WEBVTT 00:00:02.169 --> 00:00:05.309 Hi, yesterday we ended with a question. What 00:00:05.309 --> 00:00:10.289 does it mean for an infinite set to be big? Now 00:00:10.289 --> 00:00:12.730 that sounds like a simple question, but today 00:00:12.730 --> 00:00:15.449 we're going to discover that even the word big 00:00:15.449 --> 00:00:19.870 needs some fine tuning. Let's return to our number 00:00:19.870 --> 00:00:23.670 line. Pick any two different numbers, maybe one 00:00:23.670 --> 00:00:26.750 and two. There's a fraction between them, say 00:00:26.750 --> 00:00:31.160 three halves. Okay, zoom in again, pick say 1 00:00:31.160 --> 00:00:34.840 .4 and 1 .5, and there's a fraction between those 00:00:34.840 --> 00:00:38.539 two. And if we zoom in again, we'll find that 00:00:38.539 --> 00:00:40.820 no matter how close together two different numbers 00:00:40.820 --> 00:00:43.539 are, there's always a fraction between them. 00:00:44.079 --> 00:00:47.299 In fact, there are infinitely many fractions 00:00:47.299 --> 00:00:51.539 between them. Mathematicians say that the rational 00:00:51.539 --> 00:00:55.219 numbers, the fractions, are dense on the number 00:00:55.219 --> 00:00:59.390 line. You can zoom in as far as you want. and 00:00:59.390 --> 00:01:02.070 you'll never find an interval without fractions. 00:01:02.509 --> 00:01:07.650 They are everywhere. So that sounds like a pretty 00:01:07.650 --> 00:01:11.269 big collection of numbers. But here's our surprise. 00:01:11.810 --> 00:01:15.769 Despite being everywhere, get this, the rational 00:01:15.769 --> 00:01:19.890 numbers have measure zero on the number line. 00:01:20.969 --> 00:01:25.510 So how can something be everywhere and take up 00:01:25.510 --> 00:01:29.069 no length? Well, maybe you remember something 00:01:29.069 --> 00:01:31.930 that we've talked about before. The rational 00:01:31.930 --> 00:01:35.569 numbers are countable. And that means that we 00:01:35.569 --> 00:01:38.310 can arrange them in a list. We've got the first 00:01:38.310 --> 00:01:40.349 rational, we've got the second rational, we've 00:01:40.349 --> 00:01:44.109 got the third rational, and so on. So now that 00:01:44.109 --> 00:01:47.430 we have this list, imagine covering the first 00:01:47.430 --> 00:01:50.689 rational number with an interval length 1 half. 00:01:51.790 --> 00:01:55.030 Weird idea, but hang with me here. Cover the 00:01:55.030 --> 00:01:59.189 second with one of length 1 4th, the third gets 00:01:59.189 --> 00:02:03.069 1 8th, and so on. Now if we added up all those 00:02:03.069 --> 00:02:07.489 lengths, that adds up to 1. Okay, let's fine 00:02:07.489 --> 00:02:11.789 -tune our coverings. Let's give our first rational 00:02:11.789 --> 00:02:16.729 number an interval of length 1 over 2 ,000, and 00:02:16.729 --> 00:02:20.210 the second 1 over 4 ,000, and the third 1 over 00:02:20.210 --> 00:02:24.990 8 ,000, and so on. Every rational number still 00:02:24.990 --> 00:02:28.509 gets covered. But now the total length of all 00:02:28.509 --> 00:02:34.289 those intervals is only 1 over 1000. And there's 00:02:34.289 --> 00:02:37.310 nothing special about 1 over 1000. We could make 00:02:37.310 --> 00:02:41.289 that total length a millionth or a billionth 00:02:41.289 --> 00:02:46.370 or as small as we want. And every rational number 00:02:46.370 --> 00:02:50.490 would still be covered. And that is what we mean 00:02:50.490 --> 00:02:54.430 to say that the rationals have measure zero. 00:02:56.050 --> 00:02:58.150 I don't know about you, I think that's pretty 00:02:58.150 --> 00:03:01.490 cool. Yes, they're infinite. And yes, they're 00:03:01.490 --> 00:03:06.710 everywhere. And yet, together, they occupy no 00:03:06.710 --> 00:03:10.930 length at all. So let's return to our question. 00:03:11.150 --> 00:03:14.969 What does it mean for an infinite set to be big? 00:03:16.330 --> 00:03:19.650 Okay, are we asking how many numbers it contains? 00:03:20.530 --> 00:03:24.150 Or are we asking whether those numbers are everywhere, 00:03:24.310 --> 00:03:27.710 or are we asking how much of the number line 00:03:27.710 --> 00:03:31.949 they occupy? Those are different questions, and 00:03:31.949 --> 00:03:36.370 mathematics gives us different answers. So, our 00:03:36.370 --> 00:03:39.330 takeaway is that another reason fine -tuning 00:03:39.330 --> 00:03:42.689 matters is that sometimes the mathematics is 00:03:42.689 --> 00:03:49.960 not unclear. Our question is. Okay, tomorrow 00:03:49.960 --> 00:03:52.620 we'll take all those fractions, all those numbers 00:03:52.620 --> 00:03:55.879 that are everywhere, but occupy zero measure 00:03:55.879 --> 00:03:59.620 and we'll ask, what is left?