WEBVTT 00:00:02.000 --> 00:00:05.160 Hi, this week we've been exploring perspectives. 00:00:05.660 --> 00:00:08.339 And today I'd like to share a perspective that 00:00:08.339 --> 00:00:11.320 we use all the time without even thinking about 00:00:11.320 --> 00:00:16.579 it. The perspective of equality. So this morning 00:00:16.579 --> 00:00:20.500 I am running a 5K race on a standard 400 meter 00:00:20.500 --> 00:00:24.579 track. Nothing serious, just a workout. But it 00:00:24.579 --> 00:00:29.960 made me wonder, how many laps is a 5K? Now, since 00:00:29.960 --> 00:00:34.659 5 ,000 meters divided by 400 meters is 12 .5, 00:00:34.679 --> 00:00:39.299 the answer is 12 .5 laps. Now, in math, we would 00:00:39.299 --> 00:00:44.020 write 12 .5 laps equals one 5 -kilometer race. 00:00:44.460 --> 00:00:48.140 Now, at first, that seems pretty ordinary. But 00:00:48.140 --> 00:00:51.359 notice something. A distance measured in laps 00:00:51.359 --> 00:00:54.560 became equal to a distance measured in kilometers. 00:00:55.280 --> 00:00:58.469 Two different descriptions. but one reality. 00:00:59.329 --> 00:01:02.729 Now, suppose I do not like the decimal. I can 00:01:02.729 --> 00:01:05.890 multiply both sides by two, and now I have 25 00:01:05.890 --> 00:01:10.530 laps equals two 5 -kilometer races. Again, same 00:01:10.530 --> 00:01:14.349 relationships, different perspective. And now 00:01:14.349 --> 00:01:17.310 everything is a whole number. And what makes 00:01:17.310 --> 00:01:20.890 this interesting and kind of cool is on a track, 00:01:20.969 --> 00:01:25.140 if I run enough full laps around the track, I 00:01:25.140 --> 00:01:28.579 will eventually reach an integer multiple of 00:01:28.579 --> 00:01:32.719 5K distances. Now, we do this kind of thing all 00:01:32.719 --> 00:01:36.939 the time. Four quarters equals one dollar. Sixteen 00:01:36.939 --> 00:01:41.140 tablespoons equals one cup. Meters convert to 00:01:41.140 --> 00:01:44.780 miles. Inches convert to centimeters. And math 00:01:44.780 --> 00:01:47.959 gives us a way to recognize when two things that 00:01:47.959 --> 00:01:52.159 look different are actually equal. Now, what's 00:01:52.159 --> 00:01:54.659 even more interesting... is that many of these 00:01:54.659 --> 00:01:58.040 relationships are human inventions. We created 00:01:58.040 --> 00:02:02.400 dollars and cups and miles and kilometers. And 00:02:02.400 --> 00:02:05.459 because we created them, we can always find a 00:02:05.459 --> 00:02:10.120 conversion between them. If we keep scaling the 00:02:10.120 --> 00:02:13.080 relationship, eventually we can often express 00:02:13.080 --> 00:02:16.879 it using whole numbers. For example, one lap 00:02:16.879 --> 00:02:20.740 around a 400 -meter track is about one quarter 00:02:20.740 --> 00:02:24.340 of a mile. But if I keep running enough laps, 00:02:24.599 --> 00:02:27.800 eventually I can find a whole number of laps 00:02:27.800 --> 00:02:32.120 that equals a whole number of miles. Now the 00:02:32.120 --> 00:02:35.500 details may be messy, but the conversion exists. 00:02:36.599 --> 00:02:38.900 And to me, that's one of the beautiful things 00:02:38.900 --> 00:02:42.919 that mathematics does. It helps us to see equivalents. 00:02:43.259 --> 00:02:46.180 It helps us to recognize when two different descriptions 00:02:46.180 --> 00:02:50.180 are really talking about the same thing. But 00:02:50.180 --> 00:02:54.520 tomorrow. Tomorrow, we're going to look at a 00:02:54.520 --> 00:02:58.259 famous example involving a circle. And at first, 00:02:58.259 --> 00:03:00.840 it will seem like just another conversion problem, 00:03:00.979 --> 00:03:05.319 like today. But it leads to one of the most surprising 00:03:05.319 --> 00:03:09.580 discoveries in all of mathematics. We will discover 00:03:09.580 --> 00:03:13.020 that two distances that can never be converted 00:03:13.020 --> 00:03:16.979 into matching whole numbers. And that simple 00:03:16.979 --> 00:03:22.360 fact changed mathematics forever. I really hope 00:03:22.360 --> 00:03:23.060 to see you tomorrow.