﻿[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:04.42,0:00:07.22,Default,,0000,0000,0000,,Imagine we are living in prehistoric times. Dialogue: 0,0:00:07.22,0:00:09.47,Default,,0000,0000,0000,,Now, consider the following: Dialogue: 0,0:00:09.47,0:00:12.72,Default,,0000,0000,0000,,How did we keep track of time without a clock? Dialogue: 0,0:00:12.72,0:00:15.32,Default,,0000,0000,0000,,All clocks are based on some repetitive pattern Dialogue: 0,0:00:15.32,0:00:18.89,Default,,0000,0000,0000,,which divides the flow of time into equal segments. Dialogue: 0,0:00:18.89,0:00:20.69,Default,,0000,0000,0000,,To find these repetitive patterns, Dialogue: 0,0:00:20.69,0:00:22.92,Default,,0000,0000,0000,,we look towards the heavens. Dialogue: 0,0:00:22.92,0:00:24.90,Default,,0000,0000,0000,,The sun rising and falling each day Dialogue: 0,0:00:24.90,0:00:26.18,Default,,0000,0000,0000,,is the most obvious [pattern]. Dialogue: 0,0:00:26.18,0:00:28.76,Default,,0000,0000,0000,,However, to keep track of longer periods of time, Dialogue: 0,0:00:28.76,0:00:30.81,Default,,0000,0000,0000,,we looked for longer cycles. Dialogue: 0,0:00:30.81,0:00:32.51,Default,,0000,0000,0000,,For this, we looked to the moon, Dialogue: 0,0:00:32.51,0:00:33.85,Default,,0000,0000,0000,,which seemed to gradually grow Dialogue: 0,0:00:33.85,0:00:36.58,Default,,0000,0000,0000,,and shrink over many days. Dialogue: 0,0:00:36.58,0:00:37.89,Default,,0000,0000,0000,,When we count the number of days Dialogue: 0,0:00:37.89,0:00:38.98,Default,,0000,0000,0000,,between full moons, Dialogue: 0,0:00:38.98,0:00:40.91,Default,,0000,0000,0000,,we arrive at the number 29. Dialogue: 0,0:00:40.91,0:00:42.83,Default,,0000,0000,0000,,This is the origin of a month. Dialogue: 0,0:00:42.83,0:00:45.87,Default,,0000,0000,0000,,However, if we try to divide 29 into equal pieces, Dialogue: 0,0:00:45.87,0:00:49.23,Default,,0000,0000,0000,,we run into a problem: it is impossible. Dialogue: 0,0:00:49.23,0:00:51.68,Default,,0000,0000,0000,,The only way to divide 29 into equal pieces Dialogue: 0,0:00:51.68,0:00:54.82,Default,,0000,0000,0000,,is to break it back down into [29] single units. Dialogue: 0,0:00:54.82,0:00:57.10,Default,,0000,0000,0000,,29 is a 'prime number.' Dialogue: 0,0:00:57.10,0:00:59.06,Default,,0000,0000,0000,,Think of it as unbreakable. Dialogue: 0,0:00:59.06,0:01:00.88,Default,,0000,0000,0000,,If a number can be broken down into Dialogue: 0,0:01:00.88,0:01:02.81,Default,,0000,0000,0000,,equal pieces greater than one, Dialogue: 0,0:01:02.81,0:01:04.62,Default,,0000,0000,0000,,we call it a 'composite number.' Dialogue: 0,0:01:04.62,0:01:06.61,Default,,0000,0000,0000,,Now if we are curious, we may wonder, Dialogue: 0,0:01:06.61,0:01:08.45,Default,,0000,0000,0000,,"How many prime numbers are there? Dialogue: 0,0:01:08.45,0:01:10.40,Default,,0000,0000,0000,,– and how big do they get?" Dialogue: 0,0:01:10.40,0:01:13.74,Default,,0000,0000,0000,,Let's start by dividing all numbers into two categories. Dialogue: 0,0:01:13.74,0:01:15.61,Default,,0000,0000,0000,,We list the primes on the left Dialogue: 0,0:01:15.61,0:01:17.65,Default,,0000,0000,0000,,and the composites on the right. Dialogue: 0,0:01:17.65,0:01:20.38,Default,,0000,0000,0000,,At first, they seem to dance back and forth. Dialogue: 0,0:01:20.38,0:01:23.02,Default,,0000,0000,0000,,There is no obvious pattern here. Dialogue: 0,0:01:23.02,0:01:24.44,Default,,0000,0000,0000,,So let's use a modern technique Dialogue: 0,0:01:24.44,0:01:26.08,Default,,0000,0000,0000,,to see the big picture. Dialogue: 0,0:01:26.08,0:01:29.05,Default,,0000,0000,0000,,The trick is to use a "Ulam spiral." Dialogue: 0,0:01:29.05,0:01:32.01,Default,,0000,0000,0000,,First, we list all possible numbers in order Dialogue: 0,0:01:32.01,0:01:34.04,Default,,0000,0000,0000,,in a growing spiral. Dialogue: 0,0:01:34.04,0:01:37.16,Default,,0000,0000,0000,,Then, we color all the prime numbers blue. Dialogue: 0,0:01:37.16,0:01:41.29,Default,,0000,0000,0000,,Finally, we zoom out to see millions of numbers. Dialogue: 0,0:01:41.29,0:01:42.86,Default,,0000,0000,0000,,This is the pattern of primes Dialogue: 0,0:01:42.86,0:01:45.36,Default,,0000,0000,0000,,which goes on and on, forever. Dialogue: 0,0:01:45.36,0:01:47.97,Default,,0000,0000,0000,,Incredibly, the entire structure of this pattern Dialogue: 0,0:01:47.97,0:01:50.31,Default,,0000,0000,0000,,is still unsolved today. Dialogue: 0,0:01:50.31,0:01:51.84,Default,,0000,0000,0000,,We are onto something. Dialogue: 0,0:01:51.84,0:01:52.99,Default,,0000,0000,0000,,So, let's fast forward to Dialogue: 0,0:01:52.99,0:01:55.53,Default,,0000,0000,0000,,around 300 BC, in ancient Greece. Dialogue: 0,0:01:55.53,0:01:58.18,Default,,0000,0000,0000,,A philosopher known as Euclid of Alexandria Dialogue: 0,0:01:58.18,0:01:59.41,Default,,0000,0000,0000,,understood that all numbers Dialogue: 0,0:01:59.41,0:02:02.61,Default,,0000,0000,0000,,could be split into these two distinct categories. Dialogue: 0,0:02:02.61,0:02:04.90,Default,,0000,0000,0000,,He began by realizing that any number Dialogue: 0,0:02:04.90,0:02:07.08,Default,,0000,0000,0000,,can be divided down – over and over – Dialogue: 0,0:02:07.08,0:02:10.60,Default,,0000,0000,0000,,until you reach a group of smallest equal numbers. Dialogue: 0,0:02:10.60,0:02:12.92,Default,,0000,0000,0000,,And by definition, these smallest numbers Dialogue: 0,0:02:12.92,0:02:15.76,Default,,0000,0000,0000,,are always prime numbers. Dialogue: 0,0:02:15.76,0:02:17.15,Default,,0000,0000,0000,,So, he knew that all numbers are Dialogue: 0,0:02:17.15,0:02:20.54,Default,,0000,0000,0000,,somehow built out of smaller primes. Dialogue: 0,0:02:20.54,0:02:23.32,Default,,0000,0000,0000,,To be clear, imagine the universe of all numbers – Dialogue: 0,0:02:23.32,0:02:25.67,Default,,0000,0000,0000,,and ignore the primes. Dialogue: 0,0:02:25.67,0:02:28.04,Default,,0000,0000,0000,,Now, pick any composite number, Dialogue: 0,0:02:28.04,0:02:30.52,Default,,0000,0000,0000,,and break it down, Dialogue: 0,0:02:30.52,0:02:33.35,Default,,0000,0000,0000,,and you are always left with prime numbers. Dialogue: 0,0:02:33.35,0:02:34.77,Default,,0000,0000,0000,,So, Euclid knew that every number Dialogue: 0,0:02:34.77,0:02:37.68,Default,,0000,0000,0000,,could be expressed using a group of smaller primes. Dialogue: 0,0:02:37.68,0:02:40.22,Default,,0000,0000,0000,,Think of these as building blocks. Dialogue: 0,0:02:40.22,0:02:41.100,Default,,0000,0000,0000,,No matter what number you choose, Dialogue: 0,0:02:41.100,0:02:46.16,Default,,0000,0000,0000,,it can always be built with an addition of smaller primes. Dialogue: 0,0:02:46.16,0:02:48.03,Default,,0000,0000,0000,,This is the root of his discovery, Dialogue: 0,0:02:48.03,0:02:50.76,Default,,0000,0000,0000,,known as the 'Fundamental Theorem of Arithmetic' – Dialogue: 0,0:02:50.76,0:02:52.01,Default,,0000,0000,0000,,as follows: Dialogue: 0,0:02:52.01,0:02:53.93,Default,,0000,0000,0000,,Take any number – say 30 – Dialogue: 0,0:02:53.93,0:02:55.50,Default,,0000,0000,0000,,and find all the prime numbers Dialogue: 0,0:02:55.50,0:02:57.23,Default,,0000,0000,0000,,it [can be divided into] equally. Dialogue: 0,0:02:57.23,0:02:59.76,Default,,0000,0000,0000,,This we know as 'factorization.' Dialogue: 0,0:02:59.76,0:03:01.62,Default,,0000,0000,0000,,This will give us the prime factors. Dialogue: 0,0:03:01.62,0:03:05.81,Default,,0000,0000,0000,,In this case 2, 3, and 5 are the prime factors of 30. Dialogue: 0,0:03:05.81,0:03:07.91,Default,,0000,0000,0000,,Euclid realized that you could then multiply Dialogue: 0,0:03:07.91,0:03:10.71,Default,,0000,0000,0000,,these prime factors a specific number of times Dialogue: 0,0:03:10.71,0:03:12.74,Default,,0000,0000,0000,,to build the original number. Dialogue: 0,0:03:12.74,0:03:13.78,Default,,0000,0000,0000,,In this case, you simply Dialogue: 0,0:03:13.78,0:03:16.18,Default,,0000,0000,0000,,multiply each factor once to build 30. Dialogue: 0,0:03:16.18,0:03:20.16,Default,,0000,0000,0000,,2 × 3 × 5 is the prime factorization of 30. Dialogue: 0,0:03:20.16,0:03:23.15,Default,,0000,0000,0000,,Think of it as a special key or combination. Dialogue: 0,0:03:23.15,0:03:24.89,Default,,0000,0000,0000,,There is no other way to build 30, Dialogue: 0,0:03:24.89,0:03:27.11,Default,,0000,0000,0000,,using some other groups of prime numbers Dialogue: 0,0:03:27.11,0:03:28.79,Default,,0000,0000,0000,,multiplied together. Dialogue: 0,0:03:28.79,0:03:31.28,Default,,0000,0000,0000,,So every possible number has one – Dialogue: 0,0:03:31.28,0:03:34.05,Default,,0000,0000,0000,,and only one – prime factorization. Dialogue: 0,0:03:34.05,0:03:36.30,Default,,0000,0000,0000,,A good analogy is to imagine each number Dialogue: 0,0:03:36.30,0:03:38.02,Default,,0000,0000,0000,,as a different lock. Dialogue: 0,0:03:38.03,0:03:39.72,Default,,0000,0000,0000,,The unique key for each lock Dialogue: 0,0:03:39.72,0:03:42.05,Default,,0000,0000,0000,,would be its prime factorization. Dialogue: 0,0:03:42.05,0:03:43.94,Default,,0000,0000,0000,,No two locks share a key. Dialogue: 0,0:03:43.94,0:03:47.89,Default,,0000,0000,0000,,No two numbers share a prime factorization.