1 00:00:15,096 --> 00:00:16,871 This is Zeno of Elea, 2 00:00:16,871 --> 00:00:18,377 an ancient Greek philosopher 3 00:00:18,377 --> 00:00:21,042 famous for inventing a number of paradoxes, 4 00:00:21,042 --> 00:00:22,560 arguments that seem logical, 5 00:00:22,560 --> 00:00:25,779 but whose conclusion is absurd or contradictory. 6 00:00:25,779 --> 00:00:27,183 For more than 2,000 years, 7 00:00:27,183 --> 00:00:29,694 Zeno's mind-bending riddles have inspired 8 00:00:29,694 --> 00:00:31,310 mathematicians and philosophers 9 00:00:31,310 --> 00:00:33,746 to better understand the nature of infinity. 10 00:00:33,746 --> 00:00:35,525 One of the best known of Zeno's problems 11 00:00:35,525 --> 00:00:37,741 is called the dichotomy paradox, 12 00:00:37,741 --> 00:00:41,527 which means, "the paradox of cutting in two" in ancient Greek. 13 00:00:41,527 --> 00:00:43,315 It goes something like this: 14 00:00:43,315 --> 00:00:46,154 After a long day of sitting around, thinking, 15 00:00:46,154 --> 00:00:48,950 Zeno decides to walk from his house to the park. 16 00:00:48,950 --> 00:00:50,397 The fresh air clears his mind 17 00:00:50,397 --> 00:00:51,920 and help him think better. 18 00:00:51,920 --> 00:00:53,075 In order to get to the park, 19 00:00:53,075 --> 00:00:55,428 he first has to get half way to the park. 20 00:00:55,428 --> 00:00:56,601 This portion of his journey 21 00:00:56,601 --> 00:00:58,443 takes some finite amount of time. 22 00:00:58,443 --> 00:01:00,452 Once he gets to the halfway point, 23 00:01:00,452 --> 00:01:02,841 he needs to walk half the remaining distance. 24 00:01:02,841 --> 00:01:05,868 Again, this takes a finite amount of time. 25 00:01:05,868 --> 00:01:08,140 Once he gets there, he still needs to walk 26 00:01:08,140 --> 00:01:09,882 half the distance that's left, 27 00:01:09,882 --> 00:01:12,371 which takes another finite amount of time. 28 00:01:12,371 --> 00:01:15,522 This happens again and again and again. 29 00:01:15,522 --> 00:01:18,195 You can see that we can keep going like this forever, 30 00:01:18,195 --> 00:01:19,857 dividing whatever distance is left 31 00:01:19,857 --> 00:01:21,772 into smaller and smaller pieces, 32 00:01:21,772 --> 00:01:25,278 each of which takes some finite time to traverse. 33 00:01:25,278 --> 00:01:27,958 So, how long does it take Zeno to get to the park? 34 00:01:27,958 --> 00:01:30,317 Well, to find out, you need to add the times 35 00:01:30,317 --> 00:01:32,284 of each of the pieces of the journey. 36 00:01:32,284 --> 00:01:36,616 The problem is, there are infinitely many of these finite-sized pieces. 37 00:01:36,616 --> 00:01:39,750 So, shouldn't the total time be infinity? 38 00:01:39,750 --> 00:01:42,548 This argument, by the way, is completely general. 39 00:01:42,548 --> 00:01:45,092 It says that traveling from any location to any other location 40 00:01:45,092 --> 00:01:47,254 should take an infinite amount of time. 41 00:01:47,254 --> 00:01:51,006 In other words, it says that all motion is impossible. 42 00:01:51,006 --> 00:01:52,785 This conclusion is clearly absurd, 43 00:01:52,785 --> 00:01:54,784 but where is the flaw in the logic? 44 00:01:54,784 --> 00:01:55,966 To resolve the paradox, 45 00:01:55,966 --> 00:01:58,731 it helps to turn the story into a math problem. 46 00:01:58,731 --> 00:02:01,618 Let's supposed that Zeno's house is one mile from the park 47 00:02:01,618 --> 00:02:04,341 and that Zeno walks at one mile per hour. 48 00:02:04,341 --> 00:02:06,692 Common sense tells us that the time for the journey 49 00:02:06,692 --> 00:02:08,205 should be one hour. 50 00:02:08,205 --> 00:02:10,867 But, let's look at things from Zeno's point of view 51 00:02:10,867 --> 00:02:13,196 and divide up the journey into pieces. 52 00:02:13,196 --> 00:02:15,656 The first half of the journey takes half an hour, 53 00:02:15,656 --> 00:02:17,782 the next part takes quarter of an hour, 54 00:02:17,782 --> 00:02:20,064 the third part takes an eighth of an hour, 55 00:02:20,064 --> 00:02:20,969 and so on. 56 00:02:20,969 --> 00:02:22,266 Summing up all these times, 57 00:02:22,266 --> 00:02:24,372 we get a series that looks like this. 58 00:02:24,372 --> 00:02:25,624 "Now", Zeno might say, 59 00:02:25,624 --> 00:02:27,964 "since there are infinitely many of terms 60 00:02:27,964 --> 00:02:29,621 on the right side of the equation, 61 00:02:29,621 --> 00:02:31,883 and each individual term is finite, 62 00:02:31,883 --> 00:02:34,518 the sum should equal infinity, right?" 63 00:02:34,518 --> 00:02:36,670 This is the problem with Zeno's argument. 64 00:02:36,670 --> 00:02:38,855 As mathematicians have since realized, 65 00:02:38,855 --> 00:02:42,618 it is possible to add up infinitely many finite-sized terms 66 00:02:42,618 --> 00:02:44,814 and still get a finite answer. 67 00:02:44,814 --> 00:02:45,989 "How?" you ask. 68 00:02:45,989 --> 00:02:47,486 Well, let's think of it this way. 69 00:02:47,486 --> 00:02:50,390 Let's start with a square that has area of one meter. 70 00:02:50,390 --> 00:02:52,528 Now let's chop the square in half, 71 00:02:52,528 --> 00:02:54,909 and then chop the remaining half in half, 72 00:02:54,909 --> 00:02:56,172 and so on. 73 00:02:56,172 --> 00:02:57,239 While we're doing this, 74 00:02:57,239 --> 00:03:00,380 let's keep track of the areas of the pieces. 75 00:03:00,380 --> 00:03:02,169 The first slice makes two parts, 76 00:03:02,169 --> 00:03:04,028 each with an area of one-half 77 00:03:04,028 --> 00:03:06,545 The next slice divides one of those halves in half, 78 00:03:06,545 --> 00:03:07,796 and so on. 79 00:03:07,796 --> 00:03:10,227 But, no matter how many times we slice up the boxes, 80 00:03:10,227 --> 00:03:14,814 the total area is still the sum of the areas of all the pieces. 81 00:03:14,814 --> 00:03:17,442 Now you can see why we choose this particular way 82 00:03:17,442 --> 00:03:18,971 of cutting up the square. 83 00:03:18,971 --> 00:03:20,888 We've obtained the same infinite series 84 00:03:20,888 --> 00:03:23,356 as we had for the time of Zeno's journey. 85 00:03:23,356 --> 00:03:25,791 As we construct more and more blue pieces, 86 00:03:25,791 --> 00:03:27,314 to use the math jargon, 87 00:03:27,314 --> 00:03:30,742 as we take the limit as n tends to infinity, 88 00:03:30,742 --> 00:03:33,356 the entire square becomes covered with blue. 89 00:03:33,356 --> 00:03:35,427 But the area of the square is just one unit, 90 00:03:35,427 --> 00:03:38,700 and so the infinite sum must equal one. 91 00:03:38,700 --> 00:03:39,754 Going back to Zeno's journey, 92 00:03:39,754 --> 00:03:42,370 we can now see how how the paradox is resolved. 93 00:03:42,370 --> 00:03:45,713 Not only does the infinite series sum to a finite answer, 94 00:03:45,713 --> 00:03:47,745 but that finite answer is the same one 95 00:03:47,745 --> 00:03:50,172 that common sense tells us is true. 96 00:03:50,172 --> 00:03:52,877 Zeno's journey takes one hour.