 Alright, we talked about how when you change units, a lot of times the reason for changing units is to switch the number that is associated with your unit into a more understandable number. There is another way of dealing with very large and very small numbers called scientific notation, and we are going to talk about that now. So here we go. The way that I usually introduce this is I say to a student, you're familiar with the number 100, hopefully you are, and I will say do you know that 100 is the same thing as 10 times 10? And usually they say yes, that's true. So I want you to realize that 100 and 10 times 10 are just two different ways of saying the same thing. There is another way of saying the same thing as 110 times 10, and that relates to the fact that we are multiplying 10 against itself two times here. Because we are multiplying 10 against itself two times, there is another way of writing that. The way that you write it is you write the number 10, and then in the upper right next to the 10, you write a little number 2. That base of that number 2 means we multiplied 10 against itself two times. There are many ways of saying this out loud. Sometimes people say 10 to the 2, sometimes people say 10 to the second power, sometimes people say 10 squared. 10 squared is the more sophisticated way of saying it, but it's not going to make any sense to most people where that comes from. So I would like you to ignore it, and I would like you to pronounce it either as 10 to the 2 or 10 to the second power, something like that. So I want you to realize that all three things, they basically are just different ways of saying the same number. This bottom way of writing numbers is almost scientific notation, but not quite. We'll get there in a minute or in a few minutes. Another example, you're probably all familiar with the number 1,000, and you probably are aware that 1,000 is equal to 10 times 10 times 10. So how many times did you multiply 10 against itself that time? Well, three times. So what's the fancy sophisticated way of writing 1,000 using the method that we used up above? Hopefully you realize that the sophisticated way of saying this is to write the number 10 and then a little 3 in the upper right, right next to the 10. And you could pronounce that 10 to the 3 or 10 to the 3rd. Sometimes, again, people say 10 cubed. You can say that if you want, but it may not make any sense where that's coming from. So I'd prefer that you say 10 to the 3 or 10 to the 3rd power. Again, these are just everything that I'm circling is just three different ways of saying the same thing. And again, this is almost scientific notation, but not quite. We'll get there in a minute. But more examples are coming. So here we go. This number in human speaking terms is 100,000,000,000,100,000. How would we write this using the method that we were using on the previous page? Well, to show you what it is, as far as 10s are concerned, it's a whole bunch of 10s multiplied against each other. In fact, if you count the number of 10s, there are 11 10s. So the way to write 100 billion using the method that we were using on the previous slide is to write 10 to the 11th power or 10 to the 11 if you want to call it that. If you don't know who this guy is over here, his name is Bill Gates. He probably sold some software to you at some point along the line. And he's worth about $100 billion. Not quite, but he's getting there. Anyway, the point here is that, again, almost scientific notation. But you can see that eventually, numbers can get large enough, like the number that I'm circling up here, numbers can get large enough that it becomes a nuisance to write all of the digits out. And people have come up with quicker ways of writing extremely large numbers like this. And this is the beginning of the quicker way. If you had to write the number 100 billion a bunch of times in a row, would you rather do the top method or would you rather do the bottom method? Hopefully you'll appreciate that the bottom method is going to be easier on your hand. So there's an advantage to being able to write the numbers that way and that it saves you time. And that's basically why this technique was invented. It's a way of writing very, very large numbers in a more compact way. So again, let's keep going. Here, this is 600 billion, 600 billion. It's related to 100 billion. It is basically 600 billion is 6 times 100 billion. And on the previous slide, we said that this was 100 billion. And we said that there were 11 tens there. And yes, there are. So basically, another way of writing 600 billion is to write 6 times 10 to the 11th power. And again, hopefully you will appreciate that if you had to pick a way of writing this a whole bunch of times, writing the number 600 billion, you would probably prefer to write it this way just because you have to write less things. You have to type out less things. And you're not losing any information. Both of the ways, the one on the left and the one on the right, they tell you the same thing as long as you know how to do it. So that's basically why this technique was invented. It was invented to save you some time. Sometimes students think that this method was invented to torture them. It wasn't. That's just the side benefit. It's primarily because people wanted an easier way to write big and small numbers. This number over here is written in proper scientific notation. And what I mean by that is that scientific notation has to have a couple of, when you write a number in scientific notation, you have to have a couple of things written out in a certain way. You always have to have some number, which I'm just going to call blah, times 10 to the some other number, which I'm going to call blah, blah. And that's it. You have to have some number times 10 to the some other number. And if you look over here, that is actually what we have. We have some number. It's a 6 times 10 to some other number, times 10 to the 11th. That is the standard format for scientific notation. There is an extra rule that I'm going to add for scientific notation. When you write a number in scientific notation, I want you to make this first number over here. This is our first number. I want you to follow a rule that it has to be between 1 and 10. Now if you look at our first number that I'm circling over here, is it between 1 and 10? Yes, it's a 6. And 6 is between 1 and 10. To be more technically correct, our number has to be greater than or equal to 1 and less than 10. And again, we haven't broken any rules. This number 6 is greater than or equal to 1, and it is also less than 10. Now, you don't have to follow that rule for your number to be written in scientific notation, but it adds some nice features that help people out on occasion that we're not really going to go into. So I'm going to demand that you write your numbers. If you need to write a number in scientific notation, that you follow this additional rule. First number between 1 and 10. Second number actually has its own features. It has to be a whole number. I'm not sure if that's correct. For those of you who give a crap, it has to be an integer. So that means that it can be a negative number or zero or a positive number, but it has to be negative 1, negative 2, negative 3, 0, 1, 2, 3. It can't be decimals, let's say. You can't have fractions up there for it to be proper scientific notation. So this first number is sometimes called a coefficient, and the second number is sometimes called an exponent. I don't really care that you know those, but I may refer to them in the exercises coming up. So keep that in mind. Now, earlier in the previous slide, I said 10 to the 11 was, it was almost scientific notation, but not quite. The reason it's almost is because it's missing the coefficient. So if I want to write this in super proper scientific notation, the way that I do it is I just have 1 be my coefficient. So 1 times 10 to the 11th power is the same thing as 10 to the 11th power. So I didn't really add any additional information by doing the 1 times there, but I put it in the quote unquote proper format for scientific notation. So that's why it's there. Let's see. And again, why do we use scientific notation? A little bit to torture you, give you a hard time. And more to the point, it lets you write huge numbers, and as you will see, very tiny numbers. More compactly, it saves you time. And we are going to deal with some huge numbers and some very tiny numbers as the course moves on. So you have to get this under your belt. All right, more examples. How do I write 6,500 in scientific notation? What I will tell you is that 6,500 is the same thing as 1,000, here's 1,000 that I'm circling, times 6 and 1 half. 6,500 is 6 and 1 half 1 thousands. But we already know what 10 multiplied against itself three times is, that's 10 to the third power. And then if I want to write in scientific notation, I can write 6,500 as 6.5 times 10 to the third power. And that is the format that we talked about in the previous slide. You have to have blah, which is the coefficient times 10 to the sum of the number, which is called the exponent. You can see, oh, the other rule that I said was the coefficient, this guy here, it has to be between 1 and 10. And it is 6 and 1 half is somewhere in between 1 and 10. So we're not breaking any of the rules. So if you want to write 6,500 in scientific notation to impress your friends at cocktail parties or something like that, that's pretty much how you're going to do it. You're going to write 6.5 times 10 to the third power. Now I have a recipe. So if it seems like I've just been doing magic and pulling numbers out of the air and writing kind of weird things, I have a recipe that sort of lets this work. As far as I can tell, it always seems to work. And this is basically how you do it. Let's say that we want to write the number 100,000 in scientific notation. So we have to write something times 10 to something else. The way that you do it is you pretend you have a new decimal. There's my new decimal. And I need to put it somewhere. I don't know where I have to put it somewhere in this number. And I have to put it in a place that is going to give me a number between 1 and 10. So should I put it here? No, that's just going to give me 100,000 again. Should I put it there? Nope. No. No. No. Should put it right here. If I put it here, if I put the new decimal way over here, then it gives me the number 1 with a bunch of zeros after it. But that is a number between 1 and 10. So that's what you do initially is you put a new decimal somewhere in your old number so that it gives you a number between 1 and 10. So I'm going to call this the new decimal. And then your coefficient is going to be the old number with the new decimal. So I'm going to write 1. You might be asking yourself, why did I have to put the decimal here? Just a moment ago, I said that the number has to be between 1 and 10. So isn't it possible that I could have put the decimal over here as well? If you remember, I mentioned something a little bit technical earlier in this video. I basically said that the number where you put the decimal, it has to make the coefficient actually greater than or equal to. So this is what greater than or equal to means. Greater than or equal to 1. So the new number, the coefficient, can be 1 because it can be equal to 1. And it also has to be less than 10. It can't actually be equal to 10. It has to be less than 10. So if you put the decimal in the spot that I'm highlighting at the moment, that's going to make a 10 in the coefficient. And according to my technical rules, you're not allowed to. It actually has to be less than 10. It can be 9.999 with a whole bunch of nines, but it can't actually be 10. But what you are allowed to do is you are allowed to make it greater than or equal to 1. So the decimal point can go here in this position to make the number 1 be the coefficient. And now we need to figure out what the exponent is. The way that you do that is you have to find the old decimal. Now this original number didn't have a decimal point written, but there is one there that's implied to be there. And it is right here. So that's our old decimal. Now nobody bothered to write it because you didn't need to, but if you think about it, it was there all the time, just kind of hiding. And if you want to figure out the exponent, what you have to do is you have to count the number of hops in between digits that it takes to go from the new decimal to the old. So 1, 2, 3, 4, 5 hops. So 5 is going to be the exponent. So if I want to write 100,000 in scientific notation, I write 1 times 10 to the fifth power. That's the way of writing it. Now I know many students have memorized a rule that is not quite correct. And sometimes I will hear students tell me converting to scientific notation is easy. All you do is count the number of zeros in your number. And so 1, 2, 3, 4, 5 zeros. And I've got a 5 in my exponent. That works sometimes. It will not work all the time. I'll show you some examples where it does not work. So don't do that. Don't count the zeros. You're going to get screwed. Here's an example. Where counting the zeros isn't going to work because there are no zeros. We want to take this number and convert it to scientific notation. So we have to write something times 10 to something else. We've got our new decimal. There it is. We need to put it somewhere. Should I put it here? Should I put it here, here, here, or here? All right, I can hear all of you screaming. You need to put the new decimal in the first spot because we need a number between 1 and 10. And there it is. That's where the new decimal goes. And let's see that. So our coefficient is going to be 1.2345. Then we have to find the old decimal. Where's the old decimal? Oh, speak up. I can't hear you. Oh, yeah. It's implied to be after the 5. So there's the old decimal. And if we want to find the exponent, well, you count the number of hops between digits that takes you from the new to the old. 1, 2, 3, 4, 4 hops. So it's going to be times 10 to the 4. So if I wanted to write the number 12,345 in scientific notation, it would be written 1.2345 times 10 to the fourth power. And that is that.