0:00:00.923,0:00:02.470 In this video I want to talk a little bit about 0:00:02.470,0:00:06.605 what it means to be a prime number 0:00:06.605,0:00:09.605 and what you will hopefully see in this video 0:00:09.605,0:00:11.606 is this pretty straightforward concept 0:00:11.606,0:00:13.940 but as you progress through your mathematical career 0:00:13.940,0:00:16.536 you'll see that there is actually fairly sophisticated concepts 0:00:16.536,0:00:19.869 that can be built on top of the idea of the prime number 0:00:19.869,0:00:22.106 and that includes the idea of cryptography 0:00:22.106,0:00:24.271 and maybe some of the encryption that your computer 0:00:24.271,0:00:27.023 uses right now could be based on prime numbers. 0:00:27.023,0:00:28.107 If you don't know what encryption means 0:00:28.107,0:00:30.439 you don't have to worry about it right now 0:00:30.439,0:00:32.071 you just need to know that prime numbers are 0:00:32.071,0:00:35.538 pretty important. So I'll give you the definition 0:00:35.538,0:00:37.870 and the definition might be a little confusing 0:00:37.870,0:00:42.537 but when we see it with examples it should be pretty straightforward 0:00:42.537,0:00:49.203 A number is prime if it is a natural number 0:00:49.203,0:00:56.870 for example 1, 2 or 3 (the counting numbers starting at 1) 0:00:56.870,0:00:58.605 or you could also say "the positive integers" 0:00:58.605,0:01:30.580 it is a natural number divisible by exactly two natural numbers: itself and 1. 0:01:30.595,0:01:34.060 Those are the two numbers that it's divisible by. 0:01:39.135,0:01:41.864 If this does not make sense for you lets just do some examples. 0:01:41.864,0:01:44.727 Lets figure out if some numbers are prime or not. 0:01:44.727,0:01:47.459 So lets start with the smallest natural numbers. 0:01:47.459,0:01:52.268 The number 1. So you might say "1 is divisible by 1" 0:01:52.268,0:01:56.864 and "1 is divisible by itself", hey! 1 is a prime number! 0:01:56.864,0:01:59.661 But remember, part of our definition, it needs to be divisible 0:01:59.661,0:02:06.661 by exactly two natural numbers. 1 is divisible only by one natural number, only by 1. 0:02:06.661,0:02:15.698 So 1, even it may be a little counter intuitive, is not prime. 0:02:15.698,0:02:19.615 Lets move on to 2. 0:02:19.615,0:02:28.102 So 2 is divisible by 1 and by 2, and not by any other natural numbers. 0:02:28.102,0:02:29.781 So it seems to fit our constraints. 0:02:29.781,0:02:32.460 It's divisible by exactly two natural numbers. 0:02:32.460,0:02:40.866 Itself and 1. So the number 2 is prime. 0:02:40.866,0:02:52.126 I will circle the numbers that are prime. 0:02:52.126,0:02:53.614 The number 2 is interesting because 0:02:53.614,0:02:56.282 it's the only even number that is prime. 0:02:56.282,0:02:58.058 If you think about it, any other even number 0:02:58.058,0:03:03.282 is also going to be divisible by 2., so it won't be prime. 0:03:03.282,0:03:05.541 We'll think about that more in future videos. 0:03:05.541,0:03:10.866 Lets try out 3. Well, 3 is definitely divisible by 1 and 3 0:03:10.866,0:03:12.866 and it's not divisible by anything in between. 0:03:12.866,0:03:19.058 it's not divisible by 2. So 3 is also a prime number. 0:03:19.058,0:03:24.449 Lets try 4. 0:03:24.449,0:03:28.282 4 is definitely divisible by 1 and 4, but 0:03:28.282,0:03:35.394 it's also divisible by 2. So it's divisible 0:03:35.394,0:03:38.861 by three natural numbers: 1, 2 and 4. 0:03:38.861,0:03:42.861 So it does not meet our constraints for being prime. 0:03:42.861,0:03:47.061 Lets try out 5. 0:03:47.061,0:03:49.394 5 is definitely divisible by 1, 0:03:49.394,0:03:56.273 It's not divisible by 2, 3 or 4 0:03:56.273,0:03:59.274 (you could divide 5 / 4 but you would get a remainder) 0:03:59.274,0:04:02.060 And it is exactly divisible by 5, obviously. 0:04:02.060,0:04:08.527 So once again, 5 is divisible by exactly two natural numbers: 1 and 5 0:04:08.527,0:04:13.061 So once again, 5 is prime. Lets keep going, 0:04:13.061,0:04:15.527 so that we see if there is any kind of a pattern here 0:04:15.527,0:04:17.527 and then maybe I'll try a really hard one 0:04:17.527,0:04:24.394 that tends to trip people up. So lets try the number 6. 0:04:24.394,0:04:33.274 It is divisible by 1, 2, 3 and 6. 0:04:33.274,0:04:36.357 So it has four natural number "factors", 0:04:36.357,0:04:38.196 I guess you could say it that way 0:04:38.196,0:04:41.524 So it does not have exactly two numbers that it's divisible by, 0:04:41.524,0:04:44.795 it has four, so it is not prime. 0:04:44.795,0:04:47.662 Lets move on to 7. 0:04:47.662,0:04:54.440 7 is divisible by 1, not 2, 3, 4, 5 or 6, 0:04:54.440,0:04:56.272 but it's also divisible by 7 0:04:56.272,0:05:00.795 so 7 is prime. I think you get the general idea here. 0:05:00.795,0:05:05.997 How many natural numbers, numbers like 1, 2, 3, 4, 5, 0:05:05.997,0:05:08.196 the numbers that you learn when you are two years old 0:05:08.196,0:05:10.524 not including zero, not including negative numbers, 0:05:10.524,0:05:12.795 not including fractions and irrational numbers, 0:05:12.795,0:05:14.357 and decimals and all the rest, 0:05:14.357,0:05:17.397 just regular counting positive numbers. 0:05:17.397,0:05:19.061 If you have only two of them, 0:05:19.061,0:05:21.773 if you are only divisible by yourself and by 1, 0:05:21.773,0:05:23.395 then you are prime. 0:05:23.395,0:05:24.606 and the way I think about it, 0:05:24.606,0:05:26.528 if we don't think of the special case of 1, 0:05:26.528,0:05:30.462 prime numbers are kind of these building blocks of numbers. 0:05:30.462,0:05:32.129 You can't break them down anymore. 0:05:32.129,0:05:33.595 They are almost like the atoms. 0:05:33.595,0:05:34.994 If you think about what the atom is, 0:05:34.994,0:05:37.631 or what people thought atoms were when they first... 0:05:37.631,0:05:38.688 they thought they were these things 0:05:38.703,0:05:39.757 you couldn't divide anymore. 0:05:39.757,0:05:41.681 We now know we could divide atoms and actually 0:05:41.681,0:05:43.690 if you do you may create a nuclear explosion. 0:05:43.690,0:05:49.655 But it's the same idea behind prime numbers. In theory, no prime number is not a theory. 0:05:49.655,0:05:52.003 You can't break them down 0:05:52.003,0:05:56.602 into products of smaller natural numbers. 0:05:56.602,0:06:00.763 Things like 6 you can say, hey, 6 is 2 times 3, 0:06:00.763,0:06:02.001 you can break it down, and notice, we can break it down 0:06:02.001,0:06:05.467 as a product of prime numbers. 0:06:05.467,0:06:07.801 We've kind of broken it down into it's parts. 0:06:07.801,0:06:09.934 7 you can't break it down anymore. 0:06:09.934,0:06:13.836 All you can say is 7 equals 1 times 7. 0:06:13.836,0:06:15.838 And in that case you haven't really broken it down much. 0:06:15.838,0:06:18.004 You just have the 7 there again. 0:06:18.004,0:06:19.533 6 you can actually break it down. 0:06:19.533,0:06:23.011 4 you can actually break it down as 2 times 2. 0:06:23.011,0:06:25.800 Now with that out of the way lets think about 0:06:25.800,0:06:27.425 some larger numbers, and think about 0:06:27.425,0:06:29.579 whether those larger numbers are prime. 0:06:29.579,0:06:34.286 So lets try 16. 0:06:34.286,0:06:40.676 So clearly any natural number is divisible by 1 and itself. 0:06:40.676,0:06:42.000 So 16 is divisible by 1 and 16. 0:06:42.000,0:06:43.512 So you are going to start with two, 0:06:43.512,0:06:45.387 so if you can find anything else that goes into this 0:06:45.387,0:06:47.194 then you know you are not prime. 0:06:47.194,0:06:50.318 And for 16 you could have 2 times 8, 0:06:50.318,0:06:52.694 you can have 4 times 4, 0:06:52.694,0:06:54.718 so it has a ton of factors here, 0:06:54.718,0:06:56.787 above and beyond just the 1 and 16. 0:06:56.787,0:07:02.278 So 16 is not prime. What about 17? 0:07:02.278,0:07:05.611 1 and 17 will definitely go into 17, 0:07:05.611,0:07:10.052 2 doesn't go into 17, 3 doesn't go, 4, 5, 6, 7, 8, ... 0:07:10.052,0:07:14.051 none of those numbers, nothing between 1 and 17 0:07:14.051,0:07:20.253 goes into 17, so 17 is prime. 0:07:20.253,0:07:23.052 And now I'll give you a hard one. 0:07:23.052,0:07:25.527 This one can trick a lot of people. 0:07:25.527,0:07:33.611 What about 51? Is 51 prime? 0:07:33.611,0:07:36.120 And if you are interested you can pause the video here 0:07:36.120,0:07:37.918 and try to figure out by yourself 0:07:37.918,0:07:40.321 if 51 is a prime number. 0:07:40.321,0:07:44.653 If you can find anything other than 1 or 51 0:07:44.653,0:07:47.611 that is divisible into 51. It seems like... 0:07:47.611,0:07:49.252 wow this is kind of a strange number 0:07:49.252,0:07:51.585 You might be tempted to think it's prime, 0:07:51.585,0:07:53.387 but I'm now going to give you the answer. 0:07:53.387,0:07:59.119 It is not prime, because it is also divisible by 3 and 17 0:07:59.119,0:08:02.998 3 times 17 is 51. 0:08:02.998,0:08:04.119 So hopefully this gives you a good idea 0:08:04.119,0:08:05.653 of what prime numbers are all about, 0:08:05.653,0:08:08.786 and hopefully we can give you some practice on that 0:08:08.786,0:08:13.000 in future videos and maybe in some of our exercises.