WEBVTT 00:00:01.959 --> 00:00:06.080 Hi, do you have anything that you fuss with until 00:00:06.080 --> 00:00:09.580 you get it just right? Maybe it's how you bake 00:00:09.580 --> 00:00:12.900 cookies or how you mow your lawn or wash your 00:00:12.900 --> 00:00:16.539 car. I know I do this with math videos that I 00:00:16.539 --> 00:00:20.780 create. I can spend a ton of time trying to get 00:00:20.780 --> 00:00:25.059 the story just right. I'll change one of my explanations 00:00:25.059 --> 00:00:28.699 or maybe I'll rearrange the order. Or I will 00:00:28.699 --> 00:00:31.620 redo something just because, you know, it just 00:00:31.620 --> 00:00:35.439 doesn't feel quite right yet. And eventually, 00:00:35.560 --> 00:00:39.640 I have to stop, of course. But I keep fine -tuning 00:00:39.640 --> 00:00:42.700 because I care about the story that I'm trying 00:00:42.700 --> 00:00:45.500 to tell. And that's what we're going to talk 00:00:45.500 --> 00:00:48.539 about this week. Fine -tuning in mathematics. 00:00:49.659 --> 00:00:52.740 Now, before we get into the math, let's consider 00:00:52.740 --> 00:00:56.200 a simple question. Why do we fine -tune anything 00:00:56.200 --> 00:01:00.539 to begin with? I thought about it. I think we 00:01:00.539 --> 00:01:03.939 fine tune things that we care about. For example, 00:01:03.979 --> 00:01:07.340 a musician fine tunes an instrument because a 00:01:07.340 --> 00:01:11.500 small difference in pitch matters. Or an engineer 00:01:11.500 --> 00:01:15.079 fine tunes a design because a small difference 00:01:15.079 --> 00:01:19.439 in measurement matters. We even fine tune our 00:01:19.439 --> 00:01:22.159 words when we're trying to say something important 00:01:22.159 --> 00:01:25.579 because sometimes almost saying what we mean 00:01:25.579 --> 00:01:29.719 is not good enough. And sometimes we fine -tune 00:01:29.719 --> 00:01:32.620 something simply because we're trying to make 00:01:32.620 --> 00:01:36.200 it more beautiful. Think about an artist trying 00:01:36.200 --> 00:01:38.359 to create the appearance of a three -dimensional 00:01:38.359 --> 00:01:42.819 world on a flat canvas. A little more shading 00:01:42.819 --> 00:01:46.480 here, a slight change in perspective there, and 00:01:46.480 --> 00:01:50.260 how about a touch of light or shadow? The canvas 00:01:50.260 --> 00:01:53.260 is still two -dimensional, but with careful fine 00:01:53.260 --> 00:01:57.519 -tuning, suddenly we see depth. Something flat 00:01:57.519 --> 00:02:00.099 begins to feel like a world that we could step 00:02:00.099 --> 00:02:04.680 into. Mathematicians do something similar. Sometimes 00:02:04.680 --> 00:02:07.680 we fine -tune mathematics because something is 00:02:07.680 --> 00:02:10.879 just not quite right. But sometimes something 00:02:10.879 --> 00:02:14.780 is already correct and we keep working because 00:02:14.780 --> 00:02:17.939 we're searching for something clearer, something 00:02:17.939 --> 00:02:23.120 simpler, or more beautiful. For example, I just 00:02:23.120 --> 00:02:26.419 learned a new proof for root two this week on 00:02:26.419 --> 00:02:29.979 how it is irrational. Now, I know many proofs, 00:02:29.979 --> 00:02:33.699 but now this is my favorite because it is the 00:02:33.699 --> 00:02:40.120 simplest and therefore the most beautiful. There 00:02:40.120 --> 00:02:42.520 is one more reason mathematics is particularly 00:02:42.520 --> 00:02:46.719 interesting. Mathematical truths can last for 00:02:46.719 --> 00:02:51.139 thousands of years. Root two is still irrational. 00:02:52.080 --> 00:02:55.000 The geometry Euclid studied more than 2 ,000 00:02:55.000 --> 00:02:58.900 years ago has not worn out. The Pythagorean theorem 00:02:58.900 --> 00:03:03.659 is not less true with age. So we should not be 00:03:03.659 --> 00:03:06.639 surprised that mathematicians care so much about 00:03:06.639 --> 00:03:10.479 fine -tuning. Precision matters, understanding 00:03:10.479 --> 00:03:13.979 matters, and beauty matters. And we will see 00:03:13.979 --> 00:03:17.539 all three this week. But today, let's just start 00:03:17.539 --> 00:03:20.650 with the number one. Imagine traveling a distance 00:03:20.650 --> 00:03:24.370 of exactly one meter. First, go halfway there, 00:03:24.509 --> 00:03:27.289 one half a meter. Then go half of what's left, 00:03:27.469 --> 00:03:30.729 another quarter of a meter. Now we're at three 00:03:30.729 --> 00:03:33.610 quarters of a meter. Do it again, go half the 00:03:33.610 --> 00:03:36.409 remaining distance, another one eighth. Now we 00:03:36.409 --> 00:03:39.270 are seven eighths, and we just keep doing that. 00:03:39.629 --> 00:03:42.990 So here's our question. If we continue traveling 00:03:42.990 --> 00:03:46.030 half the remaining distance forever, Do we ever 00:03:46.030 --> 00:03:51.469 arrive at exactly one meter? Welcome to the strange 00:03:51.469 --> 00:03:54.969 world of fine -tuning mathematics. And we will 00:03:54.969 --> 00:03:56.069 pick up there tomorrow.