0:00:15.096,0:00:16.871 This is Zeno of Elea, 0:00:16.871,0:00:18.377 an ancient Greek philosopher 0:00:18.377,0:00:21.042 famous for inventing a number of paradoxes, 0:00:21.042,0:00:22.560 arguments that seem logical, 0:00:22.560,0:00:25.779 but whose conclusion is absurd or contradictory. 0:00:25.779,0:00:27.183 For more than 2,000 years, 0:00:27.183,0:00:29.694 Zeno's mind-bending riddles have inspired 0:00:29.694,0:00:31.310 mathematicians and philosophers 0:00:31.310,0:00:33.746 to better understand the nature of infinity. 0:00:33.746,0:00:35.525 One of the best known of Zeno's problems 0:00:35.525,0:00:37.741 is called the dichotomy paradox, 0:00:37.741,0:00:41.527 which means, "the paradox of cutting in two" in ancient Greek. 0:00:41.527,0:00:43.315 It goes something like this: 0:00:43.315,0:00:46.154 After a long day of sitting around, thinking, 0:00:46.154,0:00:48.950 Zeno decides to walk from his house to the park. 0:00:48.950,0:00:50.397 The fresh air clears his mind 0:00:50.397,0:00:51.920 and help him think better. 0:00:51.920,0:00:53.075 In order to get to the park, 0:00:53.075,0:00:55.428 he first has to get half way to the park. 0:00:55.428,0:00:56.601 This portion of his journey 0:00:56.601,0:00:58.443 takes some finite amount of time. 0:00:58.443,0:01:00.452 Once he gets to the halfway point, 0:01:00.452,0:01:02.841 he needs to walk half the remaining distance. 0:01:02.841,0:01:05.868 Again, this takes a finite amount of time. 0:01:05.868,0:01:08.140 Once he gets there, he still needs to walk 0:01:08.140,0:01:09.882 half the distance that's left, 0:01:09.882,0:01:12.371 which takes another finite amount of time. 0:01:12.371,0:01:15.522 This happens again and again and again. 0:01:15.522,0:01:18.195 You can see that we can keep going like this forever, 0:01:18.195,0:01:19.857 dividing whatever distance is left 0:01:19.857,0:01:21.772 into smaller and smaller pieces, 0:01:21.772,0:01:25.278 each of which takes some finite time to traverse. 0:01:25.278,0:01:27.958 So, how long does it take Zeno to get to the park? 0:01:27.958,0:01:30.317 Well, to find out, you need to add the times 0:01:30.317,0:01:32.284 of each of the pieces of the journey. 0:01:32.284,0:01:36.616 The problem is, there are infinitely many of these finite-sized pieces. 0:01:36.616,0:01:39.750 So, shouldn't the total time be infinity? 0:01:39.750,0:01:42.548 This argument, by the way, is completely general. 0:01:42.548,0:01:45.092 It says that traveling from any location to any other location 0:01:45.092,0:01:47.254 should take an infinite amount of time. 0:01:47.254,0:01:51.006 In other words, it says that all motion is impossible. 0:01:51.006,0:01:52.785 This conclusion is clearly absurd, 0:01:52.785,0:01:54.784 but where is the flaw in the logic? 0:01:54.784,0:01:55.966 To resolve the paradox, 0:01:55.966,0:01:58.731 it helps to turn the story into a math problem. 0:01:58.731,0:02:01.618 Let's supposed that Zeno's house is one mile from the park 0:02:01.618,0:02:04.341 and that Zeno walks at one mile per hour. 0:02:04.341,0:02:06.692 Common sense tells us that the time for the journey 0:02:06.692,0:02:08.205 should be one hour. 0:02:08.205,0:02:10.867 But, let's look at things from Zeno's point of view 0:02:10.867,0:02:13.196 and divide up the journey into pieces. 0:02:13.196,0:02:15.656 The first half of the journey takes half an hour, 0:02:15.656,0:02:17.782 the next part takes quarter of an hour, 0:02:17.782,0:02:20.064 the third part takes an eighth of an hour, 0:02:20.064,0:02:20.969 and so on. 0:02:20.969,0:02:22.266 Summing up all these times, 0:02:22.266,0:02:24.372 we get a series that looks like this. 0:02:24.372,0:02:25.624 "Now", Zeno might say, 0:02:25.624,0:02:27.964 "since there are infinitely many of terms 0:02:27.964,0:02:29.621 on the right side of the equation, 0:02:29.621,0:02:31.883 and each individual term is finite, 0:02:31.883,0:02:34.518 the sum should equal infinity, right?" 0:02:34.518,0:02:36.670 This is the problem with Zeno's argument. 0:02:36.670,0:02:38.855 As mathematicians have since realized, 0:02:38.855,0:02:42.618 it is possible to add up infinitely many finite-sized terms 0:02:42.618,0:02:44.814 and still get a finite answer. 0:02:44.814,0:02:45.989 "How?" you ask. 0:02:45.989,0:02:47.486 Well, let's think of it this way. 0:02:47.486,0:02:50.390 Let's start with a square that has area of one meter. 0:02:50.390,0:02:52.528 Now let's chop the square in half, 0:02:52.528,0:02:54.909 and then chop the remaining half in half, 0:02:54.909,0:02:56.172 and so on. 0:02:56.172,0:02:57.239 While we're doing this, 0:02:57.239,0:03:00.380 let's keep track of the areas of the pieces. 0:03:00.380,0:03:02.169 The first slice makes two parts, 0:03:02.169,0:03:04.028 each with an area of one-half 0:03:04.028,0:03:06.545 The next slice divides one of those halves in half, 0:03:06.545,0:03:07.796 and so on. 0:03:07.796,0:03:10.227 But, no matter how many times we slice up the boxes, 0:03:10.227,0:03:14.814 the total area is still the sum of the areas of all the pieces. 0:03:14.814,0:03:17.442 Now you can see why we choose this particular way 0:03:17.442,0:03:18.971 of cutting up the square. 0:03:18.971,0:03:20.888 We've obtained the same infinite series 0:03:20.888,0:03:23.356 as we had for the time of Zeno's journey. 0:03:23.356,0:03:25.791 As we construct more and more blue pieces, 0:03:25.791,0:03:27.314 to use the math jargon, 0:03:27.314,0:03:30.742 as we take the limit as n tends to infinity, 0:03:30.742,0:03:33.356 the entire square becomes covered with blue. 0:03:33.356,0:03:35.427 But the area of the square is just one unit, 0:03:35.427,0:03:38.700 and so the infinite sum must equal one. 0:03:38.700,0:03:39.754 Going back to Zeno's journey, 0:03:39.754,0:03:42.370 we can now see how how the paradox is resolved. 0:03:42.370,0:03:45.713 Not only does the infinite series sum to a finite answer, 0:03:45.713,0:03:47.745 but that finite answer is the same one 0:03:47.745,0:03:50.172 that common sense tells us is true. 0:03:50.172,0:03:52.877 Zeno's journey takes one hour.