WEBVTT 00:00:03.390 --> 00:00:06.969 Hi. Yesterday, we opened the door to infinity 00:00:06.969 --> 00:00:10.349 through prime numbers. Today, infinity appears 00:00:10.349 --> 00:00:14.390 in a familiar yet unexpected place, inside a 00:00:14.390 --> 00:00:18.089 simple fraction. In math, we often take for granted 00:00:18.089 --> 00:00:20.489 that numbers can be expressed in different ways. 00:00:21.089 --> 00:00:26.390 For example, one half equals 0 .5. No surprises 00:00:26.390 --> 00:00:30.390 and nothing mysterious. A fraction and a decimal. 00:00:30.989 --> 00:00:33.130 Two different ways to describe the same number. 00:00:33.829 --> 00:00:36.429 Now consider the simple fraction one -third. 00:00:37.450 --> 00:00:40.109 We see this number in life simply by dividing 00:00:40.109 --> 00:00:43.770 a pizza into three equal slices, then one slice 00:00:43.770 --> 00:00:47.369 is one -third of the pizza. When we convert one 00:00:47.369 --> 00:00:54.609 -third into a decimal, we get 0 .333... The threes 00:00:54.609 --> 00:00:59.469 never end. Hmm. Suddenly, hidden beneath this 00:00:59.469 --> 00:01:02.469 simple fraction, we discover an infinite process. 00:01:03.329 --> 00:01:08.109 Infinity was there the whole time. At first, 00:01:08.109 --> 00:01:10.790 this infinity may not seem too mysterious. We 00:01:10.790 --> 00:01:13.530 still have a pattern to follow. Just keep adding 00:01:13.530 --> 00:01:16.909 more threes. But now let's combine the three 00:01:16.909 --> 00:01:19.849 pieces of pizza and put them back together. In 00:01:19.849 --> 00:01:22.290 other words, one -third plus one -third plus 00:01:22.290 --> 00:01:25.909 one -third. From the fraction perspective, the 00:01:25.909 --> 00:01:29.540 answer is obviously one. Three -thirds. make 00:01:29.540 --> 00:01:33.459 one whole. Writing the same problem using decimal 00:01:33.459 --> 00:01:38.760 produces 0 .333 dot dot dot plus 0 .333 dot dot 00:01:38.760 --> 00:01:43.760 dot plus 0 .333 dot dot dot which equals 0 .99 00:01:43.760 --> 00:01:48.700 dot dot dot. And now we seem to have a paradox. 00:01:49.140 --> 00:01:52.799 The fraction representation says the answer is 00:01:52.799 --> 00:01:56.579 1, but the decimal representation appears to 00:01:56.579 --> 00:02:00.500 say the answer is something slightly less. So, 00:02:00.640 --> 00:02:05.640 which answer is correct? This is where infinity 00:02:05.640 --> 00:02:08.740 forces us to think more carefully about what 00:02:08.740 --> 00:02:12.659 numbers actually are. What happens when two correct 00:02:12.659 --> 00:02:15.460 ways of describing the same number appear to 00:02:15.460 --> 00:02:19.139 disagree? We normally think both descriptions 00:02:19.139 --> 00:02:22.539 are completely compatible. But how do we reconcile 00:02:22.539 --> 00:02:27.680 when they appear to give different answers? Historically, 00:02:27.740 --> 00:02:30.199 questions like this cause enormous confusion 00:02:30.199 --> 00:02:34.680 in mathematics. The question at play is how can 00:02:34.680 --> 00:02:37.659 something go on forever and still equal something 00:02:37.659 --> 00:02:42.099 exact? The answer eventually came through an 00:02:42.099 --> 00:02:46.659 important idea called a limit. A limit describes 00:02:46.659 --> 00:02:50.000 the value of a process approaches as it continues 00:02:50.000 --> 00:02:53.770 forever. And in this case, we continue adding 00:02:53.770 --> 00:02:59.030 more nines. We go from 0 .9 to 0 .99 to 0 .999. 00:02:59.409 --> 00:03:03.270 And as we add nines, the limit of the number 00:03:03.270 --> 00:03:08.270 approaches 1 perfectly. There is no gap left. 00:03:08.590 --> 00:03:17.270 So mathematicians define 0 .99... to equal 1 00:03:17.270 --> 00:03:22.710 exactly. Not approximately and not close enough. 00:03:22.969 --> 00:03:27.629 Exactly the same number. And this is where infinity 00:03:27.629 --> 00:03:31.629 can change the way we think. Infinity is no longer 00:03:31.629 --> 00:03:35.729 just counting forever. Now it begins reshaping 00:03:35.729 --> 00:03:40.050 what numbers themselves can mean. A simple fraction. 00:03:40.289 --> 00:03:44.229 A familiar decimal. And suddenly we discover 00:03:44.229 --> 00:03:48.189 there was an infinity iceberg. hidden beneath 00:03:48.189 --> 00:03:55.169 them all along. So, what do you think about mathematicians 00:03:55.169 --> 00:04:00.250 declaring that 0 .99 dot dot dot exactly equals 00:04:00.250 --> 00:04:05.729 1? Does it feel reasonable? Does it feel like 00:04:05.729 --> 00:04:09.530 cheating? Is it surprising that mathematicians 00:04:09.530 --> 00:04:13.030 agree on something that initially feels so strange? 00:04:14.210 --> 00:04:17.800 Maybe this feels a bit unsettling. And don't 00:04:17.800 --> 00:04:21.680 ignore that feeling. Give it some space. And 00:04:21.680 --> 00:04:25.540 welcome to the world of infinity, where logic 00:04:25.540 --> 00:04:29.459 sometimes stretches beyond intuition. And we 00:04:29.459 --> 00:04:31.720 will see that intuition stretch further this 00:04:31.720 --> 00:04:35.800 week as we uncover stranger stories of infinity.