDIBLE]? 01:13:08.030 --> 01:13:08.530 align:middle line:90% PROFESSOR: That's right. 01:13:08.530 --> 01:13:08.970 align:middle line:90% That's right. 01:13:08.970 --> 01:13:09.512 align:middle line:90% That's right. 01:13:09.512 --> 01:13:10.210 align:middle line:90% That's right. 01:13:10.210 --> 01:13:12.530 align:middle line:90% So it's this. 01:13:12.530 --> 01:13:16.350 align:middle line:84% A field is complete means that whenever 01:13:16.350 --> 01:13:20.390 align:middle line:84% you take a subset of that field that is bounded, 01:13:20.390 --> 01:13:24.110 align:middle line:84% then that subset has to have a least upper bound. 01:13:24.110 --> 01:13:27.430 align:middle line:84% But any bound, including the least upper bound, 01:13:27.430 --> 01:13:28.980 align:middle line:84% that always have to be in the fields. 01:13:28.980 --> 01:13:31.590 align:middle line:90% 01:13:31.590 --> 01:13:32.210 align:middle line:90% OK. 01:13:32.210 --> 01:13:33.170 align:middle line:90% Any other questions? 01:13:33.170 --> 01:13:34.180 align:middle line:90% Yeah? 01:13:34.180 --> 01:13:36.240 align:middle line:84% STUDENT: Well, when we tried to define it, 01:13:36.240 --> 01:13:42.300 align:middle line:84% and [INAUDIBLE] did we also prove the result 2 minus x 01:13:42.300 --> 01:13:46.900 align:middle line:84% squared over 1 plus 2x is smaller than 1 because you have 01:13:46.900 --> 01:13:51.400 align:middle line:84% assumed for [INAUDIBLE] smaller than 1 so that we can do the-- 01:13:51.400 --> 01:13:56.980 align:middle line:90% 01:13:56.980 --> 01:13:59.452 align:middle line:90% PROFESSOR: So you're saying-- 01:13:59.452 --> 01:14:01.160 align:middle line:84% so I just want to make sure I understand. 01:14:01.160 --> 01:14:05.900 align:middle line:84% So in the step two, we added something. 01:14:05.900 --> 01:14:09.100 align:middle line:84% And we wanted to prove that you could add something, 01:14:09.100 --> 01:14:11.240 align:middle line:90% and this is still less than 2. 01:14:11.240 --> 01:14:13.924 align:middle line:90% 01:14:13.924 --> 01:14:20.660 align:middle line:84% And so I was looking for h between 0 and 1. 01:14:20.660 --> 01:14:24.620 align:middle line:84% And so then I got some equation somewhere 01:14:24.620 --> 01:14:26.460 align:middle line:90% that is probably erased now. 01:14:26.460 --> 01:14:28.620 align:middle line:90% But it just require that-- 01:14:28.620 --> 01:14:31.260 align:middle line:90% 01:14:31.260 --> 01:14:34.873 align:middle line:84% h had to be positive, but it was less than something. 01:14:34.873 --> 01:14:35.790 align:middle line:90% And that was positive. 01:14:35.790 --> 01:14:37.332 align:middle line:84% STUDENT: Yeah, but that's something-- 01:14:37.332 --> 01:14:40.360 align:middle line:84% should the left something smaller than 1? 01:14:40.360 --> 01:14:41.660 align:middle line:90% PROFESSOR: Right. 01:14:41.660 --> 01:14:42.160 align:middle line:90% OK. 01:14:42.160 --> 01:14:44.820 align:middle line:84% So I don't remember what I got here. 01:14:44.820 --> 01:14:48.100 align:middle line:90% So yeah, you're right. 01:14:48.100 --> 01:14:50.020 align:middle line:90% I guess I can reconstruct it. 01:14:50.020 --> 01:14:51.200 align:middle line:90% STUDENT: Over 1 plus 2x. 01:14:51.200 --> 01:14:51.867 align:middle line:90% PROFESSOR: Yeah. 01:14:51.867 --> 01:14:54.960 align:middle line:90% 01:14:54.960 --> 01:14:59.880 align:middle line:90% So what you are saying is that-- 01:14:59.880 --> 01:15:02.080 align:middle line:84% actually, so this thing here is positive. 01:15:02.080 --> 01:15:05.920 align:middle line:84% And so you want to choose h so that it's smaller 01:15:05.920 --> 01:15:09.400 align:middle line:90% than this but also this here. 01:15:09.400 --> 01:15:11.000 align:middle line:90% So you don't. 01:15:11.000 --> 01:15:17.640 align:middle line:84% But if this is 500, well, I'm not 01:15:17.640 --> 01:15:20.900 align:middle line:84% going to choose 500 because it was supposed to be less than 1. 01:15:20.900 --> 01:15:23.640 align:middle line:84% But I just choose it so that it's less than 1 01:15:23.640 --> 01:15:25.780 align:middle line:84% and also less than this positive number. 01:15:25.780 --> 01:15:27.290 align:middle line:90% Yeah. 01:15:27.290 --> 01:15:31.000 align:middle line:90%