WEBVTT 00:00:02.029 --> 00:00:04.990 Hi, yesterday I left you with a question. Do 00:00:04.990 --> 00:00:08.849 prime numbers go on forever? Or do they eventually 00:00:08.849 --> 00:00:12.970 run out? Did you think of any ideas? Did you 00:00:12.970 --> 00:00:17.190 find a way? It is not an easy problem. Because 00:00:17.190 --> 00:00:19.469 you cannot just keep checking numbers forever. 00:00:19.890 --> 00:00:22.910 So how can you possibly prove something about 00:00:22.910 --> 00:00:27.489 an infinite collection? Over 2 ,000 years ago, 00:00:27.589 --> 00:00:31.250 a mathematician, Euclid, faced this same question. 00:00:31.899 --> 00:00:34.960 Rather than trying to chase primes out to infinity, 00:00:35.259 --> 00:00:39.579 he turned the problem around. He said, let's 00:00:39.579 --> 00:00:42.880 suppose the opposite. Let's imagine that prime 00:00:42.880 --> 00:00:46.560 numbers do run out, that there is a complete 00:00:46.560 --> 00:00:50.579 list of all the primes. Maybe it's short, maybe 00:00:50.579 --> 00:00:53.679 it's long, but suppose we've found them all. 00:00:54.039 --> 00:00:57.600 Now here's the key idea. Take every prime on 00:00:57.600 --> 00:01:00.560 that list and multiply them together. It starts 00:01:00.560 --> 00:01:04.519 with 2 times 3 times 5 times 7 times 11, and 00:01:04.519 --> 00:01:07.459 then it goes to the final number that's in that 00:01:07.459 --> 00:01:14.140 set. This answer will equal one gigantic, enormous 00:01:14.140 --> 00:01:18.659 number. So every prime number will be a factor 00:01:18.659 --> 00:01:24.379 of this ginormous number. Then take that ginormous 00:01:24.379 --> 00:01:29.780 number and add 1. Just 1. Now pause and think 00:01:29.780 --> 00:01:32.079 about this new number, one bigger than the one 00:01:32.079 --> 00:01:35.420 we calculated. We know the original number is 00:01:35.420 --> 00:01:39.079 a factor of every prime number. That means this 00:01:39.079 --> 00:01:44.200 new number is not divisible by any of the primes 00:01:44.200 --> 00:01:47.299 on the list. Because when you divide it by any 00:01:47.299 --> 00:01:50.640 of them, there is always a remainder of one. 00:01:52.010 --> 00:01:55.709 So it does not fit into any of the rectangles 00:01:55.709 --> 00:01:58.590 that's built from the primes that we thought 00:01:58.590 --> 00:02:04.530 were complete. Which means either this number 00:02:04.530 --> 00:02:09.270 is itself prime or it is built from primes that 00:02:09.270 --> 00:02:14.289 are not on our list. Either way, our list was 00:02:14.289 --> 00:02:19.469 not complete. And that is the breakthrough. That 00:02:19.469 --> 00:02:23.009 is the surprise. No matter how many primes you 00:02:23.009 --> 00:02:26.530 think you have, you can always find another. 00:02:26.990 --> 00:02:32.409 So, primes do not run out. They can't. There 00:02:32.409 --> 00:02:36.969 are infinitely many of them. And what's remarkable 00:02:36.969 --> 00:02:41.789 is how this answer comes. Not from checking forever, 00:02:42.069 --> 00:02:47.490 but from a simple idea. Assume you're done, and 00:02:47.490 --> 00:02:53.949 then show you can't be. From rocks to rectangles 00:02:53.949 --> 00:02:59.509 to infinity. A question a child can ask. And 00:02:59.509 --> 00:03:02.069 an answer that stretches beyond the horizon. 00:03:02.610 --> 00:03:07.270 And if you follow that even just a little, you 00:03:07.270 --> 00:03:11.990 have stepped into something amazing. You are 00:03:11.990 --> 00:03:15.770 sharing a line of thought with one of the great 00:03:15.770 --> 00:03:20.539 mathematical minds in history. Not just knowing 00:03:20.539 --> 00:03:26.300 a fact, but understanding why it is true. To 00:03:26.300 --> 00:03:29.759 me, that is something to value. Because it means 00:03:29.759 --> 00:03:34.280 you can do this. You can learn ideas that at 00:03:34.280 --> 00:03:38.460 first feel out of reach. And this is just the 00:03:38.460 --> 00:03:41.659 beginning. The beginning of seeing that mathematics 00:03:41.659 --> 00:03:46.060 is not just about answers, but about understanding. 00:03:47.370 --> 00:03:53.789 And that understanding is available to you. And 00:03:53.789 --> 00:03:57.349 that is the beauty of math. It enlightens our 00:03:57.349 --> 00:04:00.469 mind beyond what we thought was true.