WEBVTT 00:00:01.909 --> 00:00:05.669 Hi. Yesterday, we explored the idea of equality. 00:00:06.230 --> 00:00:08.990 We saw that math gives us a way to recognize 00:00:08.990 --> 00:00:11.869 when two things that look different are actually 00:00:11.869 --> 00:00:15.669 the same. A certain number of laps equals a 5K, 00:00:15.750 --> 00:00:20.190 or four quarters equals a dollar. A certain number 00:00:20.190 --> 00:00:23.030 of meters actually equals a certain number of 00:00:23.030 --> 00:00:26.589 miles. And even when the conversion is not obvious 00:00:26.589 --> 00:00:29.969 at first, we can usually find a way to make the 00:00:29.969 --> 00:00:33.869 relationship work. Now today, I would like to 00:00:33.869 --> 00:00:36.710 show you a situation where the intuition completely 00:00:36.710 --> 00:00:42.890 breaks down. Imagine a circular track with a 00:00:42.890 --> 00:00:47.130 diameter of 100 meters. A perfect circle. Now 00:00:47.130 --> 00:00:49.810 the diameter is simply the distance across the 00:00:49.810 --> 00:00:52.850 circle through its center. And now let's ask 00:00:52.850 --> 00:00:55.670 a question that is similar to yesterday's questions. 00:00:56.429 --> 00:01:00.939 How many diameters equal... one trip around the 00:01:00.939 --> 00:01:04.459 circle. Now, in math, we refer to the distance 00:01:04.459 --> 00:01:07.799 around the circle as its circumference. And the 00:01:07.799 --> 00:01:10.760 relationship is actually simple. Circumference 00:01:10.760 --> 00:01:14.719 equals pi times the diameter. And since pi is 00:01:14.719 --> 00:01:19.219 about 3 .14, one trip around the circle is a 00:01:19.219 --> 00:01:23.459 little more than three diameters. So far, nothing 00:01:23.459 --> 00:01:28.090 unusual. But... Now let's ask the same kind of 00:01:28.090 --> 00:01:31.670 question that we asked yesterday. If I keep going 00:01:31.670 --> 00:01:35.930 around the track, will I eventually find a whole 00:01:35.930 --> 00:01:40.170 number of circumferences that equals a whole 00:01:40.170 --> 00:01:44.349 number of diameters? Now for miles and meters, 00:01:44.489 --> 00:01:47.409 the answer is yes. For quarters and dollars, 00:01:47.650 --> 00:01:51.189 the answer is yes. For tablespoons and cups, 00:01:51.409 --> 00:01:55.750 the answer is yes. But for a circle, The answer 00:01:55.750 --> 00:02:00.510 is no. Never. No matter how many times you go 00:02:00.510 --> 00:02:03.489 around the track, and no matter how large the 00:02:03.489 --> 00:02:07.450 circle becomes, you will never find a whole number 00:02:07.450 --> 00:02:11.490 of circumferences that exactly equals a whole 00:02:11.490 --> 00:02:16.490 number of diameters. Hmm. The ancient Greeks 00:02:16.490 --> 00:02:21.250 called this idea incommensurable. And what it 00:02:21.250 --> 00:02:25.189 means is that the two distances have no common 00:02:25.189 --> 00:02:30.590 measuring stick. And to me, that is remarkable. 00:02:31.069 --> 00:02:34.650 A circle is one of the simplest shapes we can 00:02:34.650 --> 00:02:39.810 imagine. And yet, hidden inside is a relationship 00:02:39.810 --> 00:02:44.330 that can never be perfectly reconciled. The circumference 00:02:44.330 --> 00:02:49.069 and diameter are forever linked, yet never fit 00:02:49.069 --> 00:02:53.819 together exactly. And that mystery is wrapped 00:02:53.819 --> 00:03:00.439 up in a number we call Pi. Hmm. Well, this week, 00:03:00.460 --> 00:03:03.699 we've explored perspectives. We've climbed mountains, 00:03:03.800 --> 00:03:06.939 and we've moved between viewpoints. We've gone 00:03:06.939 --> 00:03:10.580 deeper into ideas, and we've discovered hidden 00:03:10.580 --> 00:03:14.259 roots. We've even built a mathematical tool belt. 00:03:14.599 --> 00:03:17.340 And we've learned to see relationships that were 00:03:17.340 --> 00:03:20.180 hidden in plain sight. But perhaps one of the 00:03:20.180 --> 00:03:24.500 greatest gifts math offers is the ability to 00:03:24.500 --> 00:03:28.099 see the world differently. Sometimes it reveals 00:03:28.099 --> 00:03:31.419 that two things are equal, and sometimes it reveals 00:03:31.419 --> 00:03:35.340 that they never can be. And sometimes, as with 00:03:35.340 --> 00:03:39.879 a circle, that realization opens the door to 00:03:39.879 --> 00:03:44.020 wonder. And for me, it raises a deeper question. 00:03:44.219 --> 00:03:48.430 Where did this relationship come from? We humans 00:03:48.430 --> 00:03:51.770 create systems of measurement, miles and meters, 00:03:51.909 --> 00:03:55.449 dollars and quarters, tablespoons and cups. And 00:03:55.449 --> 00:03:58.310 with enough conversions, we can always make them 00:03:58.310 --> 00:04:01.770 equal. But the relationship between a circle's 00:04:01.770 --> 00:04:06.370 diameter and circumference is different. We did 00:04:06.370 --> 00:04:10.530 not invent it, but we discovered it. So perhaps 00:04:10.530 --> 00:04:13.389 our final perspective for the week is simply 00:04:13.389 --> 00:04:18.680 this. Why? Why should such a simple shape contain 00:04:18.680 --> 00:04:22.019 such a beautiful and surprising relationship? 00:04:22.959 --> 00:04:26.540 And for me, it's questions like that are part 00:04:26.540 --> 00:04:30.959 of what makes math so fascinating. It's not because 00:04:30.959 --> 00:04:34.040 math answers every question, but it's because 00:04:34.040 --> 00:04:37.899 math sometimes points us towards deeper ones. 00:04:38.959 --> 00:04:41.600 This is Dave from Intersecting Us.