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Scientific Notation

Patrick Ford
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Hey, guys. So a lot of times in physics, we're gonna work with really long numbers, whether the really big or really small. For example, the mass of the earth is this crazy long number here. Imagine we had to write this out every single time we used it. Well, fortunately, we can use a type of notation called scientific notation, and we use scientific notation to compress really long, inconvenient numbers into much shorter ones. For example, we could take this long number here with a bunch of zeros and we could rewrite the mass of the Earth as 5.97 times 10 to the 24th kilograms. So this is a much shorter way of representing that number. So the general format first scientific notation is going to be a number a point B C times 10 to the d. So a point B c is just gonna be a number that's greater than or equal to one, but less than 10. So, for example, 5 97. So then that's the number, and you're gonna multiply it by 10. Raised to a power, this d is just a exponents. So how do we actually get all these numbers, I'm gonna show you really simple process for how we convert or how we rewrite standard form into scientific notation. Standard form is just normal numbers and then into scientific notation. Let's just do a bunch of examples. So we get the hang of this. So we're gonna take this number and rewrite it in scientific notation. The first thing gonna dio is you're gonna move the decimal place until you get to a number that's between greater than or equal to one, but less than 10. So here the decimal places over here and we have to move it to the left. 1234 and then five. So we're gonna move it five decimal places to the left in orderto land at this number 3.4 the next we wanna dio is we want around this number two the second decimal place. If it's a really long number with a lot of non zero numbers like this one is so we're gonna take this, I'm gonna round it to the second decimal place. So we're gonna have to look at the next digit over here and it's greater than five. So we're gonna have to round it up. So I'm gonna round this up to 3.5 and I'm multiplied by 10. And now I have to figure out what the exponents is. Well, that's the third step. The third step is the number of decimal places that you moved in. The first step is gonna be equal to your exponents. And if you came from a new original number, that's greater than 10. That exponents is positive. So, for example, we moved from a number that original number greater than 10. We move five spaces to the left, so our exponents is positive five. And that's how you rewrite this number. That's really all there is to it. Let's do a couple more examples. We get the hang of this. So now we're gonna do the same thing over here. We now have to move the decimal place until we get a number that's between one and 10. So we actually have to shift it forward to 34 So there was actually four spaces to the right that we moved here. So we end up with is a number that's 1.2 We don't have to round it or anything. Times 10. And now we have to figure out the exponents what we came for. An original number that was less than once, or exponents is negative. So it's 10 to the minus four. The four meetings did number of decimal places that we moved. And now you might also see some weird ones you might also see. Like you might have to. You know, your professor might ask you to represent seven in terms of scientific notation. So we're gonna follow the same exact steps, move the decimal place until we get to a number that's between one and 10. But if you think about it, the decimal places right here. And this number is already between one and 10. So the way you would, right? This is seven. You don't have to move it. Times 10. And the number of decimal places is equal to your exponents. But you didn't actually move any decimal places. So this is just times 10 to the zero. So just in case you see some weird stuff like this, this is actually how you would represent this in scientific notation. Um all right, that's really all there is to it. And the next thing you might see, the other kinds of questions you might see is you might actually have to go backwards. And what I mean by that is you might have to go from scientific notation out back into standard form, so you might have to rewrite scientific notation numbers as normal numbers. So let's get a few examples. Let me show you really simple process for doing that too. So imagine we wanted to take this number here, and we want to write it as a normal number. Well, really, all you're gonna do here is the exponents is just gonna be the number of decimal places that you moved just as it was up above. And if the exponents is positive, your number becomes larger. If it's negative, it becomes smaller. So, for example, 5.45 times 10 to the eighth. This eight here means move the decimal place. In order for the number of to become larger, you're gonna have to move it to the right. So 5.45 what you do is take this number and you're gonna shifted to the right eight times. 12345678 And you're gonna fill in six zeroes over here. So this is how this number would be in standard form. And for this last 19.62 times 10 to the minus five. So we're gonna do is 9. 62 and the exponents is the number of decimal places they're gonna move. And if the expo is negative, your number becomes smaller. So you're gonna have to shift to the left. You're gonna go 12345 and then fill in your zeros. 123 and four. Your decimal place ends up over here. You're gonna put another zero there. So that means that this here is how you would expand the scientific notation back out to a normal number. Our guys, that's all there is to it.