 Okay, we've talked about prime numbers. I've mentioned it through most of the videos Now if you recall the real number set is basically them the reality that you're going to deal with in high school and From the real number set the rational numbers if you remember this This is the real number set and it's broken up into two sections, right now This is the rational numbers and this is the irrational numbers, right now As far as you're concerned majority of numbers you're going to deal with in high school are going to be from this group Now this group the definition of rational numbers is Any number that can be expressed as a fraction of integers and the integers one was one Subgroup below the rational numbers so the rational numbers includes the integers and then went down to the whole numbers And then we have natural numbers, right? now if you're able to deal with Integers, which are just positive negative whole numbers Then you have this whole section covered and you're able to deal with the majority numbers They're going to they're going to throw at you in high school now whole numbers Can really be defined as prime numbers because in the whole number set in a natural number set any numbers That aren't prime numbers are made up of prime numbers. So let's take a look at the The first 20 numbers okay from 1 to 20 Now if you ask most people you say how many numbers are there between 1 and 20 they'll tell you oh, there's 20 numbers But there really isn't 20 numbers between 1 and 20. There's only 9 numbers. Okay. Now. Let's circle the prime numbers. We got 2 3 5 7 11 13 17 and 19 Okay One is a special number. It's not a prime number. But as far as you're concerned in high school You can think it was a prime number now What were prime numbers was the definition of prime numbers prime numbers are any natural number? That can only be divided evenly by 1 and itself So between 1 and 20 there aren't 20 numbers. There's only 1 2 3 4 5 6 7 8 9 numbers Okay. Now. Let's take a look at the other numbers and See why this is so now 1 2 3 They're individuals 2 and 3 are prime numbers and 1 is just 1 special number special case 4 is not a new number. Okay 4 can be broken down into 2 times 2 Okay, you got 6 6 can be broken down into 2 times 3 8 is 2 times 4 and 4 can be broken down into 2 times 2 so 8 is really 2 times 2 times 2 9 is 3 times 3 10 is 2 times 5 12 is 2 times 6 and 6 can be broken down into 2 times 3. Okay 14 is 2 times 7 15 is 3 times 5 16 is 4 times 4 and 4 can be broken down into 2 times 2 and 2 times 2 So 16 is really 2 times 2 times 2 times 2 When you're doing the tree diagrams what you do is any number that branches off you kill it So those guys would have been dead. So 16 would have just equal to this. So you take the ends of each branch 18 is 2 times 9 and 9 becomes 3 times 3 again the 9 dies 20 is 2 times 10 10 is 2 times 5 and the 10 dies. So 20 is really 2 times 2 times 5. Okay Now let's do a couple other numbers. So you get this down pack Now keep in mind. I'm going to include a worksheet on my website with the solutions and you should go through this because Prime numbers are super important. You need to be able to break numbers down To the prime factors. Okay, so let's take I don't know. Let's take 100. Okay, actually let's take 10,000 Now these should be the easiest numbers you get. If you want we'll kick this up to 9,000. Okay, or 90,000 So what you do if you get any number like this the best way to do it is take the number by itself over here and Take the all the zeros 1 2 3 4 and you put a 1 in front of it. If this was 29 you would have put 29 here and 10,000 over here. Now 9 is just 3 times 3 10,000 is 100 times 100 and 100 breaks down to 10 times 10 10 times 10. Okay, and 10 breaks down to 2 times 5 2 times 5 2 times 5 2 times 5 Okay, keep in mind. These aren't 25. It's 2 times 5. So all the numbers above this You can kill and you take all the end branches To figure out what 90,000 is as prime as expresses a prime number So 90,000 is really 3 times 3 2 times 2 times 2 times 2 5 times 5 times 5 times 5 So you have 3 times 3 4 2s 1 2 3 4 and 4 5s 1 2 3 4 That's what 90,000 is now once you start dealing with exponents You can express this more simply as 3 squared Okay times you got 4 2s there, so it becomes 2 to the power of 4 and You got 4 5 so it becomes 5 to the power of 4 and this is 90,000 in its prime 4 Simplified into its prime factors. Let's try something else try another number. Let's do 448 Now 448 that's an even number even numbers can always divided by 2 So 2 goes into this goes into 2 twice it goes into 4 twice 2 and 4 That's still an even number. You can divide it by 2 again. That becomes 2 1 1 2 that's still an even number So a hundred divided by 250 12 divided by 2 is 6 so 112 goes in is divided it becomes 56 okay, now that's still a prime number so you can break it down That's still an even number you can still divided by 2 so 56 divided by 2 is 50 divided by 2 is 25 6 divided by 2 is 3 25 plus 3 is 28 28 again is even number so this becomes 2 times 14 Now 14 is an even number becomes 2 times 7 now 7 is a prime number 2 is a prime number you're done You get to be end of the branches end of the trees now what you do is you kill these guys Whatever branched off you kill so 448 is really 1 2 3 4 5 6 2's times a 7 So this becomes 2 to the power of 6 times 7 2 to the power of 6 means to multiply it by itself 6 times easy Once you know how to do this Life becomes a lot simpler managing numbers, especially irrational numbers. Well specifically irrational numbers Because any rational number can be broken down into prime factors and Rational numbers are just integers that can express expresses fraction And that there are numbers that can be expressed as a fraction of integers and what are integers? positive negative whole numbers and whole numbers are really natural numbers, so prime numbers are Natural numbers that can only be divided evenly by one and themselves and anything That's not a prime number is made up of prime numbers. You have to keep that in mind I'm going to continue to say this anything. That's not a prime number is made up of prime numbers So When you're dealing with the rational numbers, if you can break them down to their prime numbers You know what they're made out of and that way you can manage them. You can deal with them You can simplify the numbers down I'm going to put some of stuff on my website for you to reduce these Break them down to their prime factors using the tree diagram Make sure you go through it. You have to know how to multiply To be able to do this and super important So make sure you go back and learn your multiplication table at least for the single digits For the rest of the stuff for the higher numbers, you can use the calculator But slowly try to wean yourself off the calculator. Okay. Good luck with the stuff super important. We'll talk into it