1 00:00:00,000 --> 00:00:00,040 2 00:00:00,040 --> 00:00:02,400 The following content is provided under a Creative 3 00:00:02,400 --> 00:00:03,690 Commons license. 4 00:00:03,690 --> 00:00:06,630 Your support will help MIT OpenCourseWare continue to 5 00:00:06,630 --> 00:00:09,990 offer high-quality educational resources for free. 6 00:00:09,990 --> 00:00:12,830 To make a donation or to view additional materials from 7 00:00:12,830 --> 00:00:16,760 hundreds of MIT courses, visit MIT OpenCourseWare at 8 00:00:16,760 --> 00:00:18,010 ocw.mit.edu. 9 00:00:18,010 --> 00:00:34,496 10 00:00:34,496 --> 00:00:35,468 PROFESSOR: Hi. 11 00:00:35,468 --> 00:00:39,810 Our lecture for today probably should be entitled it 12 00:00:39,810 --> 00:00:42,690 should've been functions, but it's analytic geometry 13 00:00:42,690 --> 00:00:46,950 instead, or a picture is worth a thousand words. 14 00:00:46,950 --> 00:00:50,930 What we hope to do today is to establish the fact that 15 00:00:50,930 --> 00:00:54,410 whereas in the study of calculus when we deal with 16 00:00:54,410 --> 00:00:57,340 rate of change we are interested in analytical 17 00:00:57,340 --> 00:01:01,600 terms, that more often than not, we prefer to visualize 18 00:01:01,600 --> 00:01:05,770 things more intuitively in terms of a graph or other 19 00:01:05,770 --> 00:01:10,160 suitable visual aid, and that actually, this is not quite as 20 00:01:10,160 --> 00:01:14,950 alien or as profound as it may at first glance seem. 21 00:01:14,950 --> 00:01:17,270 Consider, for example, the businessman who 22 00:01:17,270 --> 00:01:20,780 says profits rose. 23 00:01:20,780 --> 00:01:21,770 Profits rose. 24 00:01:21,770 --> 00:01:25,870 Now, you know, profits don't rise unless the safe blows up 25 00:01:25,870 --> 00:01:26,940 or something like this. 26 00:01:26,940 --> 00:01:31,650 What profits do is they increase or they decrease. 27 00:01:31,650 --> 00:01:35,940 The reason that we say profits rise is that when the profits 28 00:01:35,940 --> 00:01:39,920 are increasing, if we are plotting profit in terms of 29 00:01:39,920 --> 00:01:46,690 time, the resulting graph shows a rising tendency. 30 00:01:46,690 --> 00:01:50,670 As the profit increases, the curve rises. 31 00:01:50,670 --> 00:01:54,080 And in other words then, we begin to establish the feeling 32 00:01:54,080 --> 00:01:58,510 that we can identify the analytic term increasing with 33 00:01:58,510 --> 00:02:00,700 the geometric term rising. 34 00:02:00,700 --> 00:02:04,410 And this identification, whereby difficult arithmetic 35 00:02:04,410 --> 00:02:08,100 concepts are visualized pictorially, is something that 36 00:02:08,100 --> 00:02:12,270 begins not only very early in the history of man, but very 37 00:02:12,270 --> 00:02:15,560 early in the development of the mathematics curriculum. 38 00:02:15,560 --> 00:02:19,820 Oh, as a case in point, consider the problem of 5 39 00:02:19,820 --> 00:02:24,210 divided by 3 versus 6 divided by 3. 40 00:02:24,210 --> 00:02:26,920 I remember when I was in grade school that this particular 41 00:02:26,920 --> 00:02:30,290 problem always seemed more appealing to me than this 42 00:02:30,290 --> 00:02:34,440 problem, that 6 divided by 3 seemed natural, but 5 divided 43 00:02:34,440 --> 00:02:35,920 by 3 didn't. 44 00:02:35,920 --> 00:02:38,810 And the reason was is that in terms of visualizing tally 45 00:02:38,810 --> 00:02:44,370 marks, it was much easier to see how you divide six tallies 46 00:02:44,370 --> 00:02:47,840 into three groups than five tallies into three groups. 47 00:02:47,840 --> 00:02:52,240 However, as soon as we pick a length as our 48 00:02:52,240 --> 00:02:54,640 model, the idea is this. 49 00:02:54,640 --> 00:02:59,510 Either one can divide a line into three parts of equal 50 00:02:59,510 --> 00:03:02,210 length or one can't divide the line. 51 00:03:02,210 --> 00:03:05,135 Now, if I can geometrically divide this line into three 52 00:03:05,135 --> 00:03:09,010 equal parts, and in plain geometry we learn to do this, 53 00:03:09,010 --> 00:03:12,940 then the fact is that I can divide this line segment into 54 00:03:12,940 --> 00:03:15,570 three equal parts regardless of how long this 55 00:03:15,570 --> 00:03:16,930 line happens to be. 56 00:03:16,930 --> 00:03:22,760 Oh, to be sure, if this line happens to be 6 units long, 57 00:03:22,760 --> 00:03:25,310 this point is named 2. 58 00:03:25,310 --> 00:03:29,440 And if, on the other hand, the line happens to be only 5 59 00:03:29,440 --> 00:03:37,130 inches long, the resulting point is named 5/3. 60 00:03:37,130 --> 00:03:41,290 But notice that in either case, I can in a very natural 61 00:03:41,290 --> 00:03:47,110 way define or identify either ratio as a point on the line. 62 00:03:47,110 --> 00:03:52,240 And this idea of identifying numerical concepts called 63 00:03:52,240 --> 00:03:56,170 numbers with geometric concepts called points is a 64 00:03:56,170 --> 00:04:00,130 very old device and a device that was used and still is 65 00:04:00,130 --> 00:04:03,210 used in the curriculum today under the name of the number 66 00:04:03,210 --> 00:04:07,470 line, under the name of graphs, and what we will use 67 00:04:07,470 --> 00:04:11,620 as a fundamental building block as our course develops. 68 00:04:11,620 --> 00:04:14,940 Now, you know, in the same way that we can think of a single 69 00:04:14,940 --> 00:04:19,640 number as being a point on the line, we can think of a pair 70 00:04:19,640 --> 00:04:23,380 of numbers, an ordered pair of numbers, as being a 71 00:04:23,380 --> 00:04:25,210 point in the plane. 72 00:04:25,210 --> 00:04:29,150 This is Descartes geometry, which we can call coordinate 73 00:04:29,150 --> 00:04:33,200 geometry, the idea being that in the same way as we can 74 00:04:33,200 --> 00:04:37,170 locate a number of along the x-axis, shall we say, we could 75 00:04:37,170 --> 00:04:41,970 have located a numbered pair as a point in the plane. 76 00:04:41,970 --> 00:04:45,850 Namely, 2 comma 3 would mean the point whose x-coordinate 77 00:04:45,850 --> 00:04:49,230 was 2 and whose y-coordinate was 3. 78 00:04:49,230 --> 00:04:53,610 By the way, the reason that we say ordered pairs is, if you 79 00:04:53,610 --> 00:04:57,490 observe, 2 comma 3 and 3 comma 2 happen to 80 00:04:57,490 --> 00:04:58,950 be different pairs. 81 00:04:58,950 --> 00:05:02,580 And notice again how vividly the geometric interpretation 82 00:05:02,580 --> 00:05:03,640 of this is. 83 00:05:03,640 --> 00:05:07,130 Namely, in terms of locating a point in space, it's obvious 84 00:05:07,130 --> 00:05:10,870 that the point named 2 comma 3 is not the same as the point 85 00:05:10,870 --> 00:05:12,760 named 3 comma 2. 86 00:05:12,760 --> 00:05:14,790 Again, the important thing is this. 87 00:05:14,790 --> 00:05:18,400 When I think of 5 divided by 3, when I think of the ordered 88 00:05:18,400 --> 00:05:23,460 pair 2 comma 3, I do not have to think of a picture. 89 00:05:23,460 --> 00:05:26,010 I can think of these things analytically. 90 00:05:26,010 --> 00:05:29,590 But the picture gives me certain insights that will 91 00:05:29,590 --> 00:05:33,000 help me with my intuition, an aid that I don't want to 92 00:05:33,000 --> 00:05:34,120 relinquish. 93 00:05:34,120 --> 00:05:36,790 For example, going back to the graph again. 94 00:05:36,790 --> 00:05:41,190 Thinking of the analytic term greater than, notice how much 95 00:05:41,190 --> 00:05:49,470 easier it is to think of, for example, higher than, see, one 96 00:05:49,470 --> 00:05:53,308 point being higher than another or to the right of. 97 00:05:53,308 --> 00:05:58,680 98 00:05:58,680 --> 00:06:02,650 You see, geometric concepts to name analytic statements, or 99 00:06:02,650 --> 00:06:04,150 instead of increasing, as we mentioned 100 00:06:04,150 --> 00:06:06,450 before, to say rising. 101 00:06:06,450 --> 00:06:10,080 And there will be many, many more such identifications as 102 00:06:10,080 --> 00:06:12,260 we go along with our course. 103 00:06:12,260 --> 00:06:14,660 At any rate, let's continue to see then what is the 104 00:06:14,660 --> 00:06:18,180 relationship then between functions that we talked about 105 00:06:18,180 --> 00:06:19,720 and graphs? 106 00:06:19,720 --> 00:06:23,270 The idea is something like this. 107 00:06:23,270 --> 00:06:27,060 Let's return to our friend of the first lecture: s equals 108 00:06:27,060 --> 00:06:28,730 16t squared. 109 00:06:28,730 --> 00:06:32,720 We can think of a distance machine being the function 110 00:06:32,720 --> 00:06:40,590 where the input will be time and the output will be what? 111 00:06:40,590 --> 00:06:44,130 The square of the input multiplied by 16. 112 00:06:44,130 --> 00:06:49,150 Observe that from this, I do not have to have any picture 113 00:06:49,150 --> 00:06:51,590 to understand what's happening here. 114 00:06:51,590 --> 00:06:55,820 Namely, I can measure an input, measure an output, and 115 00:06:55,820 --> 00:06:58,420 observe analytically what is happening. 116 00:06:58,420 --> 00:07:02,620 However, as we saw last time, our graph sort of shows us at 117 00:07:02,620 --> 00:07:05,820 a glance what seems to be happening, that we can 118 00:07:05,820 --> 00:07:10,380 identify rising and falling with increasing and decreasing 119 00:07:10,380 --> 00:07:11,930 and things of this type. 120 00:07:11,930 --> 00:07:15,020 We will explore this, of course, in much more detail as 121 00:07:15,020 --> 00:07:17,820 we continue in our course. 122 00:07:17,820 --> 00:07:21,730 By the way, there's no reason why the input has to be a 123 00:07:21,730 --> 00:07:22,750 single number. 124 00:07:22,750 --> 00:07:25,450 For example, why couldn't the input be an 125 00:07:25,450 --> 00:07:27,350 ordered pair of numbers? 126 00:07:27,350 --> 00:07:30,310 Among other things, let's take a simple geometric example. 127 00:07:30,310 --> 00:07:35,380 Consider, for example, finding the volume of a cylinder in 128 00:07:35,380 --> 00:07:38,720 terms of the radius of its base and the height. 129 00:07:38,720 --> 00:07:41,790 We know from solid geometry that the volume is 130 00:07:41,790 --> 00:07:44,220 pi r squared h. 131 00:07:44,220 --> 00:07:47,970 We could therefore think of a volume machine where the input 132 00:07:47,970 --> 00:07:53,210 is the ordered pair r comma h, and the output is the single 133 00:07:53,210 --> 00:07:56,310 number pi r squared h. 134 00:07:56,310 --> 00:07:59,650 By the way, notice here the meaning of ordered pair. 135 00:07:59,650 --> 00:08:04,030 You see, if the pair 2 comma 3 goes into the machine, notice 136 00:08:04,030 --> 00:08:05,790 that the recipe here says what? 137 00:08:05,790 --> 00:08:08,120 You square the first member of the pair. 138 00:08:08,120 --> 00:08:11,420 In other words, if 2 comma 3 is the input, we square 2, 139 00:08:11,420 --> 00:08:15,790 which is 4, multiplied by 3, which is 12, and 12 times pi, 140 00:08:15,790 --> 00:08:17,950 of course, is 12 pi. 141 00:08:17,950 --> 00:08:21,820 On the other hand, if the input is 3 comma 2, the first 142 00:08:21,820 --> 00:08:22,810 number is 3. 143 00:08:22,810 --> 00:08:25,480 Our recipe squares the first number. 144 00:08:25,480 --> 00:08:31,640 That would be 9, times 2 is 18, times pi is 18 pi. 145 00:08:31,640 --> 00:08:35,659 But again, observe that I at no time needed a picture to 146 00:08:35,659 --> 00:08:37,750 visualize what was happening here. 147 00:08:37,750 --> 00:08:40,789 Of course, if I wanted a picture, I could try to plot 148 00:08:40,789 --> 00:08:44,940 this also, but notice now that my graph would probably need 149 00:08:44,940 --> 00:08:46,710 three dimensions to draw. 150 00:08:46,710 --> 00:08:48,820 And why would it need three dimensions? 151 00:08:48,820 --> 00:08:52,930 Well, notice that my input has two independent measurements r 152 00:08:52,930 --> 00:08:57,250 and h, and therefore, I would need two dimensions just to 153 00:08:57,250 --> 00:08:58,860 take care of r and h. 154 00:08:58,860 --> 00:09:02,380 Then I would need a third dimension to plot v. 155 00:09:02,380 --> 00:09:04,750 And by the way, notice the next stage. 156 00:09:04,750 --> 00:09:08,520 If I had an input that consisted of three independent 157 00:09:08,520 --> 00:09:12,210 measurements, this would still make sense, but now I would be 158 00:09:12,210 --> 00:09:14,870 at a loss for the picture. 159 00:09:14,870 --> 00:09:17,190 In other words, what I'm trying to bring out next is 160 00:09:17,190 --> 00:09:21,010 the fact that whereas pictures are a tremendous help, maybe a 161 00:09:21,010 --> 00:09:24,900 second subtitle to our lecture should've been a picture is 162 00:09:24,900 --> 00:09:27,505 worth a thousand words provided you 163 00:09:27,505 --> 00:09:28,970 can could draw it. 164 00:09:28,970 --> 00:09:32,610 Because, you see, if we needed three independent dimensions 165 00:09:32,610 --> 00:09:36,570 to locate the input and then a fourth one to locate the 166 00:09:36,570 --> 00:09:37,995 output, how would we draw the picture? 167 00:09:37,995 --> 00:09:40,820 By the way, this happens in high school algebra again, if 168 00:09:40,820 --> 00:09:42,390 you want to see the analogy. 169 00:09:42,390 --> 00:09:44,140 Look at, for example, the algebraic 170 00:09:44,140 --> 00:09:46,020 equation a plus b squared. 171 00:09:46,020 --> 00:09:50,230 We learned in algebra that this is a squared plus 2ab 172 00:09:50,230 --> 00:09:51,920 plus b squared. 173 00:09:51,920 --> 00:09:56,020 Observe that we do not need to have a picture to understand 174 00:09:56,020 --> 00:09:57,460 how this works. 175 00:09:57,460 --> 00:10:03,360 Oh, to be sure, if we had a picture, we get a tremendous 176 00:10:03,360 --> 00:10:05,450 amount of insight as to what's happening here. 177 00:10:05,450 --> 00:10:10,060 Namely, let's visualize a square whose side is a plus b. 178 00:10:10,060 --> 00:10:13,920 On the one hand, you see, the area of the square would be a 179 00:10:13,920 --> 00:10:17,080 plus b squared, you see, the side squared. 180 00:10:17,080 --> 00:10:20,020 On the other hand, if we now subdivide this figure this 181 00:10:20,020 --> 00:10:23,910 way, we see that that same square is made up of four 182 00:10:23,910 --> 00:10:29,020 pieces having one piece of area a squared, two pieces of 183 00:10:29,020 --> 00:10:32,840 area ab, and one piece of area b squared. 184 00:10:32,840 --> 00:10:35,270 And so we see, on the other hand, that the area of the 185 00:10:35,270 --> 00:10:40,470 square is a squared plus 2ab plus b squared. 186 00:10:40,470 --> 00:10:43,990 And so again, observe that whereas this stands on its own 187 00:10:43,990 --> 00:10:47,020 two legs, the picture helps us quite a bit. 188 00:10:47,020 --> 00:10:51,430 By the way, we could continue this with a cube over here. 189 00:10:51,430 --> 00:10:55,950 Namely, it turns out that a plus b cubed is a cubed plus 190 00:10:55,950 --> 00:11:01,600 3a squared b plus 3ab squared plus b cubed. 191 00:11:01,600 --> 00:11:04,450 And again, if we wanted to, and we won't take the time 192 00:11:04,450 --> 00:11:08,970 here, but if we wanted to, we could now draw a cube whose 193 00:11:08,970 --> 00:11:10,670 side is a plus b. 194 00:11:10,670 --> 00:11:14,020 Namely, we could take the same diagram that we had before and 195 00:11:14,020 --> 00:11:15,680 now make a third dimension to it. 196 00:11:15,680 --> 00:11:18,210 And if you did that, you would see what? 197 00:11:18,210 --> 00:11:21,850 That the cube whose volume is a plus b cubed is divided into 198 00:11:21,850 --> 00:11:27,080 eight pieces, one of size a by a by a, three of size a by a 199 00:11:27,080 --> 00:11:31,740 by b, three of size a by b by b, and one of 200 00:11:31,740 --> 00:11:34,660 size b by b by b. 201 00:11:34,660 --> 00:11:38,540 Again, the picture is a tremendous visual aid. 202 00:11:38,540 --> 00:11:40,620 Now, the key step is this. 203 00:11:40,620 --> 00:11:44,410 If we were to now give in to our geometric intuition and 204 00:11:44,410 --> 00:11:47,720 say, lookit, why worry about algebra when it's so much 205 00:11:47,720 --> 00:11:51,110 easier to do this thing by geometry, the counterexample 206 00:11:51,110 --> 00:11:54,410 is consider a plus b to the fourth power. 207 00:11:54,410 --> 00:11:57,800 Now, by the binomial theorem, and I might add, the same 208 00:11:57,800 --> 00:12:01,150 binomial theorem that allowed us to get these results, we 209 00:12:01,150 --> 00:12:02,880 can also write that this is what? 210 00:12:02,880 --> 00:12:08,840 It's a is the fourth plus 4a cubed b plus 6a squared b 211 00:12:08,840 --> 00:12:14,760 squared plus 4ab cubed plus b to the fourth. 212 00:12:14,760 --> 00:12:17,730 Now, it's not important how we get this result. 213 00:12:17,730 --> 00:12:20,640 The important point is that analytically, we can raise a 214 00:12:20,640 --> 00:12:23,770 number to the fourth power just as easily as we can to 215 00:12:23,770 --> 00:12:26,160 the third power or the second power. 216 00:12:26,160 --> 00:12:30,800 The only difference is that in the case of the third power, 217 00:12:30,800 --> 00:12:32,910 we had a picture that we could use. 218 00:12:32,910 --> 00:12:36,910 In the fourth power case, we didn't have a picture. 219 00:12:36,910 --> 00:12:39,660 And what a tragedy it would have been to say, hey, we 220 00:12:39,660 --> 00:12:41,740 can't solve this problem because we 221 00:12:41,740 --> 00:12:43,240 can't draw the picture. 222 00:12:43,240 --> 00:12:46,350 And by the way, as a rather interesting aside, notice the 223 00:12:46,350 --> 00:12:49,230 geometric influence on how we read this. 224 00:12:49,230 --> 00:12:52,260 This is called a plus b to the fourth power. 225 00:12:52,260 --> 00:12:55,490 But somehow or other, we don't call this one a plus b to the 226 00:12:55,490 --> 00:12:56,310 third power. 227 00:12:56,310 --> 00:13:01,440 We call it a plus b cubed, suggesting the geometric 228 00:13:01,440 --> 00:13:03,110 configuration of the cube. 229 00:13:03,110 --> 00:13:06,650 And here we don't usually say a plus b to the second power. 230 00:13:06,650 --> 00:13:10,220 We say a plus b squared. 231 00:13:10,220 --> 00:13:14,240 You see, the idea is that when the picture is available, it 232 00:13:14,240 --> 00:13:17,740 gives us a tremendous insight as to what can be done, and it 233 00:13:17,740 --> 00:13:21,440 helps us learn to visualize what's happening analytically. 234 00:13:21,440 --> 00:13:24,830 In fact, what usually happens is we use the picture to 235 00:13:24,830 --> 00:13:27,730 justify what's happening analytically when we can see 236 00:13:27,730 --> 00:13:30,490 the picture and then just carry the analytic part 237 00:13:30,490 --> 00:13:33,690 through unimpeded in the case where we 238 00:13:33,690 --> 00:13:35,140 can't draw the picture. 239 00:13:35,140 --> 00:13:37,050 Let me give you another example of this. 240 00:13:37,050 --> 00:13:38,800 Let's look at the following set. 241 00:13:38,800 --> 00:13:41,790 That also should review our language of sets for us. 242 00:13:41,790 --> 00:13:45,710 Let S be the set of all ordered pairs x comma y such 243 00:13:45,710 --> 00:13:49,150 that x squared plus y squared equals 25. 244 00:13:49,150 --> 00:13:54,910 Question: Does the ordered pair 3 comma 4 belong to S? 245 00:13:54,910 --> 00:13:56,470 Answer: Yes. 246 00:13:56,470 --> 00:13:57,360 How do we know? 247 00:13:57,360 --> 00:13:59,460 Well, we have a test for membership. 248 00:13:59,460 --> 00:14:00,940 We're supposed to do what? 249 00:14:00,940 --> 00:14:05,640 Square each of the entries, each of the numbers, add them, 250 00:14:05,640 --> 00:14:08,880 and if the answer is 25, then that ordered pair belongs to 251 00:14:08,880 --> 00:14:14,610 S. 3 squared plus 4 squared is 25, so this pair belongs to S. 252 00:14:14,610 --> 00:14:16,120 How about 1 comma 2? 253 00:14:16,120 --> 00:14:21,260 Well, 1 squared plus 2 squared is 1 plus 4, which is 5. 254 00:14:21,260 --> 00:14:26,920 5 is not equal to 25, so 1 comma 2 does not belong to S. 255 00:14:26,920 --> 00:14:30,070 Well, did we need any geometry to be able to 256 00:14:30,070 --> 00:14:31,860 visualize this result? 257 00:14:31,860 --> 00:14:35,730 Hopefully, one did not need any geometry to 258 00:14:35,730 --> 00:14:37,380 visualize this result. 259 00:14:37,380 --> 00:14:40,150 On the other hand then, what does it mean in analytic 260 00:14:40,150 --> 00:14:44,100 geometry when we say that x squared plus y squared equals 261 00:14:44,100 --> 00:14:46,580 25 is a circle? 262 00:14:46,580 --> 00:14:51,040 As badly as I draw, x squared plus y squared equals 25 looks 263 00:14:51,040 --> 00:14:54,590 less like a circle than the circle I drew over here. 264 00:14:54,590 --> 00:14:57,030 You see, what we really mean is this. 265 00:14:57,030 --> 00:15:01,670 Consider all the points in the plane x comma y for which x 266 00:15:01,670 --> 00:15:04,390 squared plus y squared equals 25. 267 00:15:04,390 --> 00:15:08,480 These are precisely the points on this particular circle. 268 00:15:08,480 --> 00:15:11,200 And the easiest way to see that, of course, is that since 269 00:15:11,200 --> 00:15:15,850 the radius of the circle is 5 and the point x comma y means 270 00:15:15,850 --> 00:15:18,980 that this length is x and this length is y, notice that from 271 00:15:18,980 --> 00:15:22,800 the Pythagorean theorem, we see it once, that x squared 272 00:15:22,800 --> 00:15:26,970 plus y squared equals 25. 273 00:15:26,970 --> 00:15:31,940 Now again, the solution set to this equation is our set S 274 00:15:31,940 --> 00:15:34,510 whether we're thinking of this thing algebraically or 275 00:15:34,510 --> 00:15:35,640 geometrically. 276 00:15:35,640 --> 00:15:39,400 On the other hand, watch what our picture seems to give us 277 00:15:39,400 --> 00:15:40,890 that we didn't have before. 278 00:15:40,890 --> 00:15:44,380 Let's return to our point 1 comma 2, which we saw didn't 279 00:15:44,380 --> 00:15:47,010 belong to S. Well, look at this. 280 00:15:47,010 --> 00:15:48,630 Where would 1 comma 2 be? 281 00:15:48,630 --> 00:15:52,560 1 comma 2 would be inside the circle. 282 00:15:52,560 --> 00:15:53,890 Why is that? 283 00:15:53,890 --> 00:16:01,080 Because, you see, if we take the point 1 comma 2, if we 284 00:16:01,080 --> 00:16:04,470 take, say, for example, the point 1 comma 2, notice that 285 00:16:04,470 --> 00:16:07,930 the distance from the origin to the point 1 comma 2 286 00:16:07,930 --> 00:16:09,080 is less than 5. 287 00:16:09,080 --> 00:16:12,460 In other words, the distance is less than 5 so the square 288 00:16:12,460 --> 00:16:14,660 of the distance is less than 25. 289 00:16:14,660 --> 00:16:18,550 In other words, not only can we say that 1 comma 2 does not 290 00:16:18,550 --> 00:16:21,420 belong to S, which we could have said without the picture, 291 00:16:21,420 --> 00:16:24,080 we can now say what? 292 00:16:24,080 --> 00:16:26,410 1 comma 2 is-- 293 00:16:26,410 --> 00:16:28,880 and notice the geometric language here-- 294 00:16:28,880 --> 00:16:30,780 is inside the circle. 295 00:16:30,780 --> 00:16:34,590 296 00:16:34,590 --> 00:16:36,310 In other words, the study of 297 00:16:36,310 --> 00:16:39,210 inequalities can now be reduced. 298 00:16:39,210 --> 00:16:42,325 Instead of talking about less than and greater than, we can 299 00:16:42,325 --> 00:16:46,060 now talk about such things as inside and outside. 300 00:16:46,060 --> 00:16:48,520 You see, inside the circle, which is a simple geometric 301 00:16:48,520 --> 00:16:50,340 concept, just means what? 302 00:16:50,340 --> 00:16:53,530 A set of all points for which x squared plus y squared is 303 00:16:53,530 --> 00:16:54,920 less than 25. 304 00:16:54,920 --> 00:16:58,250 Outside that circle, x squared plus y squared is 305 00:16:58,250 --> 00:16:59,700 greater than 25. 306 00:16:59,700 --> 00:17:04,040 On the circle, x squared plus y squared equals 25. 307 00:17:04,040 --> 00:17:09,490 Again, a nice identification between numbers and pictures, 308 00:17:09,490 --> 00:17:11,890 analysis and geometry. 309 00:17:11,890 --> 00:17:16,510 Well, this then shows us why we want to study pictures 310 00:17:16,510 --> 00:17:18,160 rather than functions. 311 00:17:18,160 --> 00:17:21,300 Now, if we look at any textbook in which we deal with 312 00:17:21,300 --> 00:17:24,500 graphs, it always seems that we start with graphs of 313 00:17:24,500 --> 00:17:25,319 straight lines. 314 00:17:25,319 --> 00:17:28,359 And the question is what is so great about a straight line? 315 00:17:28,359 --> 00:17:31,360 After all, pictures in general are going to be much more 316 00:17:31,360 --> 00:17:32,480 complicated than that. 317 00:17:32,480 --> 00:17:36,180 What advantage is there in starting with straight lines? 318 00:17:36,180 --> 00:17:39,680 And again, we begin to realize how straight lines are the 319 00:17:39,680 --> 00:17:43,540 backbone of all types of analytical procedures and all 320 00:17:43,540 --> 00:17:44,840 types of curve plotting. 321 00:17:44,840 --> 00:17:47,910 For example, let's suppose we were studying this particular 322 00:17:47,910 --> 00:17:50,860 curve, and we wanted to know what was happened to that 323 00:17:50,860 --> 00:17:55,310 curve in the neighborhood around the point p. 324 00:17:55,310 --> 00:17:59,320 Let's draw in the tangent line to the curve at the point p. 325 00:17:59,320 --> 00:18:03,090 Notice that this line that we've drawn serves as a 326 00:18:03,090 --> 00:18:07,740 wonderful approximation curve itself if we stay close enough 327 00:18:07,740 --> 00:18:09,520 to the point of tangency. 328 00:18:09,520 --> 00:18:12,590 In other words, notice how much we can deduce about this 329 00:18:12,590 --> 00:18:16,910 curve if we study only the straight line segment at the 330 00:18:16,910 --> 00:18:18,000 point of tangency. 331 00:18:18,000 --> 00:18:20,660 Of course, the approximation gets worse and worse as we 332 00:18:20,660 --> 00:18:22,180 move further and further out. 333 00:18:22,180 --> 00:18:25,000 But in the neighborhood of the point of what's going on, 334 00:18:25,000 --> 00:18:27,660 notice again then that the straight line is an important 335 00:18:27,660 --> 00:18:28,820 building block. 336 00:18:28,820 --> 00:18:32,170 By the way, again we use straight lines in a rather 337 00:18:32,170 --> 00:18:35,520 subtle way in something called interpolation. 338 00:18:35,520 --> 00:18:38,370 For example, let's suppose I go to a log table and I look 339 00:18:38,370 --> 00:18:40,030 up the log of 2. 340 00:18:40,030 --> 00:18:44,270 I find that the log of 2 is 0.301. 341 00:18:44,270 --> 00:18:45,770 I look up the log of 4. 342 00:18:45,770 --> 00:18:48,900 I find that's 0.602. 343 00:18:48,900 --> 00:18:52,080 Now I look for the log of 3, and I see that it's been 344 00:18:52,080 --> 00:18:53,660 obliterated. 345 00:18:53,660 --> 00:18:54,710 I don't know what it is. 346 00:18:54,710 --> 00:18:57,670 So I say, well, let me guess. 347 00:18:57,670 --> 00:19:00,640 3 is halfway between 2 and 4. 348 00:19:00,640 --> 00:19:04,670 Therefore, I would suspect that the log of 3 is halfway 349 00:19:04,670 --> 00:19:07,570 between the log of 2 and the log of 4. 350 00:19:07,570 --> 00:19:10,030 And so, halfway between here would be about what? 351 00:19:10,030 --> 00:19:13,880 0.452, roughly speaking. 352 00:19:13,880 --> 00:19:16,160 All of a sudden, the obliteration on 353 00:19:16,160 --> 00:19:18,080 my book clears up. 354 00:19:18,080 --> 00:19:21,760 And I look, and I don't find 0.452. 355 00:19:21,760 --> 00:19:24,660 Instead I find-- 356 00:19:24,660 --> 00:19:27,120 well, let's write it over here. 357 00:19:27,120 --> 00:19:32,960 What I find is 0.477. 358 00:19:32,960 --> 00:19:36,030 Now, you know, this is a pretty big error to attribute 359 00:19:36,030 --> 00:19:38,800 to slide rule inaccuracy or trouble in 360 00:19:38,800 --> 00:19:40,030 rounding off the tables. 361 00:19:40,030 --> 00:19:42,010 What really went wrong over here? 362 00:19:42,010 --> 00:19:44,770 And the answer comes up again that unless otherwise 363 00:19:44,770 --> 00:19:49,260 specified, the process known as interpolation hinges on 364 00:19:49,260 --> 00:19:53,320 replacing a curve by a straight line approximation. 365 00:19:53,320 --> 00:19:56,690 In fact, you see, if we were to draw the curve of the 366 00:19:56,690 --> 00:19:59,940 logarithm function, we would find that the picture is 367 00:19:59,940 --> 00:20:01,570 something like this. 368 00:20:01,570 --> 00:20:05,910 And when we looked up the log of 2, this height is what we 369 00:20:05,910 --> 00:20:07,000 found in the table. 370 00:20:07,000 --> 00:20:11,400 When we looked up the log of 4, this height is what we 371 00:20:11,400 --> 00:20:15,670 would have found in the table. 372 00:20:15,670 --> 00:20:18,800 If we had looked up in the table the log of 3, this is 373 00:20:18,800 --> 00:20:20,240 the height that we would have found. 374 00:20:20,240 --> 00:20:23,500 This is the height that's 0.477. 375 00:20:23,500 --> 00:20:27,630 Notice that in general, if we go halfway from here to here, 376 00:20:27,630 --> 00:20:29,900 we do not go halfway from here to here. 377 00:20:29,900 --> 00:20:31,770 It depends on the shape of the curve. 378 00:20:31,770 --> 00:20:34,750 The only time you can be sure that you have proportional 379 00:20:34,750 --> 00:20:38,600 parts is if the curve that joined these two points was a 380 00:20:38,600 --> 00:20:41,620 straight line. 381 00:20:41,620 --> 00:20:44,860 And notice, by the way, that by the shape of this curve, 382 00:20:44,860 --> 00:20:49,490 the straight line falls below the curve, and therefore, the 383 00:20:49,490 --> 00:20:51,550 height that we found was to the straight 384 00:20:51,550 --> 00:20:53,080 line, not to the curve. 385 00:20:53,080 --> 00:20:55,810 That was the point 0.452. 386 00:20:55,810 --> 00:20:58,470 So, in other words, notice that we got smaller than the 387 00:20:58,470 --> 00:21:01,760 right answer because we approximated as if it was a 388 00:21:01,760 --> 00:21:03,930 straight line that was joining the curve. 389 00:21:03,930 --> 00:21:06,990 You see, what interpolation hinges on is that the size of 390 00:21:06,990 --> 00:21:10,920 the interval is very small and that you can assume that for 391 00:21:10,920 --> 00:21:13,850 the accuracy that you're interested in that the 392 00:21:13,850 --> 00:21:17,120 straight line approximation to the curve is sufficiently 393 00:21:17,120 --> 00:21:20,710 accurate to represent the curve itself. 394 00:21:20,710 --> 00:21:22,200 Well, enough about that. 395 00:21:22,200 --> 00:21:25,990 Once we've talked about why straight lines are important, 396 00:21:25,990 --> 00:21:28,850 the next thing is how do we measure straight lines? 397 00:21:28,850 --> 00:21:31,770 See, another interesting point to something like this. 398 00:21:31,770 --> 00:21:35,280 Many times we know what something means subjectively, 399 00:21:35,280 --> 00:21:37,900 but we don't know what it means objectively. 400 00:21:37,900 --> 00:21:41,350 For example, one way of finding a line is to know two 401 00:21:41,350 --> 00:21:42,730 points on the line. 402 00:21:42,730 --> 00:21:45,280 Another way is to know one point and the 403 00:21:45,280 --> 00:21:46,480 slant of the line. 404 00:21:46,480 --> 00:21:48,650 The question that comes up is how do you measure 405 00:21:48,650 --> 00:21:50,180 the slant of a line? 406 00:21:50,180 --> 00:21:53,350 In other words, shall you say that the line is very slanty? 407 00:21:53,350 --> 00:21:55,820 And if the answer to that is yes, how do you distinguish 408 00:21:55,820 --> 00:22:01,460 between slanty and very slanty, steep and very steep, 409 00:22:01,460 --> 00:22:03,240 very steep and very, very steep? 410 00:22:03,240 --> 00:22:05,670 We need something more objective. 411 00:22:05,670 --> 00:22:08,450 And the way we get around this is as follows. 412 00:22:08,450 --> 00:22:11,850 Given a line, we define the slope as follows. 413 00:22:11,850 --> 00:22:14,960 We pick any two points on the line. 414 00:22:14,960 --> 00:22:18,880 And from those two points, we can measure what we call the 415 00:22:18,880 --> 00:22:22,490 run of the line, in other words, how far you've gone 416 00:22:22,490 --> 00:22:26,680 this way, and the rise of the line, how much 417 00:22:26,680 --> 00:22:28,020 it's risen this way. 418 00:22:28,020 --> 00:22:32,780 And what we do is we define the slope to be the rise 419 00:22:32,780 --> 00:22:38,790 divided by the run, or without the delta notation in here, y2 420 00:22:38,790 --> 00:22:42,680 minus y1 over x2 minus x1. 421 00:22:42,680 --> 00:22:45,050 By the way, there are little problems that come up. 422 00:22:45,050 --> 00:22:48,020 After all, our answer should not depend on the picture. 423 00:22:48,020 --> 00:22:50,520 It should be sort of self-contained analytically. 424 00:22:50,520 --> 00:22:53,790 The question comes up is what if I had labeled this point 425 00:22:53,790 --> 00:22:58,350 x2, y2 and this point x1, y1? 426 00:22:58,350 --> 00:22:59,620 What would have happened then? 427 00:22:59,620 --> 00:23:03,090 And observe that as long as we keep the pairs straight, it 428 00:23:03,090 --> 00:23:06,570 makes no difference whether you write this or whether you 429 00:23:06,570 --> 00:23:07,430 write this. 430 00:23:07,430 --> 00:23:09,600 Because, you see, in each case, all you've done is 431 00:23:09,600 --> 00:23:13,540 change the sign, and negative over negative is positive. 432 00:23:13,540 --> 00:23:17,060 So certainly our answer to what a slope is is objective 433 00:23:17,060 --> 00:23:19,930 enough, so it does not depend on how the points are labeled. 434 00:23:19,930 --> 00:23:22,240 A second objection that most people have is they say 435 00:23:22,240 --> 00:23:25,140 something like, well, who are you to say that we pick these 436 00:23:25,140 --> 00:23:25,770 two points? 437 00:23:25,770 --> 00:23:29,050 What if I came along and picked these two points and I 438 00:23:29,050 --> 00:23:33,460 now computed the slope by taking this as my delta y and 439 00:23:33,460 --> 00:23:35,450 this as my delta x? 440 00:23:35,450 --> 00:23:38,250 Obviously, it would be a tragedy if the answer to the 441 00:23:38,250 --> 00:23:41,315 problem depended on which pair of points you picked since a 442 00:23:41,315 --> 00:23:43,730 line should have but one slope. 443 00:23:43,730 --> 00:23:46,750 Again, notice that our high school training in geometry, 444 00:23:46,750 --> 00:23:50,540 similar triangles, motivates why we pick ratios. 445 00:23:50,540 --> 00:23:54,460 Namely, while this delta y and this delta y may be different 446 00:23:54,460 --> 00:23:57,510 and this delta x and this delta x may be different, what 447 00:23:57,510 --> 00:24:02,160 is true is that the ratio of this delta y to this delta x 448 00:24:02,160 --> 00:24:05,830 is the same as the ratio of this delta y to this delta x. 449 00:24:05,830 --> 00:24:08,040 And that's why we pick the ratio. 450 00:24:08,040 --> 00:24:11,110 By the way, another way of talking about ratio is if you 451 00:24:11,110 --> 00:24:14,460 look at delta y divided by delta x and you've had some 452 00:24:14,460 --> 00:24:17,330 trigonometry, it reminds you of a trigonometric 453 00:24:17,330 --> 00:24:19,090 relationship. 454 00:24:19,090 --> 00:24:23,000 Namely, you look at delta y, you look at delta x, and you 455 00:24:23,000 --> 00:24:25,510 say, my, isn't that just the tangent of 456 00:24:25,510 --> 00:24:26,960 this particular angle? 457 00:24:26,960 --> 00:24:28,650 Couldn't I define the slope? 458 00:24:28,650 --> 00:24:31,480 And by the way, the general symbol for slope, for better 459 00:24:31,480 --> 00:24:33,550 or for worse, just happens to be letter m. 460 00:24:33,550 --> 00:24:39,280 Why couldn't I define m to be the tangent of phi, where phi 461 00:24:39,280 --> 00:24:41,590 is the angle that the straight line makes with 462 00:24:41,590 --> 00:24:43,050 the positive x-axis? 463 00:24:43,050 --> 00:24:45,180 And, of course, there is a little subtlety here that we 464 00:24:45,180 --> 00:24:46,600 should pay attention to. 465 00:24:46,600 --> 00:24:49,970 This would be an ambiguous definition if the scale on the 466 00:24:49,970 --> 00:24:52,380 x- and the y-axis were not the same. 467 00:24:52,380 --> 00:24:55,360 In other words, notice that by changing the scale here, I can 468 00:24:55,360 --> 00:24:58,330 distort the same analytic information. 469 00:24:58,330 --> 00:25:02,180 So if I agree, however, that the unit on the x-axis is the 470 00:25:02,180 --> 00:25:05,980 same as the unit on the y-axis, then I can say, OK, 471 00:25:05,980 --> 00:25:09,300 the slope is also tangent of the angle phi. 472 00:25:09,300 --> 00:25:12,670 I much prefer to say it's delta y divided by delta x, 473 00:25:12,670 --> 00:25:14,400 because then if I forget the scale, 474 00:25:14,400 --> 00:25:16,010 I'm still in no trouble. 475 00:25:16,010 --> 00:25:20,180 On the other hand, if we use the tangent definition, we can 476 00:25:20,180 --> 00:25:23,670 utilize all we know about trigonometry to get some other 477 00:25:23,670 --> 00:25:25,190 interesting results. 478 00:25:25,190 --> 00:25:27,800 Namely, the question that might come up is can we study 479 00:25:27,800 --> 00:25:31,330 the slopes of two different lines very conveniently in 480 00:25:31,330 --> 00:25:33,540 terms of our definition of slope? 481 00:25:33,540 --> 00:25:35,330 And the answer is this. 482 00:25:35,330 --> 00:25:39,630 If we imagine now that our lines are drawn to scale here, 483 00:25:39,630 --> 00:25:42,560 and here are two different lines, which I'll call l1 and 484 00:25:42,560 --> 00:25:46,400 l2, and we'll call the angle that l1 makes with the 485 00:25:46,400 --> 00:25:50,280 positive x-axis phi 1, the angle that l2 makes with the 486 00:25:50,280 --> 00:25:52,760 positive x-axis phi 2. 487 00:25:52,760 --> 00:25:55,220 Therefore, what? m1 is tan phi 1. 488 00:25:55,220 --> 00:25:57,060 m2 is tan phi 2. 489 00:25:57,060 --> 00:26:01,100 Notice that our formula for the tangent of the difference 490 00:26:01,100 --> 00:26:03,760 of two angles-- you see, notice that this 491 00:26:03,760 --> 00:26:04,890 angle here is what? 492 00:26:04,890 --> 00:26:08,960 Since this angle is the sum of these two, this angle here is 493 00:26:08,960 --> 00:26:13,580 phi 2 minus phi 1 or the negative of phi 1 minus phi 2. 494 00:26:13,580 --> 00:26:16,220 I should have had this phi 2 minus phi 2, but since that 495 00:26:16,220 --> 00:26:19,020 just changes the sign, that will not have any bearing on 496 00:26:19,020 --> 00:26:20,300 the point I want to make. 497 00:26:20,300 --> 00:26:22,220 Let's continue this way. 498 00:26:22,220 --> 00:26:27,230 Tangent of phi 1 minus phi 2 is tan phi 1 minus tan phi 2 499 00:26:27,230 --> 00:26:30,570 over 1 plus tan phi 1 tan phi 2. 500 00:26:30,570 --> 00:26:33,500 On the other hand, by our definitions of m1 and m2, this 501 00:26:33,500 --> 00:26:37,080 is m1 minus m2 over 1 plus m1 m2. 502 00:26:37,080 --> 00:26:40,220 503 00:26:40,220 --> 00:26:44,360 Now, this tells me how to find the angle between two lines 504 00:26:44,360 --> 00:26:46,730 just in terms of knowing the slope. 505 00:26:46,730 --> 00:26:50,670 Two very special interesting cases as extremes suggest 506 00:26:50,670 --> 00:26:52,110 themselves right away. 507 00:26:52,110 --> 00:26:55,760 One case is what happens if the lines are parallel? 508 00:26:55,760 --> 00:27:00,020 If the lines are parallel, you see, phi 1 equals phi 2, in 509 00:27:00,020 --> 00:27:04,350 which case phi 1 minus phi 2 is 0, in which case the 510 00:27:04,350 --> 00:27:08,110 tangent of phi 1 minus phi 2 had better be 0. 511 00:27:08,110 --> 00:27:12,060 But the only way a fraction can be 0 is for the numerator 512 00:27:12,060 --> 00:27:16,120 to be 0, and that means that m1 must equal m2. 513 00:27:16,120 --> 00:27:19,420 In other words, in terms of slopes, we can study parallel 514 00:27:19,420 --> 00:27:22,310 lines just by equating their slopes. 515 00:27:22,310 --> 00:27:25,590 A less obvious relationship that's equally important is 516 00:27:25,590 --> 00:27:29,200 how do you measure whether two lines are perpendicular? 517 00:27:29,200 --> 00:27:31,700 The answer is if they're perpendicular, the angle 518 00:27:31,700 --> 00:27:33,870 between them is 90 degrees. 519 00:27:33,870 --> 00:27:37,330 The tangent of 90 degrees is infinity, as we learned. 520 00:27:37,330 --> 00:27:39,430 That's equivalent to saying what? 521 00:27:39,430 --> 00:27:42,270 That the denominator is 0. 522 00:27:42,270 --> 00:27:44,230 See, the only way a fraction blows up is for the 523 00:27:44,230 --> 00:27:45,520 denominator to be 0. 524 00:27:45,520 --> 00:27:50,400 But the only way that 1 plus m1 m2 can be 0 is for what? 525 00:27:50,400 --> 00:27:54,260 m1 m2 to be equal to minus 1. 526 00:27:54,260 --> 00:27:57,780 And this gives us the other very well-known result that in 527 00:27:57,780 --> 00:28:00,250 terms of slopes, to study whether two lines are 528 00:28:00,250 --> 00:28:03,510 perpendicular, all we need to investigate is whether one 529 00:28:03,510 --> 00:28:07,570 slope is the negative reciprocal of the other. 530 00:28:07,570 --> 00:28:10,070 Well, again, the textbook will bring out 531 00:28:10,070 --> 00:28:11,630 slopes in more detail. 532 00:28:11,630 --> 00:28:14,220 The next question that we'd like to bring up in terms of a 533 00:28:14,220 --> 00:28:19,000 picture is worth a thousand words is how do you identify 534 00:28:19,000 --> 00:28:21,780 the straight line with an algebraic equation? 535 00:28:21,780 --> 00:28:24,690 What do we mean by the equation of a straight line? 536 00:28:24,690 --> 00:28:27,790 Well, again, there are two possibilities. 537 00:28:27,790 --> 00:28:30,150 The first possibility is that the line is 538 00:28:30,150 --> 00:28:33,020 parallel to the y-axis. 539 00:28:33,020 --> 00:28:36,580 If the line is parallel to the y-axis, if the line goes 540 00:28:36,580 --> 00:28:40,710 through the point a comma 0, notice that the only criteria 541 00:28:40,710 --> 00:28:43,940 for the point to be on that line is that its 542 00:28:43,940 --> 00:28:46,530 x-coordinate equal a. 543 00:28:46,530 --> 00:28:49,940 By the way, this is often abbreviated in the textbook as 544 00:28:49,940 --> 00:28:52,030 the line x equals a. 545 00:28:52,030 --> 00:28:54,270 Many a student says how do you know this is a line? 546 00:28:54,270 --> 00:28:56,740 Why isn't this the point x equals a? 547 00:28:56,740 --> 00:28:59,750 And here again is a good review of why we stress the 548 00:28:59,750 --> 00:29:01,480 language of sets. 549 00:29:01,480 --> 00:29:05,030 Here again is a good reason why we express the language of 550 00:29:05,030 --> 00:29:06,530 sets so strongly. 551 00:29:06,530 --> 00:29:09,060 Namely, go back to the universe of discourse here. 552 00:29:09,060 --> 00:29:12,900 When you see the set of all ordered pairs x comma y for 553 00:29:12,900 --> 00:29:15,730 which x equals a, this gives you the hint that you're 554 00:29:15,730 --> 00:29:18,760 talking about pairs of points, and that tells you that you 555 00:29:18,760 --> 00:29:22,710 have numbers in the plane, not on the line, not on the 556 00:29:22,710 --> 00:29:26,650 x-axis, a two-dimensional interpretation over here. 557 00:29:26,650 --> 00:29:30,400 You see, if this said the set of all x such that x equals a, 558 00:29:30,400 --> 00:29:31,850 it would just be a point. 559 00:29:31,850 --> 00:29:33,900 But notice the hint over here. 560 00:29:33,900 --> 00:29:37,030 At any rate, this then becomes the equation of a straight 561 00:29:37,030 --> 00:29:40,660 line if the straight line is parallel to the y-axis. 562 00:29:40,660 --> 00:29:42,710 Of course, the other possibility is what if the 563 00:29:42,710 --> 00:29:45,590 line isn't parallel to the x-axis? 564 00:29:45,590 --> 00:29:48,870 And here, too, we say OK, suppose we know a point on the 565 00:29:48,870 --> 00:29:51,500 line and suppose we know the slope of the line. 566 00:29:51,500 --> 00:29:54,750 What we will do is pick any other point on the plane, 567 00:29:54,750 --> 00:29:58,640 which we will label arbitrarily x comma y, and see 568 00:29:58,640 --> 00:30:02,100 what the equation x comma y has to satisfy. 569 00:30:02,100 --> 00:30:05,340 How do the coordinates have to be related to be on this line? 570 00:30:05,340 --> 00:30:09,820 Well, we already know that slope does not depend on which 571 00:30:09,820 --> 00:30:11,200 two points you pick. 572 00:30:11,200 --> 00:30:15,690 Consequently, since the slope of this line is m, the slope 573 00:30:15,690 --> 00:30:17,020 must also be what? 574 00:30:17,020 --> 00:30:21,250 y minus y1 over x minus x1. 575 00:30:21,250 --> 00:30:24,790 And this becomes the fundamental definition for the 576 00:30:24,790 --> 00:30:29,260 equation of a line which is not parallel to the y-axis. 577 00:30:29,260 --> 00:30:34,280 And by the way, again, I think m's and x1's and y1's tend to 578 00:30:34,280 --> 00:30:37,080 give you a bit of hardship at first until 579 00:30:37,080 --> 00:30:38,250 you get used to them. 580 00:30:38,250 --> 00:30:42,300 Let's illustrate this thing with a specific example. 581 00:30:42,300 --> 00:30:46,090 Suppose I say to you I am thinking of the line whose 582 00:30:46,090 --> 00:30:50,600 slope is 3 and which passes through the point 2 comma 5. 583 00:30:50,600 --> 00:30:52,280 And notice the language of sets here. 584 00:30:52,280 --> 00:30:55,600 To say that 2 comma 5 is on the line is the same as saying 585 00:30:55,600 --> 00:30:58,910 that 2 comma 5 belongs to the set of points 586 00:30:58,910 --> 00:31:00,440 determined by the line. 587 00:31:00,440 --> 00:31:03,520 Drawing a rough sketch over here-- 588 00:31:03,520 --> 00:31:06,340 and by the way, notice something very important here. 589 00:31:06,340 --> 00:31:09,680 I never have to draw to scale. 590 00:31:09,680 --> 00:31:12,370 Because, you see, all I'm going to use is the analytic 591 00:31:12,370 --> 00:31:16,370 terms, and 2 comma 5 is still 2 and 5, no matter how I draw 592 00:31:16,370 --> 00:31:17,280 the picture. 593 00:31:17,280 --> 00:31:20,670 So, for example, if I say OK, let's see what it means for 594 00:31:20,670 --> 00:31:24,020 the point x comma y to belong here, I say, well, 595 00:31:24,020 --> 00:31:25,020 what does that mean? 596 00:31:25,020 --> 00:31:28,330 My slope is going to have to be what? y minus 5. 597 00:31:28,330 --> 00:31:29,640 That's my rise. 598 00:31:29,640 --> 00:31:35,170 My run is x minus 2, and that must equal 3. 599 00:31:35,170 --> 00:31:38,780 And if I clear this of fractions, I get what? y is 600 00:31:38,780 --> 00:31:47,610 equal to 3x minus 1. 601 00:31:47,610 --> 00:31:48,490 By the way, does this check out? 602 00:31:48,490 --> 00:31:52,680 If x is 2, 2 times 3 is 6, minus 1 is 5. 603 00:31:52,680 --> 00:31:54,960 2 comma 5 is on the line. 604 00:31:54,960 --> 00:31:56,120 You see, here's the thing. 605 00:31:56,120 --> 00:31:58,630 We talked about the line geometrically. 606 00:31:58,630 --> 00:32:00,850 Now I have an algebraic equation. 607 00:32:00,850 --> 00:32:02,950 I no longer have to refer to the picture. 608 00:32:02,950 --> 00:32:04,730 I have something analytic now. 609 00:32:04,730 --> 00:32:07,450 For example, suppose a person says to me I wonder if the 610 00:32:07,450 --> 00:32:10,810 point 8 comma 23 is on this line? 611 00:32:10,810 --> 00:32:12,470 I don't have to draw a picture to scale. 612 00:32:12,470 --> 00:32:14,040 I don't have to waste any time. 613 00:32:14,040 --> 00:32:18,650 I know that the equation of my line is y equals 3x minus 1. 614 00:32:18,650 --> 00:32:25,720 By the way, if y equals 3x minus 1, as soon as x is 8, 615 00:32:25,720 --> 00:32:27,550 what must y equal? 616 00:32:27,550 --> 00:32:31,310 y must equal what? 617 00:32:31,310 --> 00:32:34,260 23? 618 00:32:34,260 --> 00:32:35,540 Is that right? 619 00:32:35,540 --> 00:32:42,130 And so is the point 8 comma 23 on the line? 620 00:32:42,130 --> 00:32:42,900 Yes. 621 00:32:42,900 --> 00:32:46,350 How about 8 comma 12? 622 00:32:46,350 --> 00:32:51,560 8 comma 12 isn't on the line because 3 times 8 623 00:32:51,560 --> 00:32:53,370 minus 1 is not 12. 624 00:32:53,370 --> 00:32:56,800 But notice that we can even see algebraically that 8 comma 625 00:32:56,800 --> 00:33:00,170 12 must be below the line. 626 00:33:00,170 --> 00:33:04,390 In other words, our study of equations allows us not only 627 00:33:04,390 --> 00:33:07,800 to visualize lines as equations, but we can also 628 00:33:07,800 --> 00:33:11,230 visualize inequalities as pictures. 629 00:33:11,230 --> 00:33:18,000 In other words, if we have the equation of a line, if this is 630 00:33:18,000 --> 00:33:24,570 the line y equals, say, 3x plus 1 or something like this, 631 00:33:24,570 --> 00:33:26,770 then what is this region here? 632 00:33:26,770 --> 00:33:30,310 These are all those values which lie-- whose heights lie 633 00:33:30,310 --> 00:33:32,710 below the height to be on the curve. 634 00:33:32,710 --> 00:33:36,140 Again, not a very clear example in the sense of 635 00:33:36,140 --> 00:33:39,090 drawing the picture neatly for you, but our main aim is not 636 00:33:39,090 --> 00:33:40,580 to draw neat pictures here. 637 00:33:40,580 --> 00:33:43,950 Our main aim is to show how analytical terms can be 638 00:33:43,950 --> 00:33:47,350 studied very conveniently in terms of pictures. 639 00:33:47,350 --> 00:33:50,560 In fact, perhaps to conclude today's lesson, what we should 640 00:33:50,560 --> 00:33:54,550 talk about is an old algebraic concept called 641 00:33:54,550 --> 00:33:56,390 simultaneous equations. 642 00:33:56,390 --> 00:34:00,270 Suppose you're asked to solve this pair of equations. 643 00:34:00,270 --> 00:34:03,490 You say, well, let's see, if y equals 3x minus 1 and it's 644 00:34:03,490 --> 00:34:08,699 also equal to x plus 1, that says that x plus 1 645 00:34:08,699 --> 00:34:11,159 equals 3x minus 1. 646 00:34:11,159 --> 00:34:13,510 I now solve this thing algebraically. 647 00:34:13,510 --> 00:34:18,050 I get 2x equals 2, so x equals 1. 648 00:34:18,050 --> 00:34:23,630 Knowing that x equals 1, I can see that y equals 2, and I see 649 00:34:23,630 --> 00:34:28,030 that 1 comma 2 is my solution. 650 00:34:28,030 --> 00:34:30,620 In other words, if I wound up with this thing algebraically, 651 00:34:30,620 --> 00:34:35,409 1 comma 2 is the only member that belongs to both of these 652 00:34:35,409 --> 00:34:37,520 two solution sets. 653 00:34:37,520 --> 00:34:41,690 Now, again, notice how I can solve this purely analytic 654 00:34:41,690 --> 00:34:45,530 problem without recourse to a picture. 655 00:34:45,530 --> 00:34:48,510 On the other hand, if I want to think of this thing 656 00:34:48,510 --> 00:34:52,989 pictorially, notice that y equals 3x minus 1 is the 657 00:34:52,989 --> 00:34:58,040 equation of a particular straight line, and y equals x 658 00:34:58,040 --> 00:35:02,450 plus 1 is also the equation of a line. 659 00:35:02,450 --> 00:35:05,850 Notice that since these two lines are not parallel, they 660 00:35:05,850 --> 00:35:08,420 intersect at one particular point. 661 00:35:08,420 --> 00:35:12,530 And the geometric problem that I solved on the previous board 662 00:35:12,530 --> 00:35:16,820 turns out to be that the point 1 comma 2 is the point that 663 00:35:16,820 --> 00:35:19,040 both of these lines have in common. 664 00:35:19,040 --> 00:35:23,640 In fact, if we call this line as before l1 and if we name 665 00:35:23,640 --> 00:35:30,270 this l2, in the language of sets, 1 comma 2 is what? 666 00:35:30,270 --> 00:35:33,560 The point which is the intersection of the two lines 667 00:35:33,560 --> 00:35:38,190 l1 and l2, again, a geometric interpretation 668 00:35:38,190 --> 00:35:39,830 for an analytic problem. 669 00:35:39,830 --> 00:35:43,380 In fact, notice also how much mileage I can get out of the 670 00:35:43,380 --> 00:35:44,910 geometric picture. 671 00:35:44,910 --> 00:35:49,170 For example, notice that this region here has a very nice 672 00:35:49,170 --> 00:35:50,870 geometric interpretation. 673 00:35:50,870 --> 00:35:52,050 It's the set of what? 674 00:35:52,050 --> 00:35:57,620 All points which are below this line and above this line. 675 00:35:57,620 --> 00:36:00,010 In other words, what? 676 00:36:00,010 --> 00:36:04,970 To be below this line, y must be less than x plus 1, and to 677 00:36:04,970 --> 00:36:09,880 be above this line, y must be greater than 3x minus 1. 678 00:36:09,880 --> 00:36:13,430 Notice then that a pair of simultaneous inequalities, 679 00:36:13,430 --> 00:36:16,510 which may not be that easy to handle, are very easy to 680 00:36:16,510 --> 00:36:19,890 handle in terms of regions in the plane. 681 00:36:19,890 --> 00:36:23,580 Notice also that since two lines can either be parallel 682 00:36:23,580 --> 00:36:27,660 or not parallel, we also get a nice geometric interpretation 683 00:36:27,660 --> 00:36:30,270 as to why simultaneous equations 684 00:36:30,270 --> 00:36:32,080 may have one solution. 685 00:36:32,080 --> 00:36:34,240 Namely, the lines are not parallel, 686 00:36:34,240 --> 00:36:35,540 and hence, they intersect. 687 00:36:35,540 --> 00:36:38,610 Or no solutions, the lines could've been parallel without 688 00:36:38,610 --> 00:36:39,440 intersecting. 689 00:36:39,440 --> 00:36:43,500 Or infinitely many solutions, the two lines could have been 690 00:36:43,500 --> 00:36:45,350 different equations. 691 00:36:45,350 --> 00:36:47,490 In effect, I should say what? 692 00:36:47,490 --> 00:36:50,130 The two equations could've been different equations for 693 00:36:50,130 --> 00:36:51,290 the same line. 694 00:36:51,290 --> 00:36:55,300 Again, this may seem a little bit sketchy and rapid, but all 695 00:36:55,300 --> 00:36:57,250 we want to do is give the overview. 696 00:36:57,250 --> 00:37:00,460 The reading assignment in the text goes into great detail on 697 00:37:00,460 --> 00:37:02,370 the points that we've mentioned so far. 698 00:37:02,370 --> 00:37:05,760 But again, in summary, what our lesson was supposed to be 699 00:37:05,760 --> 00:37:09,640 today was to indicate the importance of being able to 700 00:37:09,640 --> 00:37:13,880 visualize and to identify analytic results with 701 00:37:13,880 --> 00:37:15,830 geometric pictures. 702 00:37:15,830 --> 00:37:17,680 And so, until next time, goodbye. 703 00:37:17,680 --> 00:37:20,840 704 00:37:20,840 --> 00:37:24,040 Funding for the publication of this video was provided by the 705 00:37:24,040 --> 00:37:28,090 Gabriella and Paul Rosenbaum Foundation. 706 00:37:28,090 --> 00:37:32,270 Help OCW continue to provide free and open access to MIT 707 00:37:32,270 --> 00:37:36,460 courses by making a donation at ocw.mit.edu/donate. 708 00:37:36,460 --> 00:37:41,204