WEBVTT 00:00:01.610 --> 00:00:05.330 Hi. This week, we have seen infinity appear in 00:00:05.330 --> 00:00:08.609 many different ways. We have seen it in decimals, 00:00:08.630 --> 00:00:12.750 in infinite sums, and even hidden inside geometry 00:00:12.750 --> 00:00:17.750 itself. But today, we ask a new question. Are 00:00:17.750 --> 00:00:23.530 all infinities the same size? Well, at first, 00:00:23.629 --> 00:00:27.629 this feels like an impossible question. But intuitively, 00:00:28.489 --> 00:00:31.289 we might guess that some infinity sets are larger 00:00:31.289 --> 00:00:35.630 than others. For example, maybe the positive 00:00:35.630 --> 00:00:38.490 integers are twice as large as the even numbers, 00:00:38.670 --> 00:00:41.429 since the even numbers are only half of the integers. 00:00:42.250 --> 00:00:46.570 That feels reasonable. But mathematics surprises 00:00:46.570 --> 00:00:50.369 us by concluding that the even numbers and the 00:00:50.369 --> 00:00:54.469 positive integers are exactly the same size of 00:00:54.469 --> 00:00:59.770 infinity. That already feels strange. After all, 00:00:59.810 --> 00:01:02.630 the even numbers appear to be only half of the 00:01:02.630 --> 00:01:07.349 positive integers. But in infinity, appearances 00:01:07.349 --> 00:01:11.269 can be misleading. And it even gets stranger. 00:01:11.829 --> 00:01:16.390 The rational numbers, all fractions, also turn 00:01:16.390 --> 00:01:19.569 out to have the same size of infinity as the 00:01:19.569 --> 00:01:24.930 integers. Wow! the rational numbers seem vastly 00:01:24.930 --> 00:01:29.049 larger than the integers. After all, between 00:01:29.049 --> 00:01:33.230 any two integers, there are infinitely many fractions. 00:01:34.829 --> 00:01:38.189 So at this point, it may begin to feel like perhaps 00:01:38.189 --> 00:01:43.370 all infinities are the same size. But they're 00:01:43.370 --> 00:01:48.390 not. Let's climb one step higher. The real numbers 00:01:48.390 --> 00:01:51.769 consist of both rational and irrational numbers. 00:01:52.480 --> 00:01:54.719 we've already encountered two famous irrational 00:01:54.719 --> 00:01:59.219 numbers, square root of 2 and pi. Those are numbers 00:01:59.219 --> 00:02:02.180 whose decimals continue forever without repeating. 00:02:03.060 --> 00:02:06.420 There are also an infinitely many irrational 00:02:06.420 --> 00:02:11.360 numbers. So here's a question maybe my grandkids 00:02:11.360 --> 00:02:15.180 would ask. If we compare the infinity of irrational 00:02:15.180 --> 00:02:18.280 numbers to the infinity of rational numbers, 00:02:18.620 --> 00:02:23.639 which infinity is bigger? Following our previous 00:02:23.639 --> 00:02:26.400 pattern, we might expect that they are the same 00:02:26.400 --> 00:02:32.000 size. But they are not. The irrational numbers 00:02:32.000 --> 00:02:37.659 form a larger infinity. And what really astonishes 00:02:37.659 --> 00:02:40.759 me is that if you could somehow choose a number 00:02:40.759 --> 00:02:44.199 at random from the number line, the probability 00:02:44.199 --> 00:02:50.800 of selecting a rational number is zero. And the 00:02:50.800 --> 00:02:53.580 probability of selecting an irrational number 00:02:53.580 --> 00:02:59.659 is 100%. Now, that does not mean rational numbers 00:02:59.659 --> 00:03:04.680 do not exist. They are still infinite. But irrational 00:03:04.680 --> 00:03:10.520 numbers are so overwhelmingly abundant that rational 00:03:10.520 --> 00:03:15.500 numbers almost disappear in comparison. The rationals 00:03:15.500 --> 00:03:20.240 include all integers and all fractions. and yet 00:03:20.240 --> 00:03:24.319 they form only a tiny part of the number line. 00:03:24.719 --> 00:03:30.400 Almost every number is irrational. This is where 00:03:30.400 --> 00:03:34.580 infinity becomes something fundamentally different 00:03:34.580 --> 00:03:41.199 from simply very large. Infinity now has hierarchy, 00:03:41.580 --> 00:03:47.949 depth, structure. Some infinities are vastly 00:03:47.949 --> 00:03:51.930 larger than others. This was one of the great 00:03:51.930 --> 00:03:55.430 turning points in mathematics. Infinity stopped 00:03:55.430 --> 00:03:59.050 being viewed as merely an endless quantity and 00:03:59.050 --> 00:04:01.650 became something mathematicians could study, 00:04:01.750 --> 00:04:07.689 compare, and explore. And if all of this feels 00:04:07.689 --> 00:04:12.650 difficult to absorb, you are not alone. Infinity 00:04:12.650 --> 00:04:17.160 stretches the mind. So what part of this story 00:04:17.160 --> 00:04:23.100 feels most difficult for you to believe? Be sure 00:04:23.100 --> 00:04:25.660 to join us tomorrow for our final reflection 00:04:25.660 --> 00:04:29.240 on infinity. We will push our growth mindset 00:04:29.240 --> 00:04:34.060 even further as we consider eternal ideas.