 The previous section was devoted to using scientific notation to write really large numbers, or at least that's what it was meant to show you. Scientific notation is also used for writing very tiny numbers. What I mean by a tiny number is a number that is still bigger than zero but much smaller than the number one. So if this is a number line this is zero, one, two, three, etc. We're talking about numbers that fall in that area bigger than zero but a lot smaller than one. So scientific notation can also be used to make your life a little easier when you're writing those kinds of tiny numbers. The first example of that type of number is the number 0.1. I will tell you that 0.1, another way of writing it, is to say that 0.1 is the same thing as one divided by ten. I'm going to write that also as a fraction. One over ten equals one divided by ten equals 0.1. These are all different ways of saying the same number. The way that this is done with an exponent is to write ten to the negative one, or ten to the minus one if you want to call it that, ten to the negative one power. That means one divided by ten and there's only one ten in the denominator. The other way of writing that is ten to the minus one. This is sort of getting ready to learn how to use scientific notation to write very tiny numbers. The number 0.01, you can test this out on your calculator. Another way of writing 0.01 is to call it one divided by a hundred, which is also one divided by ten times ten because we already said that ten times ten is a hundred. So another way of writing this is one divided by a hundred is equal to 0.01 and that is also equal to one divided by ten times ten. Now I will ask you in this representation that I'm circling how many tens are in the denominator. There are two tens, two tens multiplied against each other. So the fancy mathematical way of writing that is to write ten to the negative two because there are two tens in the denominator or ten to the minus two power if you want to call it that. So again you can see where we're going with this. 0.01 and 0.1 they're both tiny numbers. They fall in this area of the number line and instead of writing 0.1 and 0.01, if I wanted to write them in scientific notation, if I wanted to write 0.01 in scientific notation, I would write it as one, oops, I would write it as one times ten to the negative two power. Again when you write in scientific notation the form has to be some number called the coefficient times ten to the some other number and that number is called the exponent. And so 0.01, I could write it in scientific notation as one times ten to the negative two power. Then you can get to even smaller numbers and this is this is the point where scientific notation starts to become useful because it saves you a little bit of time. How do I write 0.0001 in scientific notation? If you want to break that apart it is one divided by a whole bunch of tens multiplied against each other. I think that is one, two, three, four, five, six tens in the denominator. So I could write it like this one ten times ten times ten times ten times ten times ten times ten. That's six tens so the way I would write it is ten to the negative six power. If I wanted to be formal and write it in scientific notation, I would write one times ten to the negative six power. So eventually these numbers can get so small that it's actually worth your while to write them in scientific notation instead of writing out all of these zeros and all of those digits in your number. So keep that in mind and we're actually going to use numbers in this course that do get that small and possibly even smaller. So let's see one times ten to the negative six is the way of writing that number in scientific notation. The recipe that I showed you in the previous video works here as well. If you want to write this number in scientific notation we have to write some number times ten to the some other number. This is called the coefficient and the way that we have to do it is we have to take a decimal, see my decimal. There it is. It's a very cute decimal and we need to put it somewhere in this number, somewhere over here so that we turn our number into a number between one and ten. So I have a whole bunch of places I can, a whole bunch of options I could put it here, put it here, here, right a whole bunch of different places. Where should I put it? Well hopefully you will agree that the place that it should go to turn our number into a number between one and ten so it should go right here and that's about the only place that I can put it. So this is going to be the new decimal place or the new decimal I should say. And our coefficient is going to be the old number but with the new decimal. So here's the old number, new decimal. This is going to be a one. And then the exponent, that's the second number. Exponent is going to be the number of hops in between digits that it takes to go from the new one to the old decimal. This is a little bit different because this time the old decimal was actually written down and you can see it. There it is. It's there for everybody to see and I want to know how many hops it takes between digits to go from the new one to the old one. It takes one, two, three, four, five. But this time it's five backwards and because it's backwards what I want us to do is I want us to say that this is going to be negative five. So instead of writing point zero, zero, zero, zero, one that way I could write it as one times ten to the negative five power and that's just the scientific notation way of writing that number. So again hopefully you can appreciate that this recipe works for very big numbers. It also works for very small numbers and pretty much anything in between. Here's another example. We have to write something times ten to the something else. We've got our decimal. Oops, let's go back. Where should it go? Should it go here? There. Where is it going to go? I'm sure everybody is screaming the correct answer. The new decimal should go there because if I do that then I am going to make a number between one and ten. And so this new number goes here. It's going to be 1.23. This is the new decimal. And where's the old decimal? There's the old decimal. So if we want to figure out the exponent we have to count the number of hops it takes to go from the new one to the old one. One, two, three. Three hops backwards so this is going to be a negative three. So if I wanted to write the number zero, point zero, zero, one, two, three in scientific notation, I would write it as 1.23 times ten to the negative three power. That's just how you would do it and the recipe should work all the time. I had another point that I wanted to make and now it's slipping my mind. I'm going to give you a maybe moderately challenging question at the moment and you can work on it on your own but we're going to work on it on this slide. How would you write the number six in scientific notation? Now you might be scratching your head saying, you know, that's weird. Why would I want to write the number six in scientific notation? Most people in their right minds would never want to write the number six in scientific notation but I want you to pause the video and try it. Mostly because it is a good test of whether you understand the rules for doing this. So pause the video now. You can un-pause it when you think you have it and then I'll work through it. All right, here's me working through it. We want to write starting with the number six. There it is. We want to write it in scientific notation so it has to be something times ten to the something else. The first thing we're going to do is write the coefficient. I don't know why. Coefficient there. The stylus kind of stinks at the moment. Okay. What was the rule? We had to take a decimal place. There's our beautiful lovely decimal place or decimal point and we had to put it somewhere on our number. There's our number so that we turned that number into a number between one and ten. So do I put it over here? Hell no. Do I put it over here? You bet you're something I do. There it goes. That's where the new decimal goes. I'm going to write big nice letters new. That's our new decimal. So what's the coefficient going to be over here? It's going to be six. So which is a little weird, right? We haven't really changed anything but that's why this is a tricky problem. The second part is we want to find the exponent. And I told you the exponent is the number of hops it takes to go from the new decimal to the old decimal. Well, we wrote down the new decimal. Where was the old decimal before we started messing with our number? The old decimal was in the exact same spot. The old decimal was there as well. So the new decimal and the old decimal are the same. So the question is to get the exponent, you have to count the number of hops to go from the new one to the old one. How many hops does it take? It doesn't take any. It takes zero hops. So what goes up here is a zero. And so the proper way of writing the number six in scientific notation is to write six times ten to the zero power. Like I said, almost no one in their right mind would ever write the number six in this way. But it's a decent test of whether you are following the rules correctly. So if it doesn't make sense, maybe play it through again. Those of you who've taken some fancier math classes, you may know that ten to the zero, that's a zero there, is another way of saying that equals one. Ten to the zero is just another way of saying the number one. And so instead of six times ten to the zero, I could say six times one. And six times one is six. So we got our original number back. So six times ten to the zero, again, is just a fancy way of saying six times one. It's the scientific notation way of saying that. Here's another point that I want to make. I think that many of you have a calculator that looks like a complicated trillion dollar device that can do a lot of different things that you will never have to be able to do. What you are going to have to be able to do, however, is you're going to have to be able to use your calculator to do scientific notation. The problem is that everyone has a slightly different calculator that does scientific notation in a slightly different way. So if you are having difficulty punching numbers into your calculator using scientific notation and doing calculations with scientific notation and your calculator, you have to email me, call me, send up some smoke signals, send a telegram, have somebody parachute into my house, and let me know. And I will try to help you figure out how to use your calculator to do this. All right, here's a summary of scientific notation, just in case your eyes haven't glazed over yet. Scientific notation is a way of writing big numbers and small numbers compactly. That's basically what it's for. What I want you to be able to do is I want you to be able to interconvert between sort of the standard way that we write numbers and scientific notation, backwards and forward. So if I give you the number 32 million, you have to be able to turn that into scientific notation. Or if I give you five times 10 to the 92 power, you need to be able to convert that into standard format. You have to be able to go in both directions. Oh, here's maybe a simpler example. If I gave you this number on the left, you better be able to convert it into 2.3 times 10 to the 17. And if I gave you 2.3 times 10 to the 17, you should be able to back convert it. And you're going to have to be able to do this on your calculator because certain calculations are coming that require this because the numbers that we deal with will be too big or too small to just use all of the digits on your calculator. You're going to have to write them in scientific notation.