1 00:00:00,000 --> 00:00:01,940 FEMALE VOICE: The following content is provided under a 2 00:00:01,940 --> 00:00:03,690 Creative Commons license. 3 00:00:03,690 --> 00:00:06,630 Your support will help MIT OpenCourseWare continue to 4 00:00:06,630 --> 00:00:09,990 offer high quality educational resources for free. 5 00:00:09,990 --> 00:00:12,830 To make a donation, or to view additional materials from 6 00:00:12,830 --> 00:00:16,760 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,760 --> 00:00:18,010 ocw.mit.edu. 8 00:00:32,222 --> 00:00:33,700 PROFESSOR: Hi. 9 00:00:33,700 --> 00:00:37,190 Our lecture today concerns mathematical induction, which, 10 00:00:37,190 --> 00:00:40,490 roughly speaking, is a technique that one uses to 11 00:00:40,490 --> 00:00:43,520 prove something when one already has a pretty good 12 00:00:43,520 --> 00:00:46,460 suspicion as to what the right answer is. 13 00:00:46,460 --> 00:00:48,710 Now, rather than to philosophize about this too 14 00:00:48,710 --> 00:00:52,800 long, let's tear right into a problem and see, in action, 15 00:00:52,800 --> 00:00:54,700 just what the concept means. 16 00:00:54,700 --> 00:00:57,800 Recall that, in an earlier lecture, we have proven that 17 00:00:57,800 --> 00:01:01,460 the limit of a sum is equal to the sum of the limits 18 00:01:01,460 --> 00:01:03,800 provided, of course, that there are only two 19 00:01:03,800 --> 00:01:05,600 terms in the sum. 20 00:01:05,600 --> 00:01:09,470 That is, if we have two functions, 'f 1' and 'f 2', 21 00:01:09,470 --> 00:01:13,180 the limit of 'f1 of x' plus 'f2 of x' as 'x' approaches 22 00:01:13,180 --> 00:01:16,940 'a' is the limit of 'f1 of x' as 'x' approaches 'a' plus the 23 00:01:16,940 --> 00:01:19,710 limit of 'f2 of x' as 'x' approaches 'a'. 24 00:01:19,710 --> 00:01:22,650 The question, now, is how about the limit of 'x' 25 00:01:22,650 --> 00:01:25,870 approaches 'a' of 'f1 of x', plus 'f2 of 26 00:01:25,870 --> 00:01:28,410 x', plus 'f3 of x'? 27 00:01:28,410 --> 00:01:32,060 Now, to tackle a problem like this, we do what is so often 28 00:01:32,060 --> 00:01:35,010 done in any mathematical logical procedure. 29 00:01:35,010 --> 00:01:39,960 We try to reduce an unfamiliar problem to a familiar problem 30 00:01:39,960 --> 00:01:41,800 which has already been solved. 31 00:01:41,800 --> 00:01:44,220 Let's see what I mean by that. 32 00:01:44,220 --> 00:01:46,780 If this had been only two terms in here that we're 33 00:01:46,780 --> 00:01:50,180 adding, we would have known how to handle this problem. 34 00:01:50,180 --> 00:01:53,720 So what we observe is that since the sum of two functions 35 00:01:53,720 --> 00:01:57,750 is, again, a function, we can assume that our expression is 36 00:01:57,750 --> 00:01:59,530 written this way. 37 00:01:59,530 --> 00:02:02,710 Now, you see, we've reduced our problem to the sum of two 38 00:02:02,710 --> 00:02:07,240 functions, namely, 'f1 of x' plus 'f2 of x' being one of 39 00:02:07,240 --> 00:02:11,520 our functions, 'f3 of x' being another of our functions. 40 00:02:11,520 --> 00:02:12,860 We know, now, what? 41 00:02:12,860 --> 00:02:15,350 That the limit of a sum is the sum of the limits. 42 00:02:15,350 --> 00:02:19,760 If we have two functions, you see this is, what? 43 00:02:19,760 --> 00:02:30,100 The limit 'f1 of x' plus 'f2 of x', you see, plus the limit 44 00:02:30,100 --> 00:02:33,750 as 'x' approaches 'a', 'f3 of x'. 45 00:02:33,750 --> 00:02:36,290 But now, look what's in this expression over here. 46 00:02:36,290 --> 00:02:37,160 This is now, what? 47 00:02:37,160 --> 00:02:39,650 The limit of the sum of two functions. 48 00:02:39,650 --> 00:02:42,070 And, you see, we know that the limit of a sum is the sum of 49 00:02:42,070 --> 00:02:45,520 the limits is true if we have only two functions, so, from 50 00:02:45,520 --> 00:02:46,840 here, we get to here. 51 00:02:46,840 --> 00:02:49,170 And, from here, we can say, what? 52 00:02:49,170 --> 00:02:54,310 This is the limit 'x' approaches 'a', 'f1 of x', 53 00:02:54,310 --> 00:03:02,740 plus the limit as 'x' approaches 'a', 'f2 of x', 54 00:03:02,740 --> 00:03:06,940 plus the limit as 'x' approaches 'a', 'f3 of x'. 55 00:03:06,940 --> 00:03:09,970 In other words, what we have shown is that the limit of a 56 00:03:09,970 --> 00:03:13,160 sum equals the sum of the limits is true not just the 57 00:03:13,160 --> 00:03:17,290 sum of two functions, but for the sum of 3 as well. 58 00:03:17,290 --> 00:03:22,100 And, more importantly, the truth for 3 hinged directly on 59 00:03:22,100 --> 00:03:23,410 the truth for 2. 60 00:03:23,410 --> 00:03:26,290 In other words, it wasn't just that we proved that the 61 00:03:26,290 --> 00:03:29,430 formula was true for the sum of three functions, we proved 62 00:03:29,430 --> 00:03:33,060 it on the assumption that it was already true for the sum 63 00:03:33,060 --> 00:03:34,080 of two functions. 64 00:03:34,080 --> 00:03:37,840 And, by the way, notice how we may now begin to suspect that 65 00:03:37,840 --> 00:03:39,610 this idea generalizes. 66 00:03:39,610 --> 00:03:42,640 For example, let's take a look over here. 67 00:03:42,640 --> 00:03:45,530 Suppose I now said how about the limit as 68 00:03:45,530 --> 00:03:47,980 'x' approaches 'a'? 69 00:03:47,980 --> 00:03:52,340 And we'll now take the sum of four functions: 'f1 of x', 70 00:03:52,340 --> 00:03:59,391 plus 'f2 of x', plus 'f3 of x', plus 'f4 of x'. 71 00:03:59,391 --> 00:04:01,670 See, what about something like that? 72 00:04:01,670 --> 00:04:04,730 And, again, we argue the same way as we did before. 73 00:04:04,730 --> 00:04:07,400 We say, you know, if we had only had two 74 00:04:07,400 --> 00:04:09,530 functions in here-- 75 00:04:09,530 --> 00:04:11,960 and this gives us the hint to do this-- 76 00:04:11,960 --> 00:04:14,860 see, we do know that the limit of the sum is the sum of the 77 00:04:14,860 --> 00:04:17,110 limits if we have only two functions. 78 00:04:17,110 --> 00:04:20,570 You see, now, I could write this as, what? 79 00:04:20,570 --> 00:04:24,860 It's the limit of the first one, as 'x' approaches 'a', 80 00:04:24,860 --> 00:04:26,540 but what is the first one? 81 00:04:26,540 --> 00:04:32,060 The first one is the function 'f1 of x', plus 'f2 of x', 82 00:04:32,060 --> 00:04:37,010 plus 'f3 of x', plus the limit of the second. 83 00:04:37,010 --> 00:04:41,730 The second now you see, is 'f sub four of x' as 'x' 84 00:04:41,730 --> 00:04:43,590 approaches 'a'. 85 00:04:43,590 --> 00:04:46,330 Now, you see our previous case told us, what? 86 00:04:46,330 --> 00:04:48,890 That the limit of a sum is equal to the sum of the limits 87 00:04:48,890 --> 00:04:51,210 if you have the sum of three functions. 88 00:04:51,210 --> 00:04:54,250 That's exactly what we have over here, and now you see we 89 00:04:54,250 --> 00:04:55,130 can say, what? 90 00:04:55,130 --> 00:05:01,020 Ah, this is the limit of the first as 'x' approaches 'a', 91 00:05:01,020 --> 00:05:07,170 plus the limit of the second as 'x' approaches 'a', plus 92 00:05:07,170 --> 00:05:14,300 the limit of the third as 'x' approaches 'a', plus the limit 93 00:05:14,300 --> 00:05:17,230 the fourth as 'x' approaches 'a'. 94 00:05:17,230 --> 00:05:19,520 In other words, what have we done now? 95 00:05:19,520 --> 00:05:22,980 We've shown that knowing that the limit of a sum is the sum 96 00:05:22,980 --> 00:05:26,260 of the limits was true for a sum of two functions and of 97 00:05:26,260 --> 00:05:27,300 three functions. 98 00:05:27,300 --> 00:05:31,700 We've shown, inescapably, that the same result holds for the 99 00:05:31,700 --> 00:05:34,710 sum of four functions. 100 00:05:34,710 --> 00:05:37,870 Let's take a little breather here and make a few asides. 101 00:05:37,870 --> 00:05:40,840 I think, sometimes, when one starts to work too much with 102 00:05:40,840 --> 00:05:44,340 mathematical symbolism, we lose track of the fact that 103 00:05:44,340 --> 00:05:46,640 things are not quite as difficult as they might 104 00:05:46,640 --> 00:05:49,340 otherwise have seemed. 105 00:05:49,340 --> 00:05:54,150 You see, for one thing, my claim is that we have already 106 00:05:54,150 --> 00:05:57,900 tackled this problem as recently as first grade 107 00:05:57,900 --> 00:05:59,160 arithmetic. 108 00:05:59,160 --> 00:06:00,860 Namely, we learned, what? 109 00:06:00,860 --> 00:06:02,890 We learned tables. 110 00:06:02,890 --> 00:06:04,280 Remember the addition tables? 111 00:06:04,280 --> 00:06:06,510 You learned how to add two numbers. 112 00:06:06,510 --> 00:06:09,670 All of a sudden somebody says what is the sum of 113 00:06:09,670 --> 00:06:11,810 1, 2, 3, and 4? 114 00:06:11,810 --> 00:06:13,610 What is this number? 115 00:06:13,610 --> 00:06:15,780 And what we said was, look at, we'll just add 116 00:06:15,780 --> 00:06:17,240 these two at a time. 117 00:06:17,240 --> 00:06:19,950 1 plus 2 is a number, which is 3. 118 00:06:19,950 --> 00:06:24,240 3 plus 3 is a number, namely 6. 119 00:06:24,240 --> 00:06:28,330 And 6 plus 4 is a number: 10. 120 00:06:28,330 --> 00:06:29,860 In other words, we essentially did, what? 121 00:06:29,860 --> 00:06:32,160 We added the first to the second. 122 00:06:32,160 --> 00:06:35,860 Then we added the sum of the first two to the third, the 123 00:06:35,860 --> 00:06:38,320 sum of the first three to the fourth, and that 124 00:06:38,320 --> 00:06:39,290 was how we did this. 125 00:06:39,290 --> 00:06:43,320 Of course, what we assumed in doing this was that the sum of 126 00:06:43,320 --> 00:06:45,375 two numbers was, again, a number. 127 00:06:45,375 --> 00:06:47,470 Up above, we assumed that the sum of two 128 00:06:47,470 --> 00:06:49,040 functions was a function. 129 00:06:49,040 --> 00:06:51,280 And this is not quite as trivial as it might otherwise 130 00:06:51,280 --> 00:06:52,060 have seemed. 131 00:06:52,060 --> 00:06:56,170 Namely, look at, if you add two odd numbers, do you get 132 00:06:56,170 --> 00:06:58,570 like things when you combine like things? 133 00:06:58,570 --> 00:07:03,860 The sum of two odd numbers is always even. 134 00:07:03,860 --> 00:07:05,940 You see, you can't say let's replace the sum of two odd 135 00:07:05,940 --> 00:07:07,550 numbers by another odd number. 136 00:07:07,550 --> 00:07:10,080 You can say it, but it would be wrong. 137 00:07:10,080 --> 00:07:11,830 On the other hand, another example. 138 00:07:11,830 --> 00:07:12,910 How about subtraction? 139 00:07:12,910 --> 00:07:14,360 That's a nice operation. 140 00:07:14,360 --> 00:07:17,760 If you subtract a positive number from a positive number, 141 00:07:17,760 --> 00:07:20,910 are you guaranteed that the result will be positive? 142 00:07:20,910 --> 00:07:25,640 Well, for example, what about 3 less 5? 143 00:07:25,640 --> 00:07:27,800 The answer would be negative 2. 144 00:07:27,800 --> 00:07:31,770 Positive minus positive can very well be negative, so we 145 00:07:31,770 --> 00:07:33,040 must be sure, what? 146 00:07:33,040 --> 00:07:36,160 That, when we combine like objects, we get like objects. 147 00:07:36,160 --> 00:07:38,770 And another thing that we assumed was that our answer 148 00:07:38,770 --> 00:07:41,450 did not depend upon voice inflection. 149 00:07:41,450 --> 00:07:42,600 Now, what does that mean? 150 00:07:42,600 --> 00:07:44,650 Let me show you something over here. 151 00:07:44,650 --> 00:07:49,340 Look at the expression 12 divided by 6 divided by 2. 152 00:07:49,340 --> 00:07:55,910 If you read this as if it said 12 divided by 6 divided by 2, 153 00:07:55,910 --> 00:07:58,250 the answer is 1. 154 00:07:58,250 --> 00:08:04,010 On the other hand, if you read the same thing as if it said 155 00:08:04,010 --> 00:08:09,380 12 divided by 6 divided by 2, the answer is, what? 156 00:08:09,380 --> 00:08:12,760 12 divided by 3, which is 4. 157 00:08:12,760 --> 00:08:17,680 In other words, division depends on voice inflection, 158 00:08:17,680 --> 00:08:19,580 whereas, addition doesn't. 159 00:08:19,580 --> 00:08:22,730 And I simply point out these asides to show you that, as we 160 00:08:22,730 --> 00:08:25,770 go through advanced mathematical analysis, we are 161 00:08:25,770 --> 00:08:29,500 always making use of the same assumptions that we were 162 00:08:29,500 --> 00:08:32,539 making when we dealt with more simple things. 163 00:08:32,539 --> 00:08:35,000 And I think this is the healthiest way of seeing how 164 00:08:35,000 --> 00:08:36,070 our subject develops. 165 00:08:36,070 --> 00:08:39,130 We will leap from things that we already know into 166 00:08:39,130 --> 00:08:41,970 generalizations that are less familiar to us. 167 00:08:41,970 --> 00:08:45,220 At any rate, let's now return to our main theme of 168 00:08:45,220 --> 00:08:48,750 mathematical induction and try to summarize 169 00:08:48,750 --> 00:08:50,420 what's happened so far. 170 00:08:50,420 --> 00:08:53,740 We are working with a certain conjecture, let's call it. 171 00:08:53,740 --> 00:08:56,910 The conjecture is that the limit of the sum is the sum of 172 00:08:56,910 --> 00:08:57,660 the limits. 173 00:08:57,660 --> 00:09:01,590 We know that the conjecture was true for two terms. 174 00:09:01,590 --> 00:09:04,870 We proved that, once it was true for two terms, it was 175 00:09:04,870 --> 00:09:06,220 true for three terms. 176 00:09:06,220 --> 00:09:09,540 We then prove that, if it was true for three terms, it was 177 00:09:09,540 --> 00:09:11,120 true for four terms. 178 00:09:11,120 --> 00:09:14,090 And now, if we have any imagination at all, we might 179 00:09:14,090 --> 00:09:17,590 become suspicious and say, you know, I think, if it's true, 180 00:09:17,590 --> 00:09:20,340 in general, for 'n' terms, it's going to be 181 00:09:20,340 --> 00:09:22,810 true for 'n + 1' terms. 182 00:09:22,810 --> 00:09:25,660 And that brings us to our next stage in our mathematical 183 00:09:25,660 --> 00:09:30,130 induction, namely, suppose the limit of the sum is equal to 184 00:09:30,130 --> 00:09:33,770 the sum of the limits in the case that there are 'n' terms 185 00:09:33,770 --> 00:09:35,000 in our sum. 186 00:09:35,000 --> 00:09:40,690 What can we conclude about the limit of a sum in the case of 187 00:09:40,690 --> 00:09:42,380 'n + 1' terms? 188 00:09:42,380 --> 00:09:45,040 And, without going through the proof here, we'll do these 189 00:09:45,040 --> 00:09:47,990 things in our supplementary notes in our learning 190 00:09:47,990 --> 00:09:51,270 exercises, but, here, I just want to focus our attention on 191 00:09:51,270 --> 00:09:52,630 what the main theme is. 192 00:09:52,630 --> 00:09:56,770 What we say is, since we can add without worrying about 193 00:09:56,770 --> 00:10:00,060 voice inflection, why don't we throw in a pair of braces 194 00:10:00,060 --> 00:10:05,230 here, thus reducing our problem to the limit of a sum 195 00:10:05,230 --> 00:10:08,390 when we're adding but two functions, 196 00:10:08,390 --> 00:10:09,990 use the theorem there. 197 00:10:09,990 --> 00:10:13,990 And, now, given the limit of the sum of 'n' functions, we 198 00:10:13,990 --> 00:10:15,810 know that that's the limit of a sum. 199 00:10:15,810 --> 00:10:20,330 And now, it appears that the truth for 'n' is going to 200 00:10:20,330 --> 00:10:23,490 imply the truth for 'n + 1'. 201 00:10:23,490 --> 00:10:26,320 Now, of course, there may be other problems that work 202 00:10:26,320 --> 00:10:30,270 structurally this way other than the limit of a sum equals 203 00:10:30,270 --> 00:10:31,340 the sum of the limits. 204 00:10:31,340 --> 00:10:33,670 So let's generalize that result. 205 00:10:33,670 --> 00:10:36,100 And the generalization is what is known as 206 00:10:36,100 --> 00:10:37,610 mathematical induction. 207 00:10:37,610 --> 00:10:40,900 What mathematical induction says is this, let's suppose we 208 00:10:40,900 --> 00:10:43,120 have a conjecture. 209 00:10:43,120 --> 00:10:45,420 Now, how we get the conjecture is something we'll talk about 210 00:10:45,420 --> 00:10:48,040 in a while, but let's suppose we have the conjecture. 211 00:10:48,040 --> 00:10:50,670 Well, to try to show that the conjecture is true all the 212 00:10:50,670 --> 00:10:54,170 time, we had better be sure it's true at least sometimes. 213 00:10:54,170 --> 00:10:58,450 So we say, OK, let's show that the conjecture is true 214 00:10:58,450 --> 00:11:00,130 for 'n' equals 1. 215 00:11:00,130 --> 00:11:01,530 That's just a simple verification. 216 00:11:01,530 --> 00:11:04,120 We show that it's true for 'n' equals 1. 217 00:11:04,120 --> 00:11:09,630 Then we say OK, next, prove that the truth for 'n' equals 218 00:11:09,630 --> 00:11:14,950 'k' implies the truth for 'n' equals 'k + 1'. 219 00:11:14,950 --> 00:11:17,720 In other words, we're not saying that it's true for 'k', 220 00:11:17,720 --> 00:11:21,650 all we're saying is that, if it's true for 'k', if it's 221 00:11:21,650 --> 00:11:25,690 true for 'k', the truth for 'n' equals 'k' implies the 222 00:11:25,690 --> 00:11:28,720 truth for 'n' equals 'k + 1'. 223 00:11:28,720 --> 00:11:32,650 Then, if that's true, our conjecture is true for all 224 00:11:32,650 --> 00:11:33,490 whole numbers. 225 00:11:33,490 --> 00:11:34,990 Why is that? 226 00:11:34,990 --> 00:11:38,300 Well, let's take a look, informally, here. 227 00:11:38,300 --> 00:11:40,720 Let's suppose that both of these conditions are obeyed. 228 00:11:40,720 --> 00:11:43,780 We know the conjecture is true when 'n' equals 1. 229 00:11:43,780 --> 00:11:45,560 Now, take 'k' to be 1. 230 00:11:45,560 --> 00:11:49,510 Since it's true for 1, this part tells us it's going to be 231 00:11:49,510 --> 00:11:52,940 true for one more than 1, which is 2. 232 00:11:52,940 --> 00:11:57,120 Now that the conjecture is true for 2, this says, what? 233 00:11:57,120 --> 00:11:59,170 It's going to be true for 3. 234 00:11:59,170 --> 00:12:01,460 And knowing that it's true for 3, this will say 235 00:12:01,460 --> 00:12:02,820 it's true for 4. 236 00:12:02,820 --> 00:12:05,690 And now I'll loosely use the word et cetera, and come back 237 00:12:05,690 --> 00:12:08,260 and reinforce that as we go along. 238 00:12:08,260 --> 00:12:11,370 I think now, perhaps, the best thing to do is to look at a 239 00:12:11,370 --> 00:12:12,520 second example. 240 00:12:12,520 --> 00:12:15,910 You see, what we did first of all was we used an example to 241 00:12:15,910 --> 00:12:18,170 lead in to what the definition would be. 242 00:12:18,170 --> 00:12:20,720 Now that we have the definition, let's proceed 243 00:12:20,720 --> 00:12:22,450 directly to use it. 244 00:12:22,450 --> 00:12:24,460 Let me give you a conjecture. 245 00:12:24,460 --> 00:12:29,530 The conjecture which I have in mind is that the sum of the 246 00:12:29,530 --> 00:12:31,770 first n positive numbers-- 247 00:12:31,770 --> 00:12:33,120 and this is an interesting formula-- 248 00:12:33,120 --> 00:12:36,650 it's the last number multiplied by one more than 249 00:12:36,650 --> 00:12:39,860 the last number divided by 2. 250 00:12:39,860 --> 00:12:42,220 Well, let's just see if that's true at all for a while. 251 00:12:42,220 --> 00:12:50,400 Look at, if 'n' is 1, the left-hand side here is 1. 252 00:12:50,400 --> 00:12:55,300 And 1 times 2 divided by 2 is also 1. 253 00:12:55,300 --> 00:12:57,400 If 'n' is 2, the sum of the first two 254 00:12:57,400 --> 00:12:58,310 numbers here is, what? 255 00:12:58,310 --> 00:12:59,940 One plus 2 is three. 256 00:12:59,940 --> 00:13:04,710 On the other hand, 2 times 3 divided by 2 is also 3. 257 00:13:04,710 --> 00:13:08,470 So, at least, our conjecture is true for 'n' equals 1 and 258 00:13:08,470 --> 00:13:09,640 'n' equals 2. 259 00:13:09,640 --> 00:13:12,040 What does mathematical induction say? 260 00:13:12,040 --> 00:13:13,890 Let's take a look again, now. 261 00:13:13,890 --> 00:13:16,050 You see, we showed that the conjecture was true 262 00:13:16,050 --> 00:13:17,180 for 'n' equals 1. 263 00:13:17,180 --> 00:13:19,730 And, for good measure, we also showed that it was true for 264 00:13:19,730 --> 00:13:20,710 'n' equals 2. 265 00:13:20,710 --> 00:13:21,940 So, we can check this off. 266 00:13:21,940 --> 00:13:23,010 We've done that. 267 00:13:23,010 --> 00:13:24,010 Now, what do we do? 268 00:13:24,010 --> 00:13:27,670 We assume the conjecture is true for 'n' equals 'k'. 269 00:13:27,670 --> 00:13:28,920 That means, what? 270 00:13:31,200 --> 00:13:32,670 Well, just replace 'n' by 'k'. 271 00:13:38,190 --> 00:13:40,700 We're assuming that this is true. 272 00:13:40,700 --> 00:13:44,750 From the truth of this, what must we do next? 273 00:13:44,750 --> 00:13:48,710 Well, what we must do next is investigate what happens if 274 00:13:48,710 --> 00:13:50,030 you add, what? 275 00:13:50,030 --> 00:13:52,660 Not 'k' numbers, but 'k + 1'. 276 00:13:52,660 --> 00:13:54,960 So, in other words, what happens when you replace 'n' 277 00:13:54,960 --> 00:13:56,470 by 'k + 1'? 278 00:13:56,470 --> 00:13:58,120 Now, watch how we do this. 279 00:13:58,120 --> 00:14:01,070 The same thing that we did in theory before, we say, look 280 00:14:01,070 --> 00:14:04,620 at, we already know how to handle this amount. 281 00:14:04,620 --> 00:14:07,742 We're told that that's going to be ''k' times 'k + 1' over 282 00:14:07,742 --> 00:14:12,200 2', so let's rewrite this in this way. 283 00:14:12,200 --> 00:14:16,200 Now, we can replace the bracketed expression by ''k' 284 00:14:16,200 --> 00:14:18,620 times 'k + 1' over 2'. 285 00:14:18,620 --> 00:14:21,490 We add on, of course, 'k + 1' because that's the last term 286 00:14:21,490 --> 00:14:22,880 that's over here. 287 00:14:22,880 --> 00:14:29,050 Now, we factor out 'k + 1' from this factor here. 288 00:14:29,050 --> 00:14:30,950 That leaves us with, what? 289 00:14:30,950 --> 00:14:34,720 'k/2 + 1'. 290 00:14:34,720 --> 00:14:37,330 And this, in turn, says, what? 291 00:14:37,330 --> 00:14:43,560 That the sum of the first 'k + 1' numbers is ''k + 1' times 292 00:14:43,560 --> 00:14:44,810 'k + 2' over 2'. 293 00:14:48,120 --> 00:14:51,350 And notice that that's exactly what the conjecture should say 294 00:14:51,350 --> 00:14:54,790 when 'n' equals 'k + 1', namely, what? 295 00:14:54,790 --> 00:14:57,230 The sum of the first 'n' numbers, no matter how many 296 00:14:57,230 --> 00:14:58,260 you have, is, what? 297 00:14:58,260 --> 00:15:01,390 The last number times one more than the last 298 00:15:01,390 --> 00:15:03,480 number divided by 2. 299 00:15:03,480 --> 00:15:07,350 The sum of the first 'k + 1' numbers is, what? 300 00:15:07,350 --> 00:15:12,110 It's 'k + 1', the last one, times 'k + 2', which is one 301 00:15:12,110 --> 00:15:15,240 more than the last one, divided by 2. 302 00:15:15,240 --> 00:15:18,940 And now, what we've done is we have verified the second part 303 00:15:18,940 --> 00:15:23,520 of our mathematical induction setup, namely, if we go back 304 00:15:23,520 --> 00:15:26,490 to our basic definition over here, we have to show, what? 305 00:15:26,490 --> 00:15:30,170 Prove that the truth for 'n' equals 'k' implies the truth 306 00:15:30,170 --> 00:15:34,670 for 'n' equals 'k + 1', which is exactly what we did. 307 00:15:34,670 --> 00:15:37,890 And, while you're thinking about that, let's take a break 308 00:15:37,890 --> 00:15:41,170 for a few more asides which, I think, may cement down this 309 00:15:41,170 --> 00:15:43,480 idea a little bit more strongly. 310 00:15:43,480 --> 00:15:46,430 I mentioned before that induction is something that 311 00:15:46,430 --> 00:15:50,230 one uses when one already has a suspicion as to what the 312 00:15:50,230 --> 00:15:51,540 right answer is. 313 00:15:51,540 --> 00:15:54,090 I don't know how this grabs you, buy my own particular 314 00:15:54,090 --> 00:15:57,050 feeling is that you do not look at the sum of the first 315 00:15:57,050 --> 00:16:01,250 'n' numbers and say aha, it's the last one times one more 316 00:16:01,250 --> 00:16:03,800 than the last one divided by 2. 317 00:16:03,800 --> 00:16:06,050 You see, that's the nice thing about textbook problems. 318 00:16:06,050 --> 00:16:09,270 When they give you a problem on induction and they say 319 00:16:09,270 --> 00:16:13,060 prove this conjecture, notice that they've already given you 320 00:16:13,060 --> 00:16:15,920 a tremendous hint, namely, they've told you what the 321 00:16:15,920 --> 00:16:17,670 conjecture is. 322 00:16:17,670 --> 00:16:20,330 You see, in the textbook of real life, one usually has to 323 00:16:20,330 --> 00:16:23,350 find out what the conjecture is for oneself. 324 00:16:23,350 --> 00:16:26,150 In fact, in the form of a rather interesting aside, 325 00:16:26,150 --> 00:16:29,050 there is a very interesting mathematical anecdote 326 00:16:29,050 --> 00:16:31,460 connected with this particular problem. 327 00:16:31,460 --> 00:16:35,370 It's an anecdote attributed to the mathematician, Gauss, who, 328 00:16:35,370 --> 00:16:37,460 when he was a young chap, was a 329 00:16:37,460 --> 00:16:39,290 discipline problem in school. 330 00:16:39,290 --> 00:16:45,870 And the story is that his teacher, as a punishment, 331 00:16:45,870 --> 00:16:48,610 asked him to add the first 100 numbers. 332 00:16:48,610 --> 00:16:52,140 And Gauss wrote down the answer very, very rapidly. 333 00:16:52,140 --> 00:16:54,040 And what he did was, he didn't add these all up. 334 00:16:54,040 --> 00:16:56,070 What he observed was, what? 335 00:16:56,070 --> 00:17:00,010 The first one plus the last one added up to 101. 336 00:17:00,010 --> 00:17:02,110 The second plus the next to the last added 337 00:17:02,110 --> 00:17:04,970 up to 101, you see? 338 00:17:04,970 --> 00:17:08,790 And each pair going in this way added up to 101. 339 00:17:08,790 --> 00:17:11,430 And how many pairs were there, all together? 340 00:17:11,430 --> 00:17:14,400 Well, there were 100 numbers, so there were 50 pairs. 341 00:17:14,400 --> 00:17:15,970 In other words, there were, what? 342 00:17:15,970 --> 00:17:21,560 100 divided by 2 pairs, each pair adding up to 101. 343 00:17:21,560 --> 00:17:25,660 And now, notice the recipe: 100, namely the last number, 344 00:17:25,660 --> 00:17:30,540 times one more than the last number, 101, divided by 2. 345 00:17:30,540 --> 00:17:36,600 By the way, in the exercises on this assignment we have 346 00:17:36,600 --> 00:17:40,010 another problem, and that is, if you think this one was 347 00:17:40,010 --> 00:17:43,370 already cumbersome, try guessing what the recipe for 348 00:17:43,370 --> 00:17:47,000 this one is: what is the sum of the first n squares? 349 00:17:47,000 --> 00:17:50,740 In other words, not 1, plus 2, plus 3, et cetera, but 1 350 00:17:50,740 --> 00:17:54,710 squared, plus 2 squared, plus 3 squared, et cetera. 351 00:17:54,710 --> 00:17:58,280 I give you the answer, but try to think for a while as to how 352 00:17:58,280 --> 00:18:00,920 likely it is that you would have conjectured this in the 353 00:18:00,920 --> 00:18:01,950 first place. 354 00:18:01,950 --> 00:18:05,700 It turns out to be ''n' times 'n + 1' times 355 00:18:05,700 --> 00:18:08,760 '2 n + 1' over 6'. 356 00:18:08,760 --> 00:18:11,560 You see, the reason I bring this out is that I call this a 357 00:18:11,560 --> 00:18:13,090 contrived example. 358 00:18:13,090 --> 00:18:16,170 It is not the case where the mathematician would most 359 00:18:16,170 --> 00:18:19,320 likely have invented mathematical induction. 360 00:18:19,320 --> 00:18:21,440 The case where he would have invented mathematical 361 00:18:21,440 --> 00:18:24,450 induction is the case that we did earlier, for example, 362 00:18:24,450 --> 00:18:26,780 where we talk about the limit of a sum being 363 00:18:26,780 --> 00:18:27,510 the sum of the limits. 364 00:18:27,510 --> 00:18:30,690 You can actually see what's happening, how the truth that 365 00:18:30,690 --> 00:18:33,630 'k' implies the truth for 'k + 1'. 366 00:18:33,630 --> 00:18:36,340 And by the way, let me make one more aside here that I 367 00:18:36,340 --> 00:18:37,950 forgot to mention earlier. 368 00:18:37,950 --> 00:18:41,150 In our definition of mathematical induction, we 369 00:18:41,150 --> 00:18:44,090 said show the conjecture is true for 'n' equals 1. 370 00:18:44,090 --> 00:18:47,270 This was quite hypocritical because the very first example 371 00:18:47,270 --> 00:18:50,160 that I picked didn't even start until 'n' was 2. 372 00:18:50,160 --> 00:18:52,790 We talked about the limit of a sum being the sum of the 373 00:18:52,790 --> 00:18:55,250 limits, and the smallest sum we talked about was 374 00:18:55,250 --> 00:18:57,150 the sum of two terms. 375 00:18:57,150 --> 00:18:59,630 The point that I wanted to mention is, for example, 376 00:18:59,630 --> 00:19:03,230 suppose that the first number you can prove the conjecture 377 00:19:03,230 --> 00:19:05,820 for is, for example, 'n' equals 7. 378 00:19:05,820 --> 00:19:07,010 I don't know why I picked 7. 379 00:19:07,010 --> 00:19:07,720 I had to pick something. 380 00:19:07,720 --> 00:19:09,180 Let's just call it 'n' equals 7. 381 00:19:09,180 --> 00:19:12,640 Suppose I can also show that, if the conjecture is true for 382 00:19:12,640 --> 00:19:16,200 'k', it's true for 'k + 1'. 383 00:19:16,200 --> 00:19:20,060 Then, you see, what I can conclude is I can conclude 384 00:19:20,060 --> 00:19:23,390 that it's true for 'n' greater than or equal to 7. 385 00:19:23,390 --> 00:19:26,440 Namely, if it's true for 7, this says it will be true for 386 00:19:26,440 --> 00:19:28,310 one more than 7, which is 8. 387 00:19:28,310 --> 00:19:31,070 If it's true for 8, this says it will be true for 9. 388 00:19:31,070 --> 00:19:32,860 If it's true for 9, this says it will be 389 00:19:32,860 --> 00:19:34,730 true for 10, et cetera. 390 00:19:34,730 --> 00:19:36,270 And we go on this way. 391 00:19:36,270 --> 00:19:39,250 Now, again, this may be a very naive way of looking at it, 392 00:19:39,250 --> 00:19:42,630 but I always look at mathematical induction as a 393 00:19:42,630 --> 00:19:47,640 bunch of toy soldiers stacked up in a line in such a way 394 00:19:47,640 --> 00:19:50,910 that, if any one of the toy soldiers falls down, he knocks 395 00:19:50,910 --> 00:19:54,340 down the one that's immediately behind him, OK? 396 00:19:54,340 --> 00:19:56,250 You see what I'm driving at here? 397 00:19:56,250 --> 00:19:59,120 If the first falls-- and, by the way, that's a big if-- 398 00:19:59,120 --> 00:20:02,130 if the first falls, he knocks down the second. 399 00:20:02,130 --> 00:20:04,270 The second falling knocks down the third. 400 00:20:04,270 --> 00:20:06,430 The third falling knocks down the fourth. 401 00:20:06,430 --> 00:20:09,720 The fourth falling down knocks down the fifth, et cetera. 402 00:20:09,720 --> 00:20:11,920 Notice that, if the first one doesn't fall, 403 00:20:11,920 --> 00:20:12,920 none of them fall. 404 00:20:12,920 --> 00:20:16,350 Or, for that matter, going back to my previous analogy, 405 00:20:16,350 --> 00:20:19,590 if the seventh one is the first one that falls, all the 406 00:20:19,590 --> 00:20:22,330 ones behind him fall down. 407 00:20:22,330 --> 00:20:23,560 Now, be very careful. 408 00:20:23,560 --> 00:20:27,470 Mathematical induction doesn't say the first 50 fall down, or 409 00:20:27,470 --> 00:20:30,230 the first 100 fall down, it says they all 410 00:20:30,230 --> 00:20:31,490 have to fall down. 411 00:20:31,490 --> 00:20:35,120 For example, here's a case where several fall down, but, 412 00:20:35,120 --> 00:20:38,303 all of a sudden, one isn't knocked down by the one in 413 00:20:38,303 --> 00:20:38,890 front of him. 414 00:20:38,890 --> 00:20:41,220 In other words, what mathematical induction really 415 00:20:41,220 --> 00:20:46,460 involves is the idea not just that something is true, but 416 00:20:46,460 --> 00:20:50,100 that it's true because the previous one was true. 417 00:20:50,100 --> 00:20:54,570 You see, for example, is it possible that all of the 418 00:20:54,570 --> 00:20:58,700 soldiers fall down even though the one in front didn't knock 419 00:20:58,700 --> 00:21:00,040 the other guy down? 420 00:21:00,040 --> 00:21:02,510 I mean, it's possible all of these go down, but for 421 00:21:02,510 --> 00:21:03,600 different reasons. 422 00:21:03,600 --> 00:21:06,180 Mathematical induction says much more than that. 423 00:21:06,180 --> 00:21:09,910 Mathematical induction says yes, they all go down, but 424 00:21:09,910 --> 00:21:12,980 each goes down because of the one before. 425 00:21:12,980 --> 00:21:16,410 And, by the way, one more little aside. 426 00:21:16,410 --> 00:21:19,690 Notice that, in our analog, we assumed that 427 00:21:19,690 --> 00:21:21,930 there was a next one. 428 00:21:21,930 --> 00:21:22,970 You say, well, what do you mean you assume 429 00:21:22,970 --> 00:21:23,900 there was a next one? 430 00:21:23,900 --> 00:21:25,820 Obviously, there has to be a next one. 431 00:21:25,820 --> 00:21:29,610 But, the concept of next depends on whole numbers. 432 00:21:29,610 --> 00:21:33,510 For example, when you deal with fractions, 433 00:21:33,510 --> 00:21:34,840 there is no next one. 434 00:21:34,840 --> 00:21:36,260 Let me show you what I mean by that. 435 00:21:36,260 --> 00:21:38,190 Let's talk about the real numbers in general. 436 00:21:38,190 --> 00:21:39,480 On the number line, here's 0. 437 00:21:39,480 --> 00:21:42,750 What is the first fraction, the first real number? 438 00:21:42,750 --> 00:21:43,680 I don't care what you call it. 439 00:21:43,680 --> 00:21:47,190 What is the first number which is greater than 0? 440 00:21:47,190 --> 00:21:49,950 What is the first number which is greater than 0? 441 00:21:49,950 --> 00:21:52,680 And the answer is, there is none because 442 00:21:52,680 --> 00:21:54,540 whichever one you pick-- 443 00:21:54,540 --> 00:21:56,060 call it 'r'. 444 00:21:56,060 --> 00:21:59,130 Let 'r' stand for the first number which you think is 445 00:21:59,130 --> 00:22:01,430 bigger than 0, OK? 446 00:22:01,430 --> 00:22:04,855 How about 'r/2'? 447 00:22:04,855 --> 00:22:10,340 'r/2' is still bigger than 0, but 'r/2' is less than 'r'. 448 00:22:10,340 --> 00:22:14,340 In other words, given any number you pick that's bigger 449 00:22:14,340 --> 00:22:17,110 than 0, you can fit in another one. 450 00:22:17,110 --> 00:22:20,550 So, there is no number which is immediately next to 0. 451 00:22:20,550 --> 00:22:23,540 In other words, as a caution, notice that mathematical 452 00:22:23,540 --> 00:22:26,410 induction is used when we're dealing with a 453 00:22:26,410 --> 00:22:29,440 whole number of objects. 454 00:22:29,440 --> 00:22:34,280 Now, let's emphasize some of these little asides from a 455 00:22:34,280 --> 00:22:37,060 more specific point of view. 456 00:22:37,060 --> 00:22:40,150 Let me, first of all, give you an example in which something 457 00:22:40,150 --> 00:22:43,370 is true for a whole bunch of numbers, but, all of a sudden, 458 00:22:43,370 --> 00:22:45,100 isn't true in general. 459 00:22:45,100 --> 00:22:47,470 Now, how anybody ever stumbled across the example I'm going 460 00:22:47,470 --> 00:22:50,670 to give you next, I have no idea, but I find it's a very 461 00:22:50,670 --> 00:22:52,380 interesting concept. 462 00:22:52,380 --> 00:22:55,340 Let's look at this. 463 00:22:55,340 --> 00:22:59,120 Let's write down the following function: 'p' of 'n', where 464 00:22:59,120 --> 00:23:02,180 'n' is any positive whole number, will be defined to be 465 00:23:02,180 --> 00:23:05,530 'n' squared, minus 'n', plus 41. 466 00:23:05,530 --> 00:23:10,610 For example, 'p' of 1 will be 1 squared, minus 1, plus 41, 467 00:23:10,610 --> 00:23:12,510 which happens to be 41. 468 00:23:12,510 --> 00:23:15,950 41 happens to be a prime number: a number which has no 469 00:23:15,950 --> 00:23:18,780 factors other than itself and 1. 470 00:23:18,780 --> 00:23:20,540 Let's try 2 in here. 471 00:23:20,540 --> 00:23:26,570 2 squared, minus 2, plus 41, is 43: also a prime number. 472 00:23:26,570 --> 00:23:29,130 Let's try 3 in here. 473 00:23:29,130 --> 00:23:34,570 3 squared is 9, minus 3 is 6, plus 41 is 47: 474 00:23:34,570 --> 00:23:36,290 also a prime number. 475 00:23:36,290 --> 00:23:40,290 The amazing thing is that, as you go all the way through to 476 00:23:40,290 --> 00:23:44,110 40, you get nothing but prime numbers. 477 00:23:44,110 --> 00:23:47,530 And you say, ah, it's right so far, it must be 478 00:23:47,530 --> 00:23:48,790 right all the time. 479 00:23:48,790 --> 00:23:50,290 And this is wishful thinking. 480 00:23:50,290 --> 00:23:53,150 This is like the man who wants to count the deck of cards to 481 00:23:53,150 --> 00:23:55,890 see if the cards are all there, and he says one, two, 482 00:23:55,890 --> 00:23:57,090 three, four, five. 483 00:23:57,090 --> 00:23:58,620 Well, so far, so good. 484 00:23:58,620 --> 00:24:00,450 They must all be here. 485 00:24:00,450 --> 00:24:02,430 It's like falling off the Empire State building, and 486 00:24:02,430 --> 00:24:04,900 halfway down, somebody says, "How you doing?" You say, "So 487 00:24:04,900 --> 00:24:07,490 far, so good." No, we're in a little bit of trouble here 488 00:24:07,490 --> 00:24:11,510 because, as soon as we pick 'n' to be 41, watch what 489 00:24:11,510 --> 00:24:12,520 happens here. 490 00:24:12,520 --> 00:24:18,890 This becomes 41 squared, minus 41, plus 41. 491 00:24:18,890 --> 00:24:22,750 And this is 41 squared, which, obviously, is not a prime. 492 00:24:22,750 --> 00:24:25,000 It's 41 times 41. 493 00:24:25,000 --> 00:24:27,700 And here is an interesting example where a certain 494 00:24:27,700 --> 00:24:31,750 formula generates nothing but primes for the first 40 495 00:24:31,750 --> 00:24:35,190 integers, but fails on the 41st integer. 496 00:24:35,190 --> 00:24:37,020 Why did this happen? 497 00:24:37,020 --> 00:24:41,080 Because, evidently, the fact that this was a prime in no 498 00:24:41,080 --> 00:24:43,600 way depended structurally on the fact that 499 00:24:43,600 --> 00:24:44,930 this one was a prime. 500 00:24:44,930 --> 00:24:47,990 There is our mathematical induction again, that not only 501 00:24:47,990 --> 00:24:51,730 must the conjecture be true, but it must follow inescapably 502 00:24:51,730 --> 00:24:53,510 from the case before. 503 00:24:53,510 --> 00:24:56,010 If you'd like a more fascinating, realistic 504 00:24:56,010 --> 00:24:59,000 example, it's something that we call the unique 505 00:24:59,000 --> 00:25:02,890 factorization theorem of elementary number theory. 506 00:25:02,890 --> 00:25:08,470 This says that every positive whole number greater than one 507 00:25:08,470 --> 00:25:12,480 can be factored uniquely into a product of primes, unique up 508 00:25:12,480 --> 00:25:14,640 to the order in which you write them. 509 00:25:14,640 --> 00:25:17,260 For example, 2 is already a prime, it's 2. 510 00:25:17,260 --> 00:25:19,600 3 is already a prime: 3. 511 00:25:19,600 --> 00:25:22,810 4 can be factored as 2 times 2. 512 00:25:22,810 --> 00:25:24,900 5 can be factored as-- 513 00:25:24,900 --> 00:25:27,220 well, it's already a prime, it's 5. 514 00:25:27,220 --> 00:25:28,780 6 can be written as, what? 515 00:25:28,780 --> 00:25:30,440 2 times 3. 516 00:25:30,440 --> 00:25:32,300 7 can be written, of course, as 7. 517 00:25:32,300 --> 00:25:33,670 It's already a prime. 518 00:25:33,670 --> 00:25:37,740 8 is 2 times 2 times 2. 519 00:25:37,740 --> 00:25:40,090 9 is 3 times 3. 520 00:25:40,090 --> 00:25:44,860 10 is 2 times 5, and 11 is already a prime. 521 00:25:44,860 --> 00:25:46,280 Well, this doesn't prove anything. 522 00:25:46,280 --> 00:25:48,820 I'm just trying to demonstrate what the theorem says. 523 00:25:48,820 --> 00:25:50,460 The interesting thing is do you 524 00:25:50,460 --> 00:25:53,510 notice how 'n + 1' factors-- 525 00:25:53,510 --> 00:25:56,470 I don't know what phrase to use here, but let's call it 526 00:25:56,470 --> 00:25:58,860 considerably different than 'n'. 527 00:25:58,860 --> 00:26:02,060 In other words, look at what happened to 10 when I 528 00:26:02,060 --> 00:26:03,190 added 1 onto it. 529 00:26:03,190 --> 00:26:05,170 It factored into 2 times 5. 530 00:26:05,170 --> 00:26:08,710 I add 1 onto it, and, all of a sudden, the factorization 531 00:26:08,710 --> 00:26:10,420 properties change. 532 00:26:10,420 --> 00:26:11,870 Let me give you an example. 533 00:26:11,870 --> 00:26:13,770 Look at 59. 534 00:26:13,770 --> 00:26:16,240 59 happens to be a prime. 535 00:26:16,240 --> 00:26:18,390 Look at 61. 536 00:26:18,390 --> 00:26:21,140 61 also happens to be a prime. 537 00:26:21,140 --> 00:26:24,350 By the way, for those of you are number theory buffs, these 538 00:26:24,350 --> 00:26:26,410 are called twin primes. 539 00:26:26,410 --> 00:26:29,770 Consecutive odd numbers, which are both prime, are called 540 00:26:29,770 --> 00:26:31,580 twin primes. 541 00:26:31,580 --> 00:26:34,670 By the way, 2 is the only even prime, of course, because, if 542 00:26:34,670 --> 00:26:38,550 a number is greater than 2, and it's even, it's 543 00:26:38,550 --> 00:26:39,540 divisible by 2. 544 00:26:39,540 --> 00:26:41,525 So, 2 is the only even prime. 545 00:26:41,525 --> 00:26:42,850 Twin primes are, what? 546 00:26:42,850 --> 00:26:45,520 Consecutive odd numbers both of which are primes. 547 00:26:45,520 --> 00:26:49,230 So, 59 and 61 are a pair of twin primes. 548 00:26:49,230 --> 00:26:52,620 One might intuitively suspect, therefore, that the number 549 00:26:52,620 --> 00:26:56,070 between them must be sort of a prime too, or 550 00:26:56,070 --> 00:26:57,200 whatever that means. 551 00:26:57,200 --> 00:26:59,830 Obviously, it can't be a prime because in between them comes 552 00:26:59,830 --> 00:27:02,600 60, which is, in particular, an even number. 553 00:27:02,600 --> 00:27:06,200 But look at all the nice factors that sixty has. 554 00:27:06,200 --> 00:27:07,050 60 is, what? 555 00:27:07,050 --> 00:27:11,070 It's one more than 59, one less than 61, but look at how 556 00:27:11,070 --> 00:27:12,320 different it factors. 557 00:27:12,320 --> 00:27:14,990 In fact, a pseudo induction-type thing is look 558 00:27:14,990 --> 00:27:25,150 at the factors of 60: 1, 2, 3, 4, 5, 6. 559 00:27:25,150 --> 00:27:25,600 We say, what? 560 00:27:25,600 --> 00:27:27,700 7, experimental error? 561 00:27:27,700 --> 00:27:28,120 No, no. 562 00:27:28,120 --> 00:27:30,850 But, this is not induction, by the way. 563 00:27:30,850 --> 00:27:35,730 The fact that 1, 2, 3, 4, 5, and 6 are all factors of 60 564 00:27:35,730 --> 00:27:38,420 does not mean that 7 is going to be a factor of 60. 565 00:27:38,420 --> 00:27:41,370 We do not say just because it works so far, it's going to 566 00:27:41,370 --> 00:27:42,230 keep working. 567 00:27:42,230 --> 00:27:44,290 And I'm not going to belabor this point anymore. 568 00:27:44,290 --> 00:27:48,610 Suffice it to say that look at how differently 60 factors 569 00:27:48,610 --> 00:27:51,230 compare to the number the came just before it and the number 570 00:27:51,230 --> 00:27:52,660 that came just after it. 571 00:27:52,660 --> 00:27:55,610 In other words, assuming that the unique factorization 572 00:27:55,610 --> 00:28:00,350 theorem is true, the truth for 'n + 1', somehow or other, has 573 00:28:00,350 --> 00:28:02,900 nothing to do with the truth for 'n'. 574 00:28:02,900 --> 00:28:05,760 And you see this is, again, another weakness of, what's 575 00:28:05,760 --> 00:28:07,180 called, induction. 576 00:28:07,180 --> 00:28:10,650 Well, so, that's a weakness of induction. 577 00:28:10,650 --> 00:28:13,710 The point that I'm making is why shouldn't induction have 578 00:28:13,710 --> 00:28:14,770 some weaknesses? 579 00:28:14,770 --> 00:28:17,900 After all, if it could solve every problem, what we would 580 00:28:17,900 --> 00:28:21,040 do is have a calculus book that was three pages long. 581 00:28:21,040 --> 00:28:22,280 It would be called, The Principle of 582 00:28:22,280 --> 00:28:23,840 Mathematical Induction. 583 00:28:23,840 --> 00:28:27,740 When we solved that problem by induction, everything else 584 00:28:27,740 --> 00:28:28,940 would be done. 585 00:28:28,940 --> 00:28:30,450 No, there are problems that do not lend 586 00:28:30,450 --> 00:28:31,820 themselves to induction. 587 00:28:31,820 --> 00:28:35,130 In summary, induction is a particularly effective 588 00:28:35,130 --> 00:28:39,090 technique which one uses to prove that something is true 589 00:28:39,090 --> 00:28:43,870 for all whole numbers provided that one has a suspicion that 590 00:28:43,870 --> 00:28:45,600 this thing is true in the first place. 591 00:28:45,600 --> 00:28:48,990 And secondly that, even if the suspicion is true, the truth 592 00:28:48,990 --> 00:28:52,890 for the 'n' plus first case follows inescapably from the 593 00:28:52,890 --> 00:28:55,350 truth for the n-th case. 594 00:28:55,350 --> 00:28:58,940 At any rate, this completes our lecture for today. 595 00:28:58,940 --> 00:29:02,965 And, until next time, good bye. 596 00:29:02,965 --> 00:29:05,940 MALE VOICE: Funding for the publication of this video was 597 00:29:05,940 --> 00:29:10,660 provided by the Gabriella and Paul Rosenbaum Foundation. 598 00:29:10,660 --> 00:29:14,830 Help OCW continue to provide free and open access to MIT 599 00:29:14,830 --> 00:29:19,030 courses by making a donation at ocw.mit.edu/donate.