WEBVTT 00:00:03.650 --> 00:00:06.710 Hi, yesterday we discovered a surprising thing 00:00:06.710 --> 00:00:11.349 that an infinite decimal like 0 .999 dot dot 00:00:11.349 --> 00:00:15.509 dot could equal something exact. Today, let's 00:00:15.509 --> 00:00:18.850 take another kind of infinite process. This time, 00:00:18.929 --> 00:00:22.690 not with decimals, but with addition. Consider 00:00:22.690 --> 00:00:26.210 the sum, 1 half plus 1 fourth plus 1 eighth plus 00:00:26.210 --> 00:00:29.870 1 sixteenth plus dot dot dot. The pattern continues 00:00:29.870 --> 00:00:33.390 forever. Each term is half of the previous term. 00:00:34.030 --> 00:00:37.229 Now, at first, you may wonder, how can we possibly 00:00:37.229 --> 00:00:40.990 add an infinite number of terms? And even if 00:00:40.990 --> 00:00:44.189 we could, would the total continue growing forever? 00:00:44.549 --> 00:00:48.990 Or would it approach a number? Well, this becomes 00:00:48.990 --> 00:00:51.890 easier to see if we imagine something familiar, 00:00:52.070 --> 00:00:56.570 like another pizza. So first, cut the pizza in 00:00:56.570 --> 00:01:00.159 half and eat one half. Then take the remaining 00:01:00.159 --> 00:01:03.399 half, cut it in half again, and eat one of those 00:01:03.399 --> 00:01:07.439 pieces. And then repeat. And keep going forever. 00:01:07.879 --> 00:01:11.579 Each time you are adding another piece. But each 00:01:11.579 --> 00:01:15.319 new piece becomes smaller and smaller. Notice 00:01:15.319 --> 00:01:18.659 this is the same math problem. One half plus 00:01:18.659 --> 00:01:22.120 one fourth plus one eighth plus dot dot dot. 00:01:23.180 --> 00:01:26.700 So here is our question. If this process never 00:01:26.700 --> 00:01:32.200 ends, What does the total equal? Since there 00:01:32.200 --> 00:01:34.700 is always another piece left for the next step, 00:01:34.840 --> 00:01:37.780 it feels like we should never completely arrive 00:01:37.780 --> 00:01:41.840 at a whole pizza. And yet, as we continue adding 00:01:41.840 --> 00:01:45.379 those smaller and smaller pieces, the total gets 00:01:45.379 --> 00:01:49.500 closer and closer to one. After one step, one 00:01:49.500 --> 00:01:53.900 half. After two, three quarters. Then seven eighths. 00:01:53.900 --> 00:01:58.790 Then 15 over 16. Each step moves closer to 1. 00:02:00.189 --> 00:02:02.269 This connects to the idea that we introduced 00:02:02.269 --> 00:02:06.310 yesterday, a limit. A limit describes the value 00:02:06.310 --> 00:02:09.310 a process approaches as it continues forever. 00:02:10.090 --> 00:02:14.210 And in this case, the process approaches 1 perfectly. 00:02:14.729 --> 00:02:19.349 So mathematicians say 1 half plus 1 fourth plus 00:02:19.349 --> 00:02:23.370 1 eighth plus 1 sixteenth plus dot dot dot equals 00:02:23.370 --> 00:02:28.469 1 exactly. Not because the process eventually 00:02:28.469 --> 00:02:32.830 stops, but because the infinite pattern points 00:02:32.830 --> 00:02:37.110 precisely toward one. And this is where infinity 00:02:37.110 --> 00:02:40.110 begins to feel different from ordinary experience. 00:02:41.009 --> 00:02:44.610 In everyday life, we often think if something 00:02:44.610 --> 00:02:47.650 never finishes, then it never truly arrives. 00:02:48.409 --> 00:02:54.050 But infinity challenges that intuition. In mathematics, 00:02:54.370 --> 00:02:57.520 an endless process can still describe something 00:02:57.520 --> 00:03:02.860 complete. That idea became one of the foundations 00:03:02.860 --> 00:03:07.639 of calculus and physics, engineering, and much 00:03:07.639 --> 00:03:12.759 of modern science. And once again, infinity reveals 00:03:12.759 --> 00:03:17.740 hidden structure beneath something simple. Not 00:03:17.740 --> 00:03:22.159 chaos or randomness, but direction or precision, 00:03:22.500 --> 00:03:28.199 maybe meaning. Infinity is no longer just more 00:03:28.199 --> 00:03:32.580 and more. Now it becomes something we can approach, 00:03:32.860 --> 00:03:38.259 describe, and understand. And that became one 00:03:38.259 --> 00:03:41.159 of the great turning points in human understanding. 00:03:43.379 --> 00:03:46.240 So here's a thought that extends beyond math 00:03:46.240 --> 00:03:49.879 that I want to leave with you today. The theme 00:03:49.879 --> 00:03:52.340 that we've discovered this week is that infinity 00:03:52.889 --> 00:03:57.370 is not simply a lot more. It often becomes something 00:03:57.370 --> 00:04:01.210 fundamentally different. And that makes me wonder, 00:04:01.370 --> 00:04:05.949 if we imagine a life that does not end, should 00:04:05.949 --> 00:04:11.169 we think of it as merely more time? Or could 00:04:11.169 --> 00:04:14.870 it be something different in quality altogether? 00:04:17.089 --> 00:04:20.069 Give this idea some thought, and we will return 00:04:20.069 --> 00:04:21.490 to it at the end of the week.