WEBVTT 00:00:00.040 --> 00:00:02.460 The following content is provided under a Creative 00:00:02.460 --> 00:00:03.870 Commons license. 00:00:03.870 --> 00:00:06.320 Your support will help MIT OpenCourseWare 00:00:06.320 --> 00:00:10.560 continue to offer high quality educational resources for free. 00:00:10.560 --> 00:00:13.300 To make a donation or view additional materials 00:00:13.300 --> 00:00:17.210 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.210 --> 00:00:17.862 at ocw.mit.edu. 00:00:33.270 --> 00:00:36.960 PROFESSOR: Hi, welcome to Calculus Part 2, 00:00:36.960 --> 00:00:39.450 where our theme for the entire course 00:00:39.450 --> 00:00:42.760 will essentially be functions of several variables. 00:00:42.760 --> 00:00:46.950 But a more underlying theme, a theme which will not only 00:00:46.950 --> 00:00:49.620 permeate this course, but virtually 00:00:49.620 --> 00:00:52.920 every course in the mathematics curriculum, and perhaps 00:00:52.920 --> 00:00:55.550 other curricula as well, is the idea 00:00:55.550 --> 00:00:58.920 of what we mean by a mathematical structure. 00:00:58.920 --> 00:01:03.350 And if this sounds a little bit ominous and frightening, 00:01:03.350 --> 00:01:05.910 the idea of a mathematical structure 00:01:05.910 --> 00:01:10.030 can be compared very nicely in terms of a game, 00:01:10.030 --> 00:01:12.180 and for this reason, I have chosen 00:01:12.180 --> 00:01:17.960 to entitle our first lesson "The 'Game' of Mathematics". 00:01:17.960 --> 00:01:19.880 "The 'Game' of Mathematics." 00:01:19.880 --> 00:01:23.480 And let's emphasize the word "game" here. 00:01:23.480 --> 00:01:26.590 We do not mean game in a trivial sense, 00:01:26.590 --> 00:01:30.260 where in elementary school, the first row races the second row 00:01:30.260 --> 00:01:32.960 to see who finishes the addition problem first. 00:01:32.960 --> 00:01:36.830 The idea that we want to talk about is what is a game. 00:01:36.830 --> 00:01:38.850 And I remember an old riddle when 00:01:38.850 --> 00:01:41.810 I was in about the fourth or fifth grade, when somebody said 00:01:41.810 --> 00:01:46.000 to me: "what is it that looks like a box, smells like cheese, 00:01:46.000 --> 00:01:47.110 and flies?" 00:01:47.110 --> 00:01:49.970 And the answer was a flying cheese box. 00:01:49.970 --> 00:01:51.940 And the interesting point is that this 00:01:51.940 --> 00:01:55.970 is how, in the scientific world, we often make up definitions. 00:01:55.970 --> 00:01:59.590 Namely, to define a game, we try to think 00:01:59.590 --> 00:02:04.020 of every single ingredient that is common to every game, 00:02:04.020 --> 00:02:06.240 and then we roll all of these ingredients 00:02:06.240 --> 00:02:08.550 into one long definition, and that 00:02:08.550 --> 00:02:10.949 becomes our final definition. 00:02:10.949 --> 00:02:16.060 And with that in mind, let me take the following tack. 00:02:16.060 --> 00:02:19.890 And again, let me point out that I am going through this rather 00:02:19.890 --> 00:02:23.740 hurriedly because the main aim of the lesson 00:02:23.740 --> 00:02:26.520 is to give an overview, with the idea 00:02:26.520 --> 00:02:31.260 that the supplementary notes and the exercises in the unit 00:02:31.260 --> 00:02:34.570 will give you the computational drill that you need. 00:02:34.570 --> 00:02:38.930 But for a first approximation, let us say that, in any game, 00:02:38.930 --> 00:02:43.950 you must have definitions, so to know what terminology you have. 00:02:43.950 --> 00:02:46.650 Then you must have rules of the game, 00:02:46.650 --> 00:02:48.630 and notice that the rules of the game 00:02:48.630 --> 00:02:51.800 are relationships between the terms. 00:02:51.800 --> 00:02:54.460 Oh, as a trivial example, in playing cards, 00:02:54.460 --> 00:02:56.860 there are many different card games 00:02:56.860 --> 00:02:59.850 that can be played with the same deck of cards. 00:02:59.850 --> 00:03:04.100 What makes the game different are not 00:03:04.100 --> 00:03:07.520 the definitions involved, but the rules of the game. 00:03:07.520 --> 00:03:10.010 And finally, there are objectives. 00:03:10.010 --> 00:03:12.650 Well, obviously, the objective of any game is to win. 00:03:12.650 --> 00:03:15.300 What we really mean by an objective 00:03:15.300 --> 00:03:19.290 is the art of carrying out a winning situation 00:03:19.290 --> 00:03:24.550 by successfully employing the definitions and the rules. 00:03:24.550 --> 00:03:28.220 And the way we do that is usually called strategy. 00:03:28.220 --> 00:03:31.450 Strategy is the art of using the definitions and rules 00:03:31.450 --> 00:03:35.120 to carry out the objective in an inescapable manner. 00:03:35.120 --> 00:03:37.620 And the reason I like this particular little setup 00:03:37.620 --> 00:03:39.560 is it gives me a way to show you, 00:03:39.560 --> 00:03:44.790 in juxtaposition, the role of rote verses reason, logic 00:03:44.790 --> 00:03:48.180 verses memory, in any mathematical situation 00:03:48.180 --> 00:03:49.540 or real-life situation. 00:03:49.540 --> 00:03:55.230 Namely, the strategy part of a game is logic, 00:03:55.230 --> 00:03:58.532 and things like the definitions and the rules 00:03:58.532 --> 00:04:00.560 are things that we memorize. 00:04:03.160 --> 00:04:05.820 OK, now, the thing that we're going 00:04:05.820 --> 00:04:09.770 to do, in our particular course, is always come back 00:04:09.770 --> 00:04:11.070 to this particular structure. 00:04:11.070 --> 00:04:13.700 In other words, our definition of a game 00:04:13.700 --> 00:04:17.790 is any system which consists of definitions, rules, 00:04:17.790 --> 00:04:20.940 and objectives, where the objective is carried out 00:04:20.940 --> 00:04:24.830 as an inescapable consequence of the definitions and the rules 00:04:24.830 --> 00:04:27.540 by means of strategy. 00:04:27.540 --> 00:04:30.590 And by the way, don't take this lightly. 00:04:30.590 --> 00:04:32.950 This is a very serious topic. 00:04:32.950 --> 00:04:35.420 Later in the course, the computation 00:04:35.420 --> 00:04:37.660 will become sufficiently difficult 00:04:37.660 --> 00:04:41.030 that we may lose sight of the forest because of the trees. 00:04:41.030 --> 00:04:44.080 Right now, what I want to do is emphasize this to you 00:04:44.080 --> 00:04:47.060 in terms of topics that you're already familiar with, 00:04:47.060 --> 00:04:49.220 so that you can see what the overall structure 00:04:49.220 --> 00:04:51.260 of mathematics is, so that you will not 00:04:51.260 --> 00:04:53.490 be preoccupied with this when we're learning 00:04:53.490 --> 00:04:55.290 more computational things. 00:04:55.290 --> 00:04:58.140 Let me just take a look at, well, 00:04:58.140 --> 00:05:00.980 a relatively trivial example. 00:05:00.980 --> 00:05:02.930 I call this a new look at counting. 00:05:02.930 --> 00:05:05.130 We all know how to count. 00:05:05.130 --> 00:05:07.390 We start with a number called 1, and we 00:05:07.390 --> 00:05:12.400 have various definitions, 1 plus 1 is 2, 2 plus 1 is 3, 00:05:12.400 --> 00:05:18.550 3 plus 1 is 4, 4 plus 1 is 5, 5 plus 1 is 6, 6 plus 1 00:05:18.550 --> 00:05:20.090 is 7, et cetera. 00:05:20.090 --> 00:05:22.480 OK, so far, so good. 00:05:22.480 --> 00:05:26.940 Now, we ask the question: "how much is 4 plus 3?" 00:05:26.940 --> 00:05:31.130 Now, obviously, we all know that 4 plus 3 is 7. 00:05:31.130 --> 00:05:34.140 We're not saying what are we looking at this problem for. 00:05:34.140 --> 00:05:38.030 What we're trying to show now is a very important aspect 00:05:38.030 --> 00:05:39.620 of the game of mathematics. 00:05:39.620 --> 00:05:42.570 You see, notice that in our list of definitions, 00:05:42.570 --> 00:05:46.340 no place do we have the sum 4 plus 3 defined. 00:05:46.340 --> 00:05:48.640 What we are interested in, now, is not 00:05:48.640 --> 00:05:52.160 so much the truthful statement that 4 plus 3 is 7, 00:05:52.160 --> 00:05:55.370 but whether the result 4 plus 3 equals 7 00:05:55.370 --> 00:06:00.260 follows inescapably from the definitions that we've listed. 00:06:00.260 --> 00:06:03.510 Now, you see, the point is all we've listed are definitions. 00:06:03.510 --> 00:06:07.220 We haven't told any particular rules of the game yet. 00:06:07.220 --> 00:06:10.620 Well, let's make up some rules as we go along, and as I say, 00:06:10.620 --> 00:06:13.110 we'll discuss these in more details in our notes 00:06:13.110 --> 00:06:14.960 and in the exercises. 00:06:14.960 --> 00:06:17.020 We essentially do something like this. 00:06:17.020 --> 00:06:22.670 We write down 4 plus 3, then we say, OK, we 00:06:22.670 --> 00:06:24.940 do know how to add by ones; that's 00:06:24.940 --> 00:06:26.400 what our definition says. 00:06:26.400 --> 00:06:29.090 So we rewrite 3 as 2 plus 1. 00:06:29.090 --> 00:06:32.990 In other words, we substitute 2 plus 1, which is equal to 3 00:06:32.990 --> 00:06:34.560 by definition. 00:06:34.560 --> 00:06:39.860 Then we say, OK, 2 plus 1 is the same as 1 plus 2. 00:06:39.860 --> 00:06:42.100 Now, by the way, notice the tacit assumption 00:06:42.100 --> 00:06:42.950 that we're making. 00:06:42.950 --> 00:06:45.970 We're assuming that the order in which you add two numbers 00:06:45.970 --> 00:06:47.330 makes no difference. 00:06:47.330 --> 00:06:50.430 Obviously, this is not a rule in every game of life. 00:06:50.430 --> 00:06:53.400 In most things in life, order does make a difference. 00:06:53.400 --> 00:06:55.360 Consider, for example, the statements 00:06:55.360 --> 00:06:59.065 first I undress, and then I take a shower; or first I shower, 00:06:59.065 --> 00:07:00.430 and then I undress. 00:07:00.430 --> 00:07:04.000 Without meaning to pass judgment as to which is proper, 00:07:04.000 --> 00:07:06.899 at least notice there is a difference between the two. 00:07:06.899 --> 00:07:08.690 What we're saying is that, somehow or other 00:07:08.690 --> 00:07:10.690 in the game of arithmetic, we assume 00:07:10.690 --> 00:07:13.820 that addition has the property that the order in which you add 00:07:13.820 --> 00:07:15.060 makes no difference. 00:07:15.060 --> 00:07:18.630 So we say, OK, let's accept that as a rule of the game. 00:07:18.630 --> 00:07:21.200 If we accept that as a rule the game, 2 plus 1 00:07:21.200 --> 00:07:24.880 could then be substituted for by 1 plus 2. 00:07:24.880 --> 00:07:26.900 We make the additional assumption 00:07:26.900 --> 00:07:28.780 that, when you add three numbers, 00:07:28.780 --> 00:07:31.590 the answer does not depend on voice inflection. 00:07:31.590 --> 00:07:34.610 In other words that 4 plus 1 plus 2 00:07:34.610 --> 00:07:38.400 is equal to 4 plus 1 plus 2. 00:07:38.400 --> 00:07:40.450 And the strategy behind doing that 00:07:40.450 --> 00:07:44.070 is that we know that another name for 4 plus 1 is 5. 00:07:44.070 --> 00:07:47.880 In other words, we now arrive at the fact that 4 plus 3 00:07:47.880 --> 00:07:50.240 is equal to 5 plus 2. 00:07:50.240 --> 00:07:54.210 We now rewrite 2 as 1 plus 1, and we now 00:07:54.210 --> 00:07:59.340 have the 5 plus 2 is the same as 5 plus 1 plus 1. 00:07:59.340 --> 00:08:02.110 We now again use the fact that voice inflection 00:08:02.110 --> 00:08:05.230 makes no difference, and we rewrite this as 5 00:08:05.230 --> 00:08:08.360 plus 1 plus 1. 00:08:08.360 --> 00:08:13.230 5 plus 1, we know by definition, is 6, and 6 plus 1, 00:08:13.230 --> 00:08:16.190 we know by definition, is seven. 00:08:16.190 --> 00:08:19.920 In other words, subject to the rules that we've talked about 00:08:19.920 --> 00:08:23.400 implicitly here but have not stated in our game format 00:08:23.400 --> 00:08:26.560 explicitly, what we have shown is 00:08:26.560 --> 00:08:32.299 that if we accept certain rules, it follows from our definitions 00:08:32.299 --> 00:08:36.690 that 4 plus 3 equals 7 is an inescapable conclusion. 00:08:36.690 --> 00:08:39.679 Notice that the inescapability of the conclusion 00:08:39.679 --> 00:08:44.350 hinges on the fact that we've accepted certain rules. 00:08:44.350 --> 00:08:47.840 If we change the rules, we can change the conclusions. 00:08:47.840 --> 00:08:49.800 In other words, this thing called 00:08:49.800 --> 00:08:54.000 drawing inescapable conclusions is something called validity, 00:08:54.000 --> 00:08:58.490 and validity involves the art of drawing inescapable conclusions 00:08:58.490 --> 00:09:02.010 using given rules and given definitions. 00:09:02.010 --> 00:09:04.500 We'll talk about that in more detail, 00:09:04.500 --> 00:09:07.380 but for now, notice the difference here. 00:09:07.380 --> 00:09:13.470 Before we did this, we knew as a conjecture, a past experience 00:09:13.470 --> 00:09:17.150 thing, that 4 plus 3 equals 7 is a true statement. 00:09:17.150 --> 00:09:21.310 What we now know is in terms of certain rules of the game, 00:09:21.310 --> 00:09:25.020 coupled with the definitions that we've accepted, 4 plus 3 00:09:25.020 --> 00:09:29.160 equals 7 is an inescapable conclusion, henceforth to be 00:09:29.160 --> 00:09:32.250 called a "theorem" in our game. 00:09:32.250 --> 00:09:35.250 Let's see if we can't get away from this rather simple 00:09:35.250 --> 00:09:38.180 example, and again, let me emphasize that as simple 00:09:38.180 --> 00:09:40.860 as this example is, throughout our course, 00:09:40.860 --> 00:09:44.050 we will be using the same technique, only 00:09:44.050 --> 00:09:48.080 at a more sophisticated level of computational skill. 00:09:48.080 --> 00:09:52.040 But let's take a look here and see what we're really saying. 00:09:52.040 --> 00:09:56.310 What mathematical structure really involves 00:09:56.310 --> 00:09:59.200 is a logic machine type of thing. 00:09:59.200 --> 00:10:02.280 We have a logic machine, a machine 00:10:02.280 --> 00:10:06.150 that, being fed any kinds of definitions, rules, 00:10:06.150 --> 00:10:12.340 assumptions, et cetera, grinds these things through and has, 00:10:12.340 --> 00:10:16.450 as its output, inescapable conclusions. 00:10:16.450 --> 00:10:20.570 In other words, you feed in definitions, rules, et cetera, 00:10:20.570 --> 00:10:23.070 which by the way, don't really have 00:10:23.070 --> 00:10:26.150 to be true in the real-life sense of being true. 00:10:26.150 --> 00:10:28.030 I mean, for example, in the baseball game, 00:10:28.030 --> 00:10:30.120 to say the rule is three strikes is an out, 00:10:30.120 --> 00:10:32.190 there was certainly no basic truth 00:10:32.190 --> 00:10:34.000 the said that had to be the case. 00:10:34.000 --> 00:10:37.350 What is true is that, in the world of science, 00:10:37.350 --> 00:10:39.600 the scientist is the interpreter of nature. 00:10:39.600 --> 00:10:43.770 What happens in real life happens whether the scientist 00:10:43.770 --> 00:10:44.740 predicts it or not. 00:10:44.740 --> 00:10:48.720 What he tries to do is to make up definitions and rules, which 00:10:48.720 --> 00:10:51.710 are compatible with his experience, things 00:10:51.710 --> 00:10:53.640 which he calls truth. 00:10:53.640 --> 00:10:56.650 He feeds those through his logic machine, 00:10:56.650 --> 00:11:02.150 draws inescapable conclusions, and if the conclusions 00:11:02.150 --> 00:11:05.750 follow inescapably from the definitions and the rules, 00:11:05.750 --> 00:11:09.200 we call the resulting argument valid. 00:11:09.200 --> 00:11:12.430 In other words, truth is a value judgment 00:11:12.430 --> 00:11:14.940 that we make about a particular statement. 00:11:14.940 --> 00:11:17.940 Validity is a more objective thing 00:11:17.940 --> 00:11:20.030 that we attribute to an argument. 00:11:20.030 --> 00:11:23.080 In other words, a statement is either true or false. 00:11:23.080 --> 00:11:26.030 An argument is either valid or invalid, 00:11:26.030 --> 00:11:29.040 meaning that we only judge whether the conclusion 00:11:29.040 --> 00:11:31.760 of the argument follows inescapably from the given 00:11:31.760 --> 00:11:35.030 assumptions, independently of whether those assumptions 00:11:35.030 --> 00:11:37.090 happen to be true or not. 00:11:37.090 --> 00:11:40.030 By the way, as an aside, one of the reasons 00:11:40.030 --> 00:11:43.790 that the scientist prefers to dodge issues 00:11:43.790 --> 00:11:48.160 such as "what is truth" and leaves that to the philosopher, 00:11:48.160 --> 00:11:51.210 is that truth, in many cases, is a relative thing. 00:11:51.210 --> 00:11:53.670 It's based on the available knowledge. 00:11:53.670 --> 00:11:57.070 It's also based on the situation that we want to handle. 00:11:57.070 --> 00:12:00.200 In other words, what is truth? 00:12:00.200 --> 00:12:04.130 And my claim is that the answer depends on the situation. 00:12:04.130 --> 00:12:07.970 Well, let me give you again a trivial arithmetic situation. 00:12:07.970 --> 00:12:15.100 Does 1/2 plus 1/3 equal 5/6, or does 1/2 plus 1/3 equal 2/5? 00:12:15.100 --> 00:12:19.050 And again, the answer is it depends 00:12:19.050 --> 00:12:22.050 on what real-life problem you're dealing with. 00:12:22.050 --> 00:12:26.150 For example, if a person is working for you by the hour, 00:12:26.150 --> 00:12:29.460 and he works for you for a half hour one day 00:12:29.460 --> 00:12:33.250 and a third of an hour the next day, the total time he's worked 00:12:33.250 --> 00:12:34.740 is 5/6 of an hour. 00:12:34.740 --> 00:12:35.290 Why? 00:12:35.290 --> 00:12:37.960 Because this agrees with our real-life experience 00:12:37.960 --> 00:12:42.830 that 30 minutes plus 20 minutes is 50 minutes. 00:12:42.830 --> 00:12:45.420 On the other hand, if a baseball player 00:12:45.420 --> 00:12:48.130 goes one for two in the first game of a doubleheader 00:12:48.130 --> 00:12:51.690 and one for three in the second game of a doubleheader, 00:12:51.690 --> 00:12:55.730 he has batted two hits in five times at bat. 00:12:55.730 --> 00:12:57.830 In fact, if you could convince the public 00:12:57.830 --> 00:13:00.300 that he had five hits and six times at bat, 00:13:00.300 --> 00:13:04.110 you could become a director of any economy program 00:13:04.110 --> 00:13:05.550 in the nation, I guess. 00:13:05.550 --> 00:13:07.340 The point, however, is this. 00:13:07.340 --> 00:13:10.390 In most arithmetic questions that we deal with, 00:13:10.390 --> 00:13:12.930 we are used to the physical interpretation 00:13:12.930 --> 00:13:17.680 in which 1/2 plus 1/3 equals 5/6 reflects the real life 00:13:17.680 --> 00:13:18.700 situation. 00:13:18.700 --> 00:13:22.060 Whereas this example here may seem trivial, namely, 00:13:22.060 --> 00:13:25.390 how often are you going to be involved with batting averages 00:13:25.390 --> 00:13:27.730 unless you're doing sixth grade arithmetic. 00:13:27.730 --> 00:13:31.520 The point remains, ironically or whatever you want to call it, 00:13:31.520 --> 00:13:34.950 that this little batting average problem is not trivial. 00:13:34.950 --> 00:13:38.090 In the world of engineering, we know this example 00:13:38.090 --> 00:13:40.820 as the weighted average problem. 00:13:40.820 --> 00:13:44.940 In other words, you'll notice that 2 over 5, as a fraction, 00:13:44.940 --> 00:13:49.750 is more nearly equal to 1/3 that it is equal to 1/2, 00:13:49.750 --> 00:13:53.250 and the reason for that is that the player batted 00:13:53.250 --> 00:13:56.480 at the low average, 1 over 3, one hit in three times 00:13:56.480 --> 00:13:58.590 at bat, for more times at bat. 00:13:58.590 --> 00:14:01.840 In other words, the three has a heavier weighting factor 00:14:01.840 --> 00:14:05.810 than the two, and every time that you use a weighted average 00:14:05.810 --> 00:14:09.170 in a scientific engineering oriented investigation, 00:14:09.170 --> 00:14:12.460 this is the truth, not this. 00:14:12.460 --> 00:14:16.690 In other words, it is neither true or false that 1/2 plus 1/3 00:14:16.690 --> 00:14:20.540 equals 2/5 or that 1/2 plus 1/3 equals 5/6. 00:14:20.540 --> 00:14:24.644 Which is true depends on the particular physical situation. 00:14:24.644 --> 00:14:26.310 You see a rather interesting point here, 00:14:26.310 --> 00:14:28.620 that we sometimes allow truth to be 00:14:28.620 --> 00:14:32.420 based on what particular problem we're trying to solve. 00:14:32.420 --> 00:14:35.120 Another way that we manipulate truth 00:14:35.120 --> 00:14:38.760 is that we sometimes have a rule that we like to be true. 00:14:38.760 --> 00:14:40.420 In fact, I guess this is probably 00:14:40.420 --> 00:14:43.680 what happens with most political theories 00:14:43.680 --> 00:14:45.740 that one starts with the objectives, 00:14:45.740 --> 00:14:48.080 knows what it is that he wants to be true, 00:14:48.080 --> 00:14:50.590 and then invents the definitions and the rules 00:14:50.590 --> 00:14:52.350 to conform with this. 00:14:52.350 --> 00:14:55.160 In much the same way as around the fourth grade again, 00:14:55.160 --> 00:14:58.490 we learn such adages as "Look before you leap," 00:14:58.490 --> 00:15:02.030 and two minutes later you learn "He who hesitates is lost," 00:15:02.030 --> 00:15:04.240 and suddenly you come to the conclusion 00:15:04.240 --> 00:15:07.950 that you can make your assumptions validify 00:15:07.950 --> 00:15:11.130 any conclusion you want just by choosing your assumptions 00:15:11.130 --> 00:15:12.180 appropriately. 00:15:12.180 --> 00:15:14.140 Now, if that sounds degrading, let 00:15:14.140 --> 00:15:18.730 me show you how it's used effectively in mathematics. 00:15:18.730 --> 00:15:22.110 In other words, let me just say that predetermined rules may 00:15:22.110 --> 00:15:23.480 control truth. 00:15:23.480 --> 00:15:25.460 Let me give you an example. 00:15:25.460 --> 00:15:30.620 Going back to exponents, why does b to the 0 equal 1? 00:15:30.620 --> 00:15:33.570 The answer is very simple, that when 00:15:33.570 --> 00:15:37.810 we use positive exponents, positive whole number 00:15:37.810 --> 00:15:42.810 exponents, we talked about that many factors of b. 00:15:42.810 --> 00:15:45.140 And one of the interesting rules that we 00:15:45.140 --> 00:15:48.540 saw that was obeyed by whole number exponents 00:15:48.540 --> 00:15:51.490 was that if you multiply b to the m-th power 00:15:51.490 --> 00:15:55.000 by b to the n-th power, you got as an answer b 00:15:55.000 --> 00:15:59.650 to the m plus n power. 00:15:59.650 --> 00:16:01.390 b to the m plus n. 00:16:01.390 --> 00:16:06.360 Now, the interesting thing is that after a while, 00:16:06.360 --> 00:16:10.440 you never even paid attention to why this rule worked. 00:16:10.440 --> 00:16:13.680 What you did know was that this was a pretty darn 00:16:13.680 --> 00:16:15.600 convenient rule to use. 00:16:15.600 --> 00:16:18.630 Computationally, this rule simplified 00:16:18.630 --> 00:16:23.380 many particular computations that you were doing. 00:16:23.380 --> 00:16:27.460 In particular, then, as soon as n is 0, 00:16:27.460 --> 00:16:30.130 you would still like to be able to use this rule. 00:16:30.130 --> 00:16:33.120 In other words, we want this nice rule 00:16:33.120 --> 00:16:36.430 to apply even when n equals 0. 00:16:36.430 --> 00:16:39.810 Now, let's look at this from a computational point of view. 00:16:39.810 --> 00:16:45.190 If we want this rule to apply when n equals 0, 00:16:45.190 --> 00:16:48.160 let's simply rewrite this exactly 00:16:48.160 --> 00:16:53.190 as it appears here with n equal to 0. 00:16:53.190 --> 00:16:55.820 We then have, what? 00:16:55.820 --> 00:16:58.600 We have b-- we'll just repeat everything here. 00:16:58.600 --> 00:17:01.860 b to the m times b. 00:17:01.860 --> 00:17:06.079 Now, we're replacing n by 0, so that's b to the 0, 00:17:06.079 --> 00:17:15.390 and that must equal b to the m plus n, that's m plus 0. 00:17:15.390 --> 00:17:17.990 But the interesting point is that we 00:17:17.990 --> 00:17:23.089 know how to add numbers, and for numbers, m plus 0 is m. 00:17:23.089 --> 00:17:26.849 In other words, this is still b to the m. 00:17:26.849 --> 00:17:30.560 Now, we look at this, and what we're saying 00:17:30.560 --> 00:17:36.090 is if we want the rule b to the m times b to the n 00:17:36.090 --> 00:17:40.710 to equal b to the m plus n to be true even when n is 0, 00:17:40.710 --> 00:17:44.730 this says that b to the 0 must be that number such 00:17:44.730 --> 00:17:49.340 that when we multiply it by b to the m, we get b to the m. 00:17:49.340 --> 00:17:51.770 Now, what number has the property 00:17:51.770 --> 00:17:55.940 that when you multiply it by b to the m you get b to the m? 00:17:55.940 --> 00:17:58.300 And if you're real quick, you say 1, 00:17:58.300 --> 00:17:59.970 and if you're algebra-oriented, you 00:17:59.970 --> 00:18:05.110 say b to the 0 is therefore equal b to the m divided 00:18:05.110 --> 00:18:08.980 by b to the m, and again you say 1, 00:18:08.980 --> 00:18:12.890 except of course that b must be unequal to zero because, 00:18:12.890 --> 00:18:15.910 hopefully, by this time we understand why we must never 00:18:15.910 --> 00:18:18.100 divide by 0. 00:18:18.100 --> 00:18:20.900 In other words, what we now have is the old high school 00:18:20.900 --> 00:18:30.700 rule that b to the 0 equals 1 provided b is unequal to 0. 00:18:30.700 --> 00:18:32.600 But here's the important point. 00:18:32.600 --> 00:18:35.120 If I have never defined b to the 0, 00:18:35.120 --> 00:18:36.880 and somebody says to me: "make up 00:18:36.880 --> 00:18:40.540 a definition for b to the 0," and I say, OK, b to the 0 00:18:40.540 --> 00:18:44.350 is going to be 36, I have every right in the world 00:18:44.350 --> 00:18:45.770 to make up that definition. 00:18:45.770 --> 00:18:48.450 What I don't have a right to do is, 00:18:48.450 --> 00:18:51.420 when I'm ever using an expression like b to the 0, 00:18:51.420 --> 00:18:53.780 is to assume that I have the right 00:18:53.780 --> 00:18:56.030 to use this particular recipe. 00:18:56.030 --> 00:19:00.040 In other words, if I want to be able to use the nice recipe, 00:19:00.040 --> 00:19:05.290 even when n is 0, I have no choice but to define b to the 0 00:19:05.290 --> 00:19:06.840 to equal 1. 00:19:06.840 --> 00:19:08.990 Therefore, we make up that definition 00:19:08.990 --> 00:19:11.820 because we want that recipe to apply, 00:19:11.820 --> 00:19:14.050 and that's exactly what we're going to be doing 00:19:14.050 --> 00:19:15.420 through most of this course. 00:19:15.420 --> 00:19:18.310 We are going to look at certain real-life situations, 00:19:18.310 --> 00:19:21.610 we are going to look at certain recipes that we want to apply, 00:19:21.610 --> 00:19:23.980 and we are going to make up definitions this way, 00:19:23.980 --> 00:19:28.100 and then see how our inescapable conclusions follow 00:19:28.100 --> 00:19:29.760 from these definitions. 00:19:29.760 --> 00:19:33.870 More about that will be said in the remainder of this unit, 00:19:33.870 --> 00:19:35.770 and our next lecture will pick up 00:19:35.770 --> 00:19:38.450 the game of mathematics in a different context. 00:19:38.450 --> 00:19:40.300 But until next time, good bye. 00:19:46.710 --> 00:19:49.080 Funding for the publication of this video 00:19:49.080 --> 00:19:53.960 was provided by the Gabriella and Paul Rosenbaum Foundation. 00:19:53.960 --> 00:19:58.130 Help OCW continue to provide free and open access to MIT 00:19:58.130 --> 00:20:02.548 courses by making a donation at ocw.mit.edu/donate.