WEBVTT 00:00:01.849 --> 00:00:05.469 Hi. From my observation, there are essentially 00:00:05.469 --> 00:00:09.109 two different mindsets. One mindset is fixed, 00:00:09.289 --> 00:00:13.130 where problems get bigger and possibilities get 00:00:13.130 --> 00:00:16.570 smaller. The other is a growth mindset, where 00:00:16.570 --> 00:00:19.789 we ask, what is a bigger reality that I'm not 00:00:19.789 --> 00:00:23.910 yet seeing? This week is for that second mindset. 00:00:24.250 --> 00:00:29.089 And our topic for the week is infinity. On one 00:00:29.089 --> 00:00:33.689 hand, infinity feels vague. What even is infinity? 00:00:34.170 --> 00:00:36.950 But on the other hand, it feels straightforward. 00:00:37.229 --> 00:00:40.850 The positive integers go on forever. One, two, 00:00:41.109 --> 00:00:45.090 three, four, dot, dot, dot. Given any positive 00:00:45.090 --> 00:00:47.750 integer, we can always create the next integer 00:00:47.750 --> 00:00:51.490 simply by adding one. And because we can always 00:00:51.490 --> 00:00:55.750 add one more, the integers continue without bound. 00:00:56.950 --> 00:01:01.960 That idea feels intuitive. We can also look at 00:01:01.960 --> 00:01:05.099 subset of integers, like the odd numbers. The 00:01:05.099 --> 00:01:08.099 odd numbers are also infinite because we always 00:01:08.099 --> 00:01:11.719 know how to create the next odd number. Simply 00:01:11.719 --> 00:01:16.359 add two. Last week, we studied prime numbers. 00:01:17.040 --> 00:01:19.879 Prime numbers are another special set of integers. 00:01:20.459 --> 00:01:24.000 But unlike the odd numbers, we do not have a 00:01:24.000 --> 00:01:26.719 simple pattern for generating the next prime 00:01:26.719 --> 00:01:30.709 number. In fact, prime numbers become more spread 00:01:30.709 --> 00:01:35.650 out as numbers get larger. So this raises a natural 00:01:35.650 --> 00:01:40.030 question. Is there a largest prime number? Or 00:01:40.030 --> 00:01:44.530 do the primes always continue forever? This question 00:01:44.530 --> 00:01:47.329 is much more difficult because knowing that, 00:01:47.390 --> 00:01:51.609 say, 107 is prime tells us almost nothing about 00:01:51.609 --> 00:01:55.409 what the next prime number will be. So how could 00:01:55.409 --> 00:01:58.700 we possibly know whether the primes are infinite. 00:02:00.060 --> 00:02:03.799 Well, more than 2 ,000 years ago, Euclid proved 00:02:03.799 --> 00:02:06.700 that there are infinitely many prime numbers. 00:02:07.239 --> 00:02:11.520 And this opened the door to a new kind of infinity. 00:02:12.780 --> 00:02:16.219 Not an infinity based on a simple repeating pattern 00:02:16.219 --> 00:02:21.460 like adding 1 or adding 2, but an infinity hidden 00:02:21.460 --> 00:02:25.889 inside something unpredictable. Suddenly, The 00:02:25.889 --> 00:02:29.129 positive integers no longer look like one simple 00:02:29.129 --> 00:02:32.729 collection. Hidden structure began to emerge. 00:02:33.469 --> 00:02:36.150 It's a little like looking at Earth from far 00:02:36.150 --> 00:02:40.129 away. From a distance, it appears as one sphere. 00:02:40.550 --> 00:02:44.550 But as we move closer, we begin to notice oceans 00:02:44.550 --> 00:02:49.310 and continents, mountains and deserts. Structure 00:02:49.310 --> 00:02:53.069 hidden within what first appeared simple. And 00:02:53.069 --> 00:02:56.650 doing mathematics is often like that. We take 00:02:56.650 --> 00:03:00.009 familiar things, like numbers, and learn to see 00:03:00.009 --> 00:03:03.370 them in deeper ways. And this week, we're going 00:03:03.370 --> 00:03:06.830 to do that with infinity. We will discover that 00:03:06.830 --> 00:03:09.409 prime numbers are only the tip of the infinity 00:03:09.409 --> 00:03:15.030 iceberg. And what I find beautiful is that exploring 00:03:15.030 --> 00:03:18.710 this iceberg has been a deeply human journey. 00:03:20.270 --> 00:03:23.349 Human history is filled with many stories of 00:03:23.349 --> 00:03:28.069 war, division, destruction, and the way we tear 00:03:28.069 --> 00:03:32.189 things apart. But this week, we're going to study 00:03:32.189 --> 00:03:35.849 a fresh side of human history. A story of people 00:03:35.849 --> 00:03:39.669 across civilizations and centuries building something 00:03:39.669 --> 00:03:45.409 together. Greek and Indian mathematicians. Islamic 00:03:45.409 --> 00:03:49.650 scholars. The list goes on. Different cultures. 00:03:50.250 --> 00:03:55.030 different languages, different centuries. Despite 00:03:55.030 --> 00:03:59.289 these differences, step by step, humanity uncovered 00:03:59.289 --> 00:04:03.370 deeper truths about infinity and slowly built 00:04:03.370 --> 00:04:06.050 a shared structure of understanding together. 00:04:07.330 --> 00:04:11.189 To me, that is one of the most remarkable and 00:04:11.189 --> 00:04:16.069 beautiful parts of mathematics. So I am super 00:04:16.069 --> 00:04:19.740 excited. to explore a few pieces of that larger 00:04:19.740 --> 00:04:24.879 infinity story with you this week. So, are you 00:04:24.879 --> 00:04:26.800 ready to grow your mind?