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Dialogue: 0,0:00:00.92,0:00:02.47,Default,,0000,0000,0000,,In this video I want to talk a little bit about
Dialogue: 0,0:00:02.47,0:00:06.60,Default,,0000,0000,0000,,what it means to be a prime number
Dialogue: 0,0:00:06.60,0:00:09.60,Default,,0000,0000,0000,,and what you will hopefully see in this video
Dialogue: 0,0:00:09.60,0:00:11.61,Default,,0000,0000,0000,,is this pretty straightforward concept
Dialogue: 0,0:00:11.61,0:00:13.94,Default,,0000,0000,0000,,but as you progress through your mathematical career
Dialogue: 0,0:00:13.94,0:00:16.54,Default,,0000,0000,0000,,you'll see that there is actually fairly sophisticated concepts
Dialogue: 0,0:00:16.54,0:00:19.87,Default,,0000,0000,0000,,that can be built on top of the idea of the prime number
Dialogue: 0,0:00:19.87,0:00:22.11,Default,,0000,0000,0000,,and that includes the idea of cryptography
Dialogue: 0,0:00:22.11,0:00:24.27,Default,,0000,0000,0000,,and maybe some of the encryption that your computer
Dialogue: 0,0:00:24.27,0:00:27.02,Default,,0000,0000,0000,,uses right now could be based on prime numbers.
Dialogue: 0,0:00:27.02,0:00:28.11,Default,,0000,0000,0000,,If you don't know what encryption means
Dialogue: 0,0:00:28.11,0:00:30.44,Default,,0000,0000,0000,,you don't have to worry about it right now
Dialogue: 0,0:00:30.44,0:00:32.07,Default,,0000,0000,0000,,you just need to know that prime numbers are
Dialogue: 0,0:00:32.07,0:00:35.54,Default,,0000,0000,0000,,pretty important. So I'll give you the definition
Dialogue: 0,0:00:35.54,0:00:37.87,Default,,0000,0000,0000,,and the definition might be a little confusing
Dialogue: 0,0:00:37.87,0:00:42.54,Default,,0000,0000,0000,,but when we see it with examples it should be pretty straightforward
Dialogue: 0,0:00:42.54,0:00:49.20,Default,,0000,0000,0000,,A number is prime if it is a natural number
Dialogue: 0,0:00:49.20,0:00:56.87,Default,,0000,0000,0000,,for example 1, 2 or 3 (the counting numbers starting at 1)
Dialogue: 0,0:00:56.87,0:00:58.60,Default,,0000,0000,0000,,or you could also say "the positive integers"
Dialogue: 0,0:00:58.60,0:01:30.58,Default,,0000,0000,0000,,it is a natural number divisible by exactly two natural numbers: itself and 1.
Dialogue: 0,0:01:30.60,0:01:34.06,Default,,0000,0000,0000,,Those are the two numbers that it's divisible by.
Dialogue: 0,0:01:39.14,0:01:41.86,Default,,0000,0000,0000,,If this does not make sense for you lets just do some examples.
Dialogue: 0,0:01:41.86,0:01:44.73,Default,,0000,0000,0000,,Lets figure out if some numbers are prime or not.
Dialogue: 0,0:01:44.73,0:01:47.46,Default,,0000,0000,0000,,So lets start with the smallest natural numbers.
Dialogue: 0,0:01:47.46,0:01:52.27,Default,,0000,0000,0000,,The number 1. So you might say "1 is divisible by 1"
Dialogue: 0,0:01:52.27,0:01:56.86,Default,,0000,0000,0000,,and "1 is divisible by itself", hey! 1 is a prime number!
Dialogue: 0,0:01:56.86,0:01:59.66,Default,,0000,0000,0000,,But remember, part of our definition, it needs to be divisible
Dialogue: 0,0:01:59.66,0:02:06.66,Default,,0000,0000,0000,,by exactly two natural numbers. 1 is divisible only by one natural number, only by 1.
Dialogue: 0,0:02:06.66,0:02:15.70,Default,,0000,0000,0000,,So 1, even it may be a little counter intuitive, is not prime.
Dialogue: 0,0:02:15.70,0:02:19.62,Default,,0000,0000,0000,,Lets move on to 2.
Dialogue: 0,0:02:19.62,0:02:28.10,Default,,0000,0000,0000,,So 2 is divisible by 1 and by 2, and not by any other natural numbers.
Dialogue: 0,0:02:28.10,0:02:29.78,Default,,0000,0000,0000,,So it seems to fit our constraints.
Dialogue: 0,0:02:29.78,0:02:32.46,Default,,0000,0000,0000,,It's divisible by exactly two natural numbers.
Dialogue: 0,0:02:32.46,0:02:40.87,Default,,0000,0000,0000,,Itself and 1. So the number 2 is prime.
Dialogue: 0,0:02:40.87,0:02:52.13,Default,,0000,0000,0000,,I will circle the numbers that are prime.
Dialogue: 0,0:02:52.13,0:02:53.61,Default,,0000,0000,0000,,The number 2 is interesting because
Dialogue: 0,0:02:53.61,0:02:56.28,Default,,0000,0000,0000,,it's the only even number that is prime.
Dialogue: 0,0:02:56.28,0:02:58.06,Default,,0000,0000,0000,,If you think about it, any other even number
Dialogue: 0,0:02:58.06,0:03:03.28,Default,,0000,0000,0000,,is also going to be divisible by 2., so it won't be prime.
Dialogue: 0,0:03:03.28,0:03:05.54,Default,,0000,0000,0000,,We'll think about that more in future videos.
Dialogue: 0,0:03:05.54,0:03:10.87,Default,,0000,0000,0000,,Lets try out 3. Well, 3 is definitely divisible by 1 and 3
Dialogue: 0,0:03:10.87,0:03:12.87,Default,,0000,0000,0000,,and it's not divisible by anything in between.
Dialogue: 0,0:03:12.87,0:03:19.06,Default,,0000,0000,0000,,it's not divisible by 2. So 3 is also a prime number.
Dialogue: 0,0:03:19.06,0:03:24.45,Default,,0000,0000,0000,,Lets try 4.
Dialogue: 0,0:03:24.45,0:03:28.28,Default,,0000,0000,0000,,4 is definitely divisible by 1 and 4, but
Dialogue: 0,0:03:28.28,0:03:35.39,Default,,0000,0000,0000,,it's also divisible by 2. So it's divisible
Dialogue: 0,0:03:35.39,0:03:38.86,Default,,0000,0000,0000,,by three natural numbers: 1, 2 and 4.
Dialogue: 0,0:03:38.86,0:03:42.86,Default,,0000,0000,0000,,So it does not meet our constraints for being prime.
Dialogue: 0,0:03:42.86,0:03:47.06,Default,,0000,0000,0000,,Lets try out 5.
Dialogue: 0,0:03:47.06,0:03:49.39,Default,,0000,0000,0000,,5 is definitely divisible by 1,
Dialogue: 0,0:03:49.39,0:03:56.27,Default,,0000,0000,0000,,It's not divisible by 2, 3 or 4
Dialogue: 0,0:03:56.27,0:03:59.27,Default,,0000,0000,0000,,(you could divide 5 / 4 but you would get a remainder)
Dialogue: 0,0:03:59.27,0:04:02.06,Default,,0000,0000,0000,,And it is exactly divisible by 5, obviously.
Dialogue: 0,0:04:02.06,0:04:08.53,Default,,0000,0000,0000,,So once again, 5 is divisible by exactly two natural numbers: 1 and 5
Dialogue: 0,0:04:08.53,0:04:13.06,Default,,0000,0000,0000,,So once again, 5 is prime. Lets keep going,
Dialogue: 0,0:04:13.06,0:04:15.53,Default,,0000,0000,0000,,so that we see if there is any kind of a pattern here
Dialogue: 0,0:04:15.53,0:04:17.53,Default,,0000,0000,0000,,and then maybe I'll try a really hard one
Dialogue: 0,0:04:17.53,0:04:24.39,Default,,0000,0000,0000,,that tends to trip people up. So lets try the number 6.
Dialogue: 0,0:04:24.39,0:04:33.27,Default,,0000,0000,0000,,It is divisible by 1, 2, 3 and 6.
Dialogue: 0,0:04:33.27,0:04:36.36,Default,,0000,0000,0000,,So it has four natural number "factors",
Dialogue: 0,0:04:36.36,0:04:38.20,Default,,0000,0000,0000,,I guess you could say it that way
Dialogue: 0,0:04:38.20,0:04:41.52,Default,,0000,0000,0000,,So it does not have exactly two numbers that it's divisible by,
Dialogue: 0,0:04:41.52,0:04:44.80,Default,,0000,0000,0000,,it has four, so it is not prime.
Dialogue: 0,0:04:44.80,0:04:47.66,Default,,0000,0000,0000,,Lets move on to 7.
Dialogue: 0,0:04:47.66,0:04:54.44,Default,,0000,0000,0000,,7 is divisible by 1, not 2, 3, 4, 5 or 6,
Dialogue: 0,0:04:54.44,0:04:56.27,Default,,0000,0000,0000,,but it's also divisible by 7
Dialogue: 0,0:04:56.27,0:05:00.80,Default,,0000,0000,0000,,so 7 is prime. I think you get the general idea here.
Dialogue: 0,0:05:00.80,0:05:05.100,Default,,0000,0000,0000,,How many natural numbers, numbers like 1, 2, 3, 4, 5,
Dialogue: 0,0:05:05.100,0:05:08.20,Default,,0000,0000,0000,,the numbers that you learn when you are two years old
Dialogue: 0,0:05:08.20,0:05:10.52,Default,,0000,0000,0000,,not including zero, not including negative numbers,
Dialogue: 0,0:05:10.52,0:05:12.80,Default,,0000,0000,0000,,not including fractions and irrational numbers,
Dialogue: 0,0:05:12.80,0:05:14.36,Default,,0000,0000,0000,,and decimals and all the rest,
Dialogue: 0,0:05:14.36,0:05:17.40,Default,,0000,0000,0000,,just regular counting positive numbers.
Dialogue: 0,0:05:17.40,0:05:19.06,Default,,0000,0000,0000,,If you have only two of them,
Dialogue: 0,0:05:19.06,0:05:21.77,Default,,0000,0000,0000,,if you are only divisible by yourself and by 1,
Dialogue: 0,0:05:21.77,0:05:23.40,Default,,0000,0000,0000,,then you are prime.
Dialogue: 0,0:05:23.40,0:05:24.61,Default,,0000,0000,0000,,and the way I think about it,
Dialogue: 0,0:05:24.61,0:05:26.53,Default,,0000,0000,0000,,if we don't think of the special case of 1,
Dialogue: 0,0:05:26.53,0:05:30.46,Default,,0000,0000,0000,,prime numbers are kind of these building blocks of numbers.
Dialogue: 0,0:05:30.46,0:05:32.13,Default,,0000,0000,0000,,You can't break them down anymore.
Dialogue: 0,0:05:32.13,0:05:33.60,Default,,0000,0000,0000,,They are almost like the atoms.
Dialogue: 0,0:05:33.60,0:05:34.99,Default,,0000,0000,0000,,If you think about what the atom is,
Dialogue: 0,0:05:34.99,0:05:37.63,Default,,0000,0000,0000,,or what people thought atoms were when they first...
Dialogue: 0,0:05:37.63,0:05:38.69,Default,,0000,0000,0000,,they thought they were these things
Dialogue: 0,0:05:38.70,0:05:39.76,Default,,0000,0000,0000,,you couldn't divide anymore.
Dialogue: 0,0:05:39.76,0:05:41.68,Default,,0000,0000,0000,,We now know we could divide atoms and actually
Dialogue: 0,0:05:41.68,0:05:43.69,Default,,0000,0000,0000,,if you do you may create a nuclear explosion.
Dialogue: 0,0:05:43.69,0:05:49.66,Default,,0000,0000,0000,,But it's the same idea behind prime numbers. In theory, no prime number is not a theory.
Dialogue: 0,0:05:49.66,0:05:52.00,Default,,0000,0000,0000,,You can't break them down
Dialogue: 0,0:05:52.00,0:05:56.60,Default,,0000,0000,0000,,into products of smaller natural numbers.
Dialogue: 0,0:05:56.60,0:06:00.76,Default,,0000,0000,0000,,Things like 6 you can say, hey, 6 is 2 times 3,
Dialogue: 0,0:06:00.76,0:06:02.00,Default,,0000,0000,0000,,you can break it down, and notice, we can break it down
Dialogue: 0,0:06:02.00,0:06:05.47,Default,,0000,0000,0000,,as a product of prime numbers.
Dialogue: 0,0:06:05.47,0:06:07.80,Default,,0000,0000,0000,,We've kind of broken it down into it's parts.
Dialogue: 0,0:06:07.80,0:06:09.93,Default,,0000,0000,0000,,7 you can't break it down anymore.
Dialogue: 0,0:06:09.93,0:06:13.84,Default,,0000,0000,0000,,All you can say is 7 equals 1 times 7.
Dialogue: 0,0:06:13.84,0:06:15.84,Default,,0000,0000,0000,,And in that case you haven't really broken it down much.
Dialogue: 0,0:06:15.84,0:06:18.00,Default,,0000,0000,0000,,You just have the 7 there again.
Dialogue: 0,0:06:18.00,0:06:19.53,Default,,0000,0000,0000,,6 you can actually break it down.
Dialogue: 0,0:06:19.53,0:06:23.01,Default,,0000,0000,0000,,4 you can actually break it down as 2 times 2.
Dialogue: 0,0:06:23.01,0:06:25.80,Default,,0000,0000,0000,,Now with that out of the way lets think about
Dialogue: 0,0:06:25.80,0:06:27.42,Default,,0000,0000,0000,,some larger numbers, and think about
Dialogue: 0,0:06:27.42,0:06:29.58,Default,,0000,0000,0000,,whether those larger numbers are prime.
Dialogue: 0,0:06:29.58,0:06:34.29,Default,,0000,0000,0000,,So lets try 16.
Dialogue: 0,0:06:34.29,0:06:40.68,Default,,0000,0000,0000,,So clearly any natural number is divisible by 1 and itself.
Dialogue: 0,0:06:40.68,0:06:42.00,Default,,0000,0000,0000,,So 16 is divisible by 1 and 16.
Dialogue: 0,0:06:42.00,0:06:43.51,Default,,0000,0000,0000,,So you are going to start with two,
Dialogue: 0,0:06:43.51,0:06:45.39,Default,,0000,0000,0000,,so if you can find anything else that goes into this
Dialogue: 0,0:06:45.39,0:06:47.19,Default,,0000,0000,0000,,then you know you are not prime.
Dialogue: 0,0:06:47.19,0:06:50.32,Default,,0000,0000,0000,,And for 16 you could have 2 times 8,
Dialogue: 0,0:06:50.32,0:06:52.69,Default,,0000,0000,0000,,you can have 4 times 4,
Dialogue: 0,0:06:52.69,0:06:54.72,Default,,0000,0000,0000,,so it has a ton of factors here,
Dialogue: 0,0:06:54.72,0:06:56.79,Default,,0000,0000,0000,,above and beyond just the 1 and 16.
Dialogue: 0,0:06:56.79,0:07:02.28,Default,,0000,0000,0000,,So 16 is not prime. What about 17?
Dialogue: 0,0:07:02.28,0:07:05.61,Default,,0000,0000,0000,,1 and 17 will definitely go into 17,
Dialogue: 0,0:07:05.61,0:07:10.05,Default,,0000,0000,0000,,2 doesn't go into 17, 3 doesn't go, 4, 5, 6, 7, 8, ...
Dialogue: 0,0:07:10.05,0:07:14.05,Default,,0000,0000,0000,,none of those numbers, nothing between 1 and 17
Dialogue: 0,0:07:14.05,0:07:20.25,Default,,0000,0000,0000,,goes into 17, so 17 is prime.
Dialogue: 0,0:07:20.25,0:07:23.05,Default,,0000,0000,0000,,And now I'll give you a hard one.
Dialogue: 0,0:07:23.05,0:07:25.53,Default,,0000,0000,0000,,This one can trick a lot of people.
Dialogue: 0,0:07:25.53,0:07:33.61,Default,,0000,0000,0000,,What about 51? Is 51 prime?
Dialogue: 0,0:07:33.61,0:07:36.12,Default,,0000,0000,0000,,And if you are interested you can pause the video here
Dialogue: 0,0:07:36.12,0:07:37.92,Default,,0000,0000,0000,,and try to figure out by yourself
Dialogue: 0,0:07:37.92,0:07:40.32,Default,,0000,0000,0000,,if 51 is a prime number.
Dialogue: 0,0:07:40.32,0:07:44.65,Default,,0000,0000,0000,,If you can find anything other than 1 or 51
Dialogue: 0,0:07:44.65,0:07:47.61,Default,,0000,0000,0000,,that is divisible into 51. It seems like...
Dialogue: 0,0:07:47.61,0:07:49.25,Default,,0000,0000,0000,,wow this is kind of a strange number
Dialogue: 0,0:07:49.25,0:07:51.58,Default,,0000,0000,0000,,You might be tempted to think it's prime,
Dialogue: 0,0:07:51.58,0:07:53.39,Default,,0000,0000,0000,,but I'm now going to give you the answer.
Dialogue: 0,0:07:53.39,0:07:59.12,Default,,0000,0000,0000,,It is not prime, because it is also divisible by 3 and 17
Dialogue: 0,0:07:59.12,0:08:02.100,Default,,0000,0000,0000,,3 times 17 is 51.
Dialogue: 0,0:08:02.100,0:08:04.12,Default,,0000,0000,0000,,So hopefully this gives you a good idea
Dialogue: 0,0:08:04.12,0:08:05.65,Default,,0000,0000,0000,,of what prime numbers are all about,
Dialogue: 0,0:08:05.65,0:08:08.79,Default,,0000,0000,0000,,and hopefully we can give you some practice on that
Dialogue: 0,0:08:08.79,0:08:13.00,Default,,0000,0000,0000,,in future videos and maybe in some of our exercises.