Transcript for: Metric Unit Prefixes and Conversion Tips
Welcome back. We have talked about conversions now-- dimensional analysis, unit conversions, whatever you want to call them. And we've put together a system that's fairly methodical about how to set up those conversions. And what we're going to do today is we are going to talk about a specific type of conversion which are called unit prefixes. And these unit prefixes, you're going to see in life all the time. When you see a water bottle, it's got a volume in milliliters. That milli- is a prefix to the word liter that you can actually use in front of any unit you want, not just in front of liters. You can use in front of grams, you can use in front of seconds, you could use it in front of calculators if you could have a millicalculator. And what we're going to see is how can we use those unit conversions or those unit prefixes, I should say, and all of our different ways how can we convert between them, understand what they mean. And part of it goes back to thinking about what do we do when we want to change the size of a unit in English language, in English units? and so if we have something that's about an inch long and yea big, and we want to have something that's bigger, what's the next thing up from an inch, right? It's about a foot. And we know that there are 12 inches in 1 foot, which is great. I mean it's just it's just a number. But it means if you want a convert between feet and inches you've got to either multiply by 12 or divide by 12. And so if you have 10 inches, it's not equal to one foot. It's equal to 10/12 of a foot, 5/6 a foot, whatever that is. You have to put it into a calculator and find the decimal, unless you happen to know those off the top of your head. Now if you want something larger than a foot, you're probably going to go for a yard. Now in a world that was consistent, just for consistency you'd go ahead and make a yard 12 feet. But we know that's not what it is. A yard is three feet. And so if you want to get from inches to feet to yard, one time you go from 12 and one time you use a three, and it's not very consistent, but it's what we got. Now if you want something larger than a yard, wouldn't it make sense to have something that was somewhere between three and 12 bigger? No, right? Because we go from a yard, we go to a mile. And a mile in yards-- I would have to look up this number-- is 1,760 yards. These numbers are fairly random-looking. There is reason behind them. You can look up the history of these numbers and where they came from. But they are fairly random, and they make conversions fairly difficult because you have to do a different type of conversion every time. So a while back ago, people got together and they said, hey, let's try to set up a system of units where the conversions make sense and see what happens, and what can we do? So they developed the metric system. And you're probably familiar with the metric system. We talked about it a little bit in one of the previous videos where a meter is about three feet long. And then they said, hey, how would we change the size of a meter? And they said, let's put some prefixes in front of it to change it. And so you probably are also familiar with a millimeter. A millimeter is a very, very small distance. You probably can't even see that my fingers are separated. About the height of an ant is a millimeter. But if you want something larger than a millimeter you might go ahead and get a decimeter. So a decimeter is 10 a centimeter. A centimeter is 10 millimeters. And you can see what's coming next. If you want something larger than that you're going to go to a decimeter. Now in the English system, there is another random number here, but here, hey, guess what? The number is 10. One decimeter is 10 centimeters. And if you want to go something bigger than that, you're going to get to the meter, And that is, lo and behold, 10 decimeters. And so you can see all of our units here are related by a factor of 10? And why is a factor of 10 cool? Well if you want to change 123 from centimeters into millimeters, what you're going to do is you're going to multiply by 10. And so you get 1230 millimeters. And we'll see how to do that in a very methodical way later to get 1,230 millimeters. And you'll notice something is the numbers here are the same. And what you're doing is just moving the decimal back and forth when you convert using unit prefixes, which is great. It makes it really easy to understand whether you're doing it correctly because these numbers don't keep changing. You're not multiplying by three and dividing by 1760-- all these fairly random numbers. Now if we tried to have a unit that went every 10 for all the different things we do in science, it would be crazy, and we'd end up with 40 or 50 different unit prefixes. And that's hard to keep track of. So at some point they said, OK, for common size things that we run across on a daily basis, let's have lots of little tiny different things. We have millimeters, and centimeters, and decimeters, milligrams and centigrams and decigrams grams. But for things that start to get to different scales, let's go by thousands instead of by tens. And so if you want something larger than that, you're going to say that one kilometer, a kilometer, is equal to 1,000 meters. But again, since it's 1,000 you're just moving the decimal point three spaces instead of one space. You're still going to end up with the same numbers, but the decimal point has moved. It's a very nice very tidy little system. What I need you to know is I need you to know all of the different prefixes that we might commonly run across when we're doing science. And they are in this list here. Starting at the top, there is tera-. Now you might be familiar with this one because you go out and buy a hard drive nowadays, you might buy a terabyte, or two terabytes, or eight terabytes or whatever it is to store all those cat videos that you take. And that's a big number. The big number terabyte is about 10 to the 12th. Now it turns out there's a slight difference between computer science tera- and scientists' tera-. But it's about 10 to the 12th. For a scientist, it's exactly 10 to the 12th. It's a huge number. That's a lot of little bytes in there. And if you look at the memory in your computer, you're probably measuring the memory in your computer or the memory in your phone in gigabytes. You might have 128 gigabytes of memory in your phone if you're lucky. Or if you've got only 16 gigabytes, you probably have to manage things a little bit better. So you're measuring those in gigabytes. And giga-, anytime you put giga- in front of anything it means you've got 10 to the ninth of them. So I can actually put giga- in front of pandas. And if I had gigapandas, that means I've got 10 to the ninth pandas. I don't know what you'd do with that many pandas. But that's what it means. Giga- in front of anything means I've got 10 to the 9th. And 10th to the 9th is a billion. So I've got a billion of them. Mega-- you've probably heard that, too. Mega- means 10 to the sixth or a million. I can put that in front of any unit I want. I can have megagrams, I can have megaseconds, I can have megadogs. I don't know what you'd do with a million dogs, but you could. So whatever you want, these prefixes are very, very flexible. And you can use them in any sort of way. Kilo- we just talked about, a kilometer, which is 1,000 meters. Kilo- is 1,000 of anything. And you'll notice that-- I wish they'd been a little more consistent-- that all these big numbers, 10 to the 12th, 10 to the 9th, 10 to the sixth, have capital letters unfortunately, that is, except for kilo-, which is a lowercase letter. But you have to do the capitalization correctly because as we'll see in a moment, there's a capital M and a lower case m. And they mean very, very different things. For things that make a unit smaller is deci-. We talked about that. Deci- is a tenth. So if you have a decimeter it's a tenth of a meter. If you have a decigram, it's a tenth of a gram. Centi- is 100th. Centimeter is 100th of a meter. Milli- is 1,000th. A milliliter is a thousandth of a liter, 10 to the minus third if you want to represent it in scientific notation. Then we get to other ones that you've probably heard maybe, but you aren't as familiar with. Micro-- You've probably heard of micro things. Even just the use of the term microbreweries if they're talking about very small batches of beers that are cooked up. Micro- is a very small number. It means 10 the minus 6th. It means one one-millionth of something. So a micrometer is one one-millionth of a meter. And it's hard to imagine a number that's small, but we need to be able to use numbers like that in chemistry and science all the time. Because we measure things that are the size of molecules and atoms which are less than a micrometer in size. Crazy stuff. Nano-- you've probably heard of nanotechnology. People talk about that. It's kind of a buzz word in many different places. What is nano? It's 10 to the minus ninth. That means you're looking at a billionth of something, and again, you can put it in front of any unit. You can have a nanometer. You can have nanogram. You could have a nanosecond, and you could have nano chickens. I don't know what you'd do with a millionth of a chicken. It would be a very small amount of chicken. It wouldn't fill you up very much-- but you could. That's the thing about these prefixes. You can put them in front of anything you want, and all they do is modify it to say, oh, I've got a millionth of that, or I've got a billionth of that. And then down here at the very bottom, and it's not the actual very bottom, it's just the very bottom for this class, is pico, which is 10 to the minus 12th, very, very small numbers, 10 to the minus 12th of something. If you're trying to memorize these-- and I encourage you to, I really encourage you to memorize these things-- the mnemonic that I think it works really well is if you're going down from tera to nano, you have the great mighty king died choking on mini metal nails. OK, so you can actually memorize that just by using the mnemonic down there. You'd still have to keep track of the numbers, but if you look at the numbers, if you can remember this guy is 10 to the 12th, down here for giga, mega, and kilo, they go by threes. So it goes 12, 9, 6, 3. Down here and deci, centi, and milli, it goes by ones, and so that one just requires a little more memorization, but down here in these last three, for micro, nano, pico, they again go by threes, 6, 9, and 12. So we're going to use these all the time. So you definitely need to be familiar with them, be comfortable with using them, and I'll show you how to do conversions with them in just a moment. So we can put them on as SI units, like I said, meters, grams, all those seconds. We can do all those things, but what's the point of using these prefixes? The point using these prefixes is, again, back to our understanding of numbers that we talked about. We can very easily picture numbers between 1 and 1,000, and we have a hard time picturing numbers outside of that range. And you'll notice that those prefixes are every 1,000 or so, which allows us to always represent a number in a way that we can picture the number and then think about the prefix. So our goal when we're using the prefixes is to get a number between 1 and 1,000 in front of our measurement. So you wouldn't go ahead and say, "Oh, I'm going to take a measurement, and I'm going to weigh myself and figure out that I weigh 65,317,000 milligrams." It doesn't make a lot of sense to do that, because now you've got this number of 65 million. It's correct. It's not a wrong way of measuring. It's just not a useful way of measuring, and these prefixes are designed to be useful. So even better, we report ourselves as 65.317 kilograms. Now we've got a number between 1 and 1,000. We can kind of picture 65 in our head, and then if we know what a kilogram is, we just say, oh, we've got 65 of those. So we're going to try to use those kind of things, and if I ask you to ever write something in a proper way using unit prefixes, you want that number to be between 1 and 1,000. And again, we'll talk about how to convert between those in just a moment. So good reporting of numbers, 15 milliliters, 100 kilometers, 42.82 seconds-- bad reporting will say 0.00329 meters. You don't want these very, very small numbers, or 10,300 milliliters, or a million microseconds. You don't want these very large numbers in there. But they're-- again, just as a reminder, they're just conversion factors. We can use them with any unit that we want, not just the SI units, not just grams, and liters, and seconds, and things like that. We can use them literally on anything you want. You could have mega balls, right? And that would just mean you had a million balls. You could have a micro ball, which means you took a millionth of this ball. That's fine. They're just a way of modifying any unit that you want. Now, why is it important to understand these? When I first made this lecture, I was literally typing up that last slide. I got ping from my email, and back in those days, there was this thing called Google Offers, and Google Offers was trying to sell you stuff, because you know, that's what Google does. Believe it or not, they tried to sell you stuff, and they came up with this offer. It said, "Hey, here's $8 for 30 ML of shea butter hand cream," and I looked at that and I just laughed, because what they were trying to sell me are ML with a capital M, and while-- as I was doing that last slide, I knew that a capital M meant one million, and so what they were actually trying to sell me is 30 milliliters of shea butter hand cream. That's millions of gallons of hand cream for only $12. Now, I was tempted to take them up on their offer, but you know, I was worried that if someone had a sense of humor, and they actually delivered millions of gallons of hand cream to my house, I would have no idea what to do with it. So I didn't take them up on their offer. But you'll see it quite often that even though we have these prefixes, even though we try to use them correctly, a lot of times in the world, people don't use them correctly, and so we just have to be aware of those things. All right, so we're going to practice converting these things, and the easiest way is to set everything up like a conversion, like we've done in our class. We're going to put what we start with on the left, what we're trying to get to on the right, write our conversion factor, and then fill in the numbers. So if I start with 400 microliters and I want to convert that into liters, I'm going to set it up just like I talked about before. So what I start with on the left, what I'm looking for on the right-- now, what do we have to do first? Numbers or units? Right, we always do units first in our conversion factor. So where do microliters have to go? Well, we've got microliters on the top. We want to get rid of them. So we better put a microliter on the bottom. And by the way, micro, when you're trying to write it, it's kind of like an M but it's a little curvy. So there's an M, and micro has got these kind of little curves on it, and also a little more curved in the middle. And you can look up how to write these things. If you're somebody who has nice handwriting, that's good enough for me. But so we've got a microliter there. If we want liters at the end, we better put liters on the top here. And so we know that our units cancel to give us what we want. So we've got the correct units here, which is our first priority. Now we've got to get the numbers in there, and how do we know the conversion between liters and microliters? Well, any time you're looking in this class and you see a unit and a unit with a prefix, we know we're going to be using those definitions that we have. And so if we look back on your chart-- and I'm going to post the chart on Canvas so you can print it out for yourself. I highly encourage you to do that-- micro means 10 to the minus 6. And again, you can put micro in front of any unit that you want, and so what I'm going to do is I'm going to put it in front of liters, which means microliter is 10 to the minus 6 liters, and I'm going to put the 1 there explicitly, right? When you just say microliter, you really do mean 1 microliter. And so here I've got my conversion equality. Now, you remember, I can always make two conversion factors off of that? And so you could write down those conversion factors and then use the right one, or you could use that rule I said, which is numbers stay attached to their units. And so in front of microliters, I've got a 1. So up here, when I find microliters, what am I going to put there? I'm going to put a 1. And then in front of liters, I've got a 10 to the minus 6, so what am I going to put up here? I'm going to put a 10 to the minus 6. You type all that in your calculator. Your calculator is probably going to tell you this, 0.0004 liters. I'm going to go ahead and write that in scientific notation, if you remember how to do that. If not, get a little practice. It's 4 times 10 to the minus 4th liters. What do we do about significant figures here? Well, the starting number didn't have a decimal in it, so it only had one sig fig. These conversions here, when we're converting using the unit prefixes, are exact, and so we don't do any significant figure work with them, because they have infinite significant figures, and so we end up with one significant figure in our answer, and it's 4 times 10 to the minus 4th liters. Go ahead. Practice that. Plug that in your calculator. Make sure you can use your calculator properly to get the answer that I got. Next, I'm going to show you an alternative way of doing this exact same conversion. So I'm going to do this, solve the exact same problem. I want to the exact same answer, but I'm going to do it slightly differently, and the reason I'm going to show you this one is because the method that I do second here is the method I'm actually going to use for the rest of the class when I talk about unit prefixes. You are welcome to do it either way. Some people are more comfortable doing it the way I just did it. Most people find the other way a little more intuitive, but it's not required by any means. So I'm going to do this. I'm going to start with 400 microliters and then I'm going to go to liters. I'm going to put my conversion factor. Nothing changes up till this point. That's all the same. But what changes now is what numbers I put in, and you're like, wait, you can't change the numbers you put in. I mean, the numbers are the numbers, right? Like, 10 to the minus 6 is just 10 to the minus 6, and you've got to put that there. And to some extent that's correct, but remember, numbers are just numbers. We can also manipulate them as long as we keep them equal to each other, right? I can say 3 equals 3. I can also multiply both sides by 2 and I get 6 equals 6. Both of those are correct. So what I'm going to do is I'm going to do this again. I've got 1 microliter is 10 the minus 6 liters. Like, you're not doing anything different, but here I am. My next step is to do something slightly different. I'm going to divide both sides by 10 to the minus 6, and I can certainly do that, right? You can divide both sides by the same number, and you get the same number. So now what I've got is 1 over 10 to the minus 6 microliters is equal to-- well, these guys cancel out and give me a 1-- 1 liter, OK? Now, if you remember from algebra, if you have 1 over 1 over something, that's equal to something. You can also plug it in your calculator. Divide 1 divided by 10 to the minus 6, and hopefully you will get to 10 to the positive six. So I'm going to write this here. I encourage you to verify it for yourself that you get 10 to the 6 microliters makes 1 liter. Notice, the thing that's changed here is we now have a positive exponent. Micro is defined as a negative 6, but now we say we've got 10 to the 6, positive 6. What does that mean? It says I need a lot of microliters to make 1 liter, right? Because a microliter is tiny. It's a millionth. And so I need a lot of them, 10 to the 6 of them, in order to make 1 liter. So this is just a different way of writing it. It's exactly the same mathematically. By the way, I was just thinking as we were doing here, if I said, "Hey, why don't you plug this 1 times 10 to the minus 6 into your calculator," how would you actually plug that to your calculator? Because how do you write 10 to the minus 6 properly in your calculator so it will do those calculations correctly? Because remember, I warned you earlier that if you don't type in scientific notation properly in your calculator, it can screw things up. So how would I type that in my calculator? Well, what I would type into my calculator-- so let me bring this down here into green. What I would type in my calculator is this, 1 divided by, and then I want to 10 to the minus 6. So you could write 10 carat minus 6, and in this case, it would totally work, but we're going to see these kind of numbers all the time, and I think another good practice is to realize that 10 to the minus 6 is the same thing as 1 times 10 to the minus 6, right? Because any time you multiply by 1, you get the same number. And so here, I'm going to take, and I'm going to use whatever built-in scientific notation my calculator has. I'm using that lower or that small uppercase E that the Texas instruments calculators often use. And 1e minus 6, if you type that in your calculator, your calculator will either say 1 million, and then you have to turn that into scientific notation, or your calculator might say 10 to the 6. OK, so good thing to practice typing in your calculator. Also good to realize that whenever we have 10 to the power, we can write that as 1 times 10 to the power, and that's a good way of putting it into our calculator. Back to the problem at hand. We now have this relationship between microliters and liters, and so if I scroll back up to my problem here, I have space for numbers in front of liters, and I have space for numbers in front of microliters, but my numbers are going to change now. If numbers stay attached to their units and I've got 10 to the 6 in front of microliters here, then I want a 10 to the 6 in front of microliters up here. And again, it's positive 6. Lots of microliters, right? Remember, positive exponents are big numbers. Lots of microliters make up 1 liter. Why do I do this method? Do we do all this math that we did at the bottom? It seems like a lot of extra work to try to get to the exact same answer, because if you type this in your calculator, your calculator is going to tell you 0.0004 liters or 4 times 10 to the minus 4th liters. It's going to give you the exact same answer. It seems like a lot of extra work to do all that math. And if you did all that math every time, absolutely, it would be a lot of extra work and it totally wouldn't be worth it. But here's the part that I think is intuitive for some people, is that since micro is small, I need a lot of microliters to make a liter. And you can use that logic in order to fill these in, without having to actually write down the math every time. So the general rule, if you're going to be using this method, and like I said, this method is the one I'm going to be using this we go forward in the class, the general is this, is you put a 1 in front of the larger unit. So here, liters is larger than microliters, so I put a 1 in front of liters. And then you put a positive exponent in front of the smaller unit. If it's always one of the larger unit, I'm always going to need lots of the smaller unit to make up that bigger unit. So even though we had micro, which is by definition 10 to the minus 6, we say, hey, I need a lot of microliters, so I need 10 to the positive 6 microliters. Let's do a few more examples down here of that. What about if I was trying to convert between grams and nanograms? What would I say? I'd put a one in front of the larger unit. A gram is a larger unit than a nanogram. And I'd put a positive exponent in front of the smaller unit. A 10 to the 9th nanograms is a lot of little tiny nanograms. It makes up one big gram. This also works for the big units. If I was trying to convert between seconds and megaseconds, which is the larger unit-- mega is huge, right? So megasecond is a huge unit. It's a million seconds. And then mega is 10 to the 6, so I put a 10 to the positive 6 here. Lots of seconds make up 1 megasecond. So it's a little weird because you always use that positive exponent, but most people can do that logically in their brain, a little bit easier than they can do the writing that 1 and 10 to the minus 6, because those minus exponents are a little bit tougher to do mentally. It's tough to think about a millionth of something. It's easier to think about a million of something. Again, either way you want to do it, if you do it correctly, you'll get the right answer and you'll be OK. It's if you do it incorrectly that you've got a problem. So we'll practice with that one as we go forward. So let's go ahead and do a few problems. We've got four different problems here, and we're trying to convert the numbers to regular numbers without unit prefixes. So if we had gigaseconds, we'd convert it to seconds. So in the first one, we're going to take millimeters and we're going to convert it into meters. This is a great time for you to pause the video, write these out, and try them, plug them in your calculator, and get a final answer. Remember, watching me do it is very different than you doing it yourself. Even if you make mistakes, even if you aren't quite sure what you're doing now, the actual act of practicing them, figuring out how you're solving it, and correcting yourself, is actually the most useful way to learn this stuff. I make it look easy because I've been doing this longer than most of you have been alive. So pause the video. Try these out, then come back. OK, I'm assuming you tried them all. It's a good assumption, right? Yeah? OK, good. So 3.757 millimeters, I said I'm trying to convert those out of prefixes, so I'm just going to convert that to meters. What I start with on the left, what I'm looking for on the right-- I need millimeters to cancel, so it's got to go on the bottom. I'm looking for meters, so it's got to go on the top. Now, with meters and milli, it gets a little confusing sometimes with all these little Ms, but we've got to keep track of them. So what do we do? We put a 1 in front of the larger unit. Which is larger, a meter or a millimeter? Well, a milli is 1,000, right? A millimeter is a very a small amount, height of an ant. A meter is bigger than that. And so I put a 1 in front of the meter. And then milli, if you look it up in the table, is 10 to the minus 3rd, but in our way of doing it here, we're going to use the 10 to the positive 3rd. Lots of little tiny millimeters just stacked up on each other make a meter. Plug that into your calculator. Your calculator will probably tell you 0.003757. I'm going to go ahead and convert that to scientific notation, because I like reading scientific notation better. If I don't ask for an answer in a particular format, you could theoretically write either way. All right, how about the next one? We've got 1.99 megayears, and we're going to try to convert that into years, because our problem was to get rid of our prefixes. What's going to go on the bottom? We want megayears to cancel, and so it's got to go down here, and we're trying to get years, so it's got to go up here. So what's bigger, a year or a megayear? Remember, mega is 10 to the 6. It's huge. And so our megayear is our larger unit here, and mega is just 10 to the 6, and so we need 10 to the 6 years, a million years, to make 1 megayear. Type that in your calculator. You should get 1.99 times 10 to the 6 years. Next one, 37 kilograms, and we're going to try to convert 37 kilograms into grams, because our goal in this thing was to remove the unit prefixes. Where does grams have to go? Well, if we're trying to get grams, grams better go on the top, because here, it's on the top, right? Remember, when there's no denominator, it's kind of like there's a 1 in the denominator, and grams is in the numerator. Kilograms is in the top here, so it's got to go on the bottom here. Always, one on top, one on bottom. If you have two on top, they don't cancel. Even if you draw a line through them, they don't cancel if you have two on top. So which one's bigger, a gram or a kilogram? Kilo is 10 to the 3rd, if you look on your list, so it is bigger. And I need a lot of grams, 10 to the 3rd, a lot of grams stacked on top of each other to make one big kilogram. Type that into your calculator. Your calculator will probably tell you 37,000 grams. You could also write that as 3.7 times 10 to the 4th grams and you'd be OK. Last one, 0.357 micrometers. Now, 0.357 isn't a number between 1 and 1,000, so this isn't proper use of our unit prefixes, but it's allowable. It's perfectly allowable to write whatever number we want in front of the unit prefix, but when we say, hey, let's use these properly, we generally write a number between 1 and 1,000. So 0.3-- OK, for consistency sake, let's use that new color here. 0.357 micrometers-- some people will say micrometers and that's perfectly OK as well. We're going to convert that guy to meters because that was our goal. Hopefully, you're starting to get where units go. The micrometer has to go on the bottom, because one on top and one bottom cancels out. The meter has to go on top, because that's what we're trying to get at the end. How do we relate meters and micrometers? Now, here, we're doing a bunch of problems that are similar, but again, it's just good practice. When you see a unit and a unit with a prefix on it, then you know that you're going to-- see, we're using these definitions. Which one's larger, a meter or a micrometer? A micrometer is 10 to the minus 6, so a meter is going to be much larger in this case. And if you look at the definition of micro, it's 10 to the minus 6. And so we put 10 to the positive 6 here, because that's the method we're using. A lot of little tiny micrometers stacked on top of each other makes one big meter. Plug this into your calculator and you should get 3.57 times 10 to the minus 7th meters. Some calculators will write 0.000-- with six 0s and then the 3. I'm writing it in scientific notation because it's much easier to understand it this way. Be sure that when you do it, you are getting this minus 7 as your exponent. A lot of people think they should get a 6 because they've got a 6 here, but that's not the case, because this guy already has a decimal. So make sure you're getting that 10 to the minus 7th. What about if we have a number, and we're wanting to derive that number to be using our unit prefixes? So we've got, you know, 14,805,000. How are we going to convert that to be able to use our unit prefix? That turns out the easiest way to do that is to write the number in scientific notation, because then you have a number between 1 and 10, and an exponent after that. In ideal cases, the exponent matches one of our prefixes. In non-ideal cases, we're going to have to move things around a little bit. So let's do just a few examples, because examples are always the easiest way. So we're going to start with 0.039 meters, and we want to convert that to using a unit prefix, which means we're going to convert it to megameters, or nanometers, or micrometers, or whatever it is that's the appropriate thing to do. So the first thing you want to do is write this number in scientific notation, if you remember how to do that. We're going to move the decimal 1, 2, 3. It's a small number, so we should have a positive or a negative? Negative exponent. 3.9 times 10 to the minus 3rd meters. And again, I think we've said this before, but just as a reminder, when we are converting between numbers and decimals and scientific notation, we don't change the number of sig figs. We had two sig figs here. We have to save figs here. And that's an important thing is when you're just converting between decimal and scientific notation, you cannot change the number of significant figures. It's very important. 3.9 times 10 to the minus 3rd meters-- well, what do we know? We know that 10 to the minus 3rd is exactly the same thing as milli. They are entirely equivalent. So everywhere I write 10 to the minus 3rd, I could also write milli. It's just like if you had 3x equals 5 and you knew that x-- well, let's just do 3x equals 6. You knew that x equals 2, right? That means I could write x or I could write 2, and it would be fine. I could write 3 times 2 equals 6. So I can write x or I could write 2, and it's the exact same problem. Here I can write 10 to the minus 3rd or I can write meters, and it's the exact same way of writing it. And so I can just substitute that in. Instead of writing 10 to the minus 3rd, I'm going to write milli. I said meters just a second ago. I met milli. And I've got 3.9 millimeters. What I see students do a lot of times is instead of substituting it, they add it, and they write 3.9 times 10 to the minus 3rd millimeters, and that's not correct. You're substituting. You're saying, instead of writing 10 to the minus 3rd, I'm going to write milli because they mean the same thing. That's great. You don't just add it onto there. If you're a little uncomfortable with that, you can be like, OK, I know that milli is 10 to minus 3rd, and so I'm going to say 3.9 times 10 to the minus 3rd meters, and I want to convert this to millimeters. And you can set up a conversion like this. I'm going to go through this one really quick, and it would look something like that. Meters on the bottom, millimeters on the top. Meters is bigger, so it gets the 1. 10 to the 3rd millimeters-- and you end up with 3.9 millimeters if you plug all that into your calculator. You can do it that way. I encourage you to try to be comfortable doing it this first way because it's a little faster, but it's certainly not wrong to do it the other way, and you'll get the same answer. All right, let's do another example. Let's go ahead and do-- We've got 4.9 times 10 to the 4th liters to use unit prefixes. Now, this is great. The numbers are already written in scientific notation, so we don't have to convert to scientific notation. What did we do last time? We looked at this exponent and we said, "Hey, it matches one of our common exponents." And we look at this one and we say, "Hey, that doesn't match one of our common exponents." So we can't just substitute in milli, or mega, or whatever, because 10 to the 4th doesn't match anything on our list if you go look at your list. So what do we do? Well, we use the fact that all of these scientific notation and all these prefixes are factors of 10, and when we do factors of 10, it's just like moving a decimal around. 4.9 times 10 to the 4th-- what I'm going to do is I'm going to take this decimal point and I'm going to move it to the right. I'm going to make the number 49 here. Why did I move it to the right? Because remember, if we want to use unit prefixes, we want a number between 1 and 1,000. If I moved it to the left, I would gotten 0.49, less than 1, which is not what I want to do when I use unit prefixes. So I've got 49. Can I still write times 10 to the 4th? I can't, right? Because here, this was 4.9 times 10 to the 4th. I can't now write 49 times 10 to the 4th. That would be a different number. So I made this one bigger by a factor of 10. What I need to do it is I need to make this one smaller by a factor of 10. How do you make it smaller? You subtract 1 from the exponent. So now, since I made this 10 times bigger, I'm going to make this 10 times smaller by subtracting 1. And so I actually get 49 times 10 to the 3rd liters. Now, 10 to the 3rd does match one of our prefixes, and so we're good. 10 to the 3rd matches kilo, and so we end up with 49 kiloliters, and we can just substitute that straight in, or we could do it the math way, and doing the conversion. Cool, so there we go. If we've got a number where the exponent doesn't match, we can go ahead and just move the decimal point to the right, change our exponent by subtracting 1, and then see what happens. What about in the case that that doesn't even work? So let's do one more example here. Now we've got 5.35 times 10 to the minus 5th grams. Sorry, 5.35 times 10 to the minus 4th grams. What are we going to do here? 5.35 times 10 to the minus 4th grams. Well, we say, hey, 10 to the minus 4th doesn't match anything we know, so I'm going to move my decimal point over one. I'm going to get 53.5 times 10 to the minus 4th, and I've got to subtract 1. When I made one 10 times bigger, I need to make one 10 times smaller. Always subtract 1. That's going to be 53.5 times 10 to the-- what's minus 4 minus 1? People attempted to write minus 6 here-- I'm sorry, minus 3 here, to say, well, 4 minus 1 is 3. But minus 4 minus 1 is going to be minus 5. So make sure you understand that one. Go over it a couple of times if you need to, OK? So 53.5 times 10 to the minus 5-- you go over to your chart. You say, 10 to the minus 5, it isn't on my chart. Well, if it's not in my chart, what do I do? I do the same thing. And you'll never have to do it more than twice. Since all of our numbers are separated by powers of 1,000, you'll never have to do this more than twice, but in this case, we do have to do it a second time because our exponent didn't match. So we now have 535 times 10 to the minus 5th, but we've got to subtract 1 from that, and so we get 535 times 10 to the minus 6 grams. So every time we move the decimal point to the right, we need to subtract 1 from our exponent. But now, 10 to the minus 6 does match something in our table. It matches micro, and so we get 535 micrograms. 535 micrograms. Total aside here, for a lot of you, you can be going into health fields. Micro-- if you remember, most doctors don't have great handwriting, right? And micro and milli look awful similar when you're trying to write things out. So if you're not really careful about your micro, your micro and your milli can look very similar, especially if your handwriting is not so good. So what you'll actually find quite commonly in the health field is instead of writing micrograms, you will see people write mcg. Now, technically mcg is milli, which is 10 to the minus 3rd, and centi, which is 10 to the minus 2nd, and grams. So it's millicentigrams. If you do that out, 10 to the minus 3 times 10 to the minus 2 is 10 to the minus 5th grams, and that's not the same as micro. A millicentigram is a valid unit. You can put milli in front of anything, even centigrams. However, it's not equal to a microgram. However, it is used in the health field quite frequently as a substitute for micro, so just be aware of that. Even though millicenti does not equal micro, it is used as a substitute for micro. So when you see mcg on something, it actually means micrograms. All righty, just a few more to try here. We're going to convert each of these numbers into using unit prefixes. Again, what I highly encourage you to do is pause this video. Go ahead, write out those numbers using prefixes, following the method that we did. Convert to scientific notation. Check your exponent. If not, move the decimal point to the right, subtracting 1 from your exponent every time you do that. Pause the video. Try it out, because that makes all the difference. You doing this is what actually makes it work for you. You watching me do this does not make it work for you. You're back. OK, How did it go? Was it was good? OK, good to hear. Most of them, most of them went OK? Awesome. All right, let's try it. We'll see how you did. 0.00374 joules, I'm going to write that out as 3.74 times 10 to the minus 3rd joules. You're like, "I don't know what a joule is, professor." Doesn't matter. I can put it in front of unit I want, even ones that I have no idea what they are. Doesn't matter, OK. We'll learn what a joule is later in the class. 10 the minus 3rd does match milli. They're the same thing. So I can substitute in milli for 10 to the minus 3rd and keep this the same. Now, you'll notice I don't write the times here. I almost did just out of habit. I don't write the times here. I just write 3.74 millijoules. Implicitly, this is actually saying the same thing as 3.74 times milli times joules. That's what it means but nobody actually writes it like that. They just write 3.74 millijoules. Here we've got 1,387,000 seconds. 1.387 times 10 to the 6 seconds, and if you remember, 10 to the 6 is mega, and so we get 1.387 megaseconds. Another thing about these conversions with unit prefixes is just like converting between decimal and scientific notation, you should never change the number of significant figures. So if I start with four sig figs, I should end with four sig figs. If I start with one, I should end with one. 8.27 times 10 to minus 3rd grams. This is the easiest so far. We already gave it to you in scientific notation. This one's already in a unit that we know that exists. It's milligrams. This is 8.27 milligrams. Now, notice my handwriting. My handwriting is not great. I can't tell whether this is a lowercase M or a capital M the way I wrote it. So I'm going to go back and I'm going to correct this, and try to make sure that my M is small. There are people out there who love to write in all caps all the time. If you write that, people are going to think that you mean a million grams, and so you do have to write this guy as a small M. My difficulty is that even when I write a large M, I tend to make it just a bigger small M. Good practice in this class is to write your capital Ms in a slightly different type of script here, block script as opposed to more of a smooth script, so that people can distinguish it. 6.27 times 10 to the 8 K. Remember, K stands for Kelvin. That's a temperature unit that we'll talk a lot more about in the next chapter. What do we do? 10 to the 8th doesn't match anything on our list, so what are we going to do? We're going to move that decimal point one to the right. 62.7 times 10 to the 8 minus 1 Kelvin, or 62.7 times 10 to the 7th Kelvin. Still doesn't match anything so I'm going to do it one more time. I've got 627 times 10 to the 7 minus 1 Kelvin, which is 627 times 10 to the 6 Kelvin. 10 to the 6 does match. It is mega, and I've got 627 megakelvin. So you can see, again, my handwriting is not great, but I tried to make that M a little bit bigger. One last one, 4.2 times 10 to the minus 5th meters. 10 to the minus 5 doesn't match anything on our list, and so we're going to go ahead and move that over, and we've got 42 times 10 to the minus 5 minus 1 meters, which is the same thing as 42 times 10 to the minus 6 meters. 10 to the minus 6 does match micro, and so I get 42-- oops-- micrometers, just like that. So that was how we convert numbers from regular numbers or scientific notation numbers into using unit prefixes. Our goal is to get a number between 1 and 1,000, not outside of that range. Here's one for you to try. Well, it's the same thing. It's 18 nanometers. How many meters is that? So pause a video. Try that one out, and see what you get. All righty, 18 nanometers, try to convert to meters. What goes on the bottom? Nanometers. On the top, meters, right? For nanometers, I need one on top, one on the bottom. Which is bigger, a meter or a nanometer? Nano is 10 to the minus 9. It's a tiny number. So a meter is bigger. But then we say we need lots of nanometers, 10 to the positive 9, lots of little tiny nanometers stacked on top for each other to make one big meter. If you plug that into your calculator, you should get 1.8 times 10 to the minus 8 meters, and it would look something like that. For those of you following at home, you're like, shouldn't that be 10 to the minus 9th, because you know, I just have it in the denominator. It should be 10 to the minus 9. No, because you actually have to type these into your calculator. It's OK to use your calculator. But it turns out, right, 18, if you put that into scientific notation is 1.8 times 10 to 1. And so that's why you end up with that minus 8, because 1 minus 9 for those exponents is minus 8. All right, what about when we are trying to do conversions of these prefixes where it's not one step? If we're trying to convert, say, 187.3 micrometers into kilometers-- if you could just go ahead and write that out as a conversion-- again, it's never a bad idea just to write out your conversion and see what you get. I get micrometer is on the bottom. I get my kilometer is on the top. Those guys are going to cancel, one on top, one on bottom. I get kilometers like I expect. [HUFFS] Tired of hearing that yet? Good. The more tired you are of hearing it, the more it means you understand it, and you're like, [HUFFS] of course I know that by now. So 187.3, when you look at those numbers, you're like [HUFFS] kilometers and micrometers. Ah-- oh-- I don't know. And it's OK, because remember, when we look at a conversion, we go, I have no idea what to do with that conversion. I don't know that conversion. There's a chance it means that we have a multi-step conversion in front of us. So we ask ourselves, what do I know how to convert micrometers into? And in general, any unit with a prefix on it, what we know how to do is we know how to get rid of that prefix. So I know how to convert micrometers into meters. Remember, now that we're doing a multi-step, we're going to do a plan. What are we going to do? What's our plan in converting this? Now, once we're in meters, do we know how to convert meters to kilometers? Yes, we've been working on that for the last half an hour or so. So we do know how to turn micrometers into meters, and we know how to convert meters into kilometers. So that's our plan. Remember, each arrow in our plan is a conversion. So I'm going to do 187.3 micrometers-- I've got arrow number one, arrow number two, and I've got kilometers on the right here. What's going to go in my first conversion? Well, I need to get rid of micrometers, so it's going to go there. My first arrow tells me that I'm converting that into meters. My micrometers cancel one on top, one on bottom. Just because you draw a line through it, doesn't mean it cancels. It has to be one on top, one on the bottom. Now, in my next one, I'm to convert meters to kilometers, which means I need meters to go away, one on top, one on the bottom. And I need kilometers on the top, so I get kilometers at the end. I've got my units right now. I can put in numbers. What's bigger, a meter or a micrometer? Micro is tiny, and it's 10 to the minus 6, but I need a lot of tiny micrometers stacked up on each other to make one big meter. Kilometers and meters, a kilometer is bigger in this case. Kilo is 10 to the 3rd. It's a lot, so I need a lot of meters to make one big kilometer. I need to stack meter, meter, meter, meter, meter to get this big thing called a kilometer. You put that into your calculator. Excuse me. And what do you get? You get 1.87 time-- sorry, 1.873 times 10 to the 2nd, divided by 10 to the 6th, divided by 10 to the 3rd. So what do you end up with? You end up with 10 to the-- [CLEARS THROAT] excuse me. That's-- 6 plus 3 is 9. 2 minus nine is minus 7. 1.873 times 10 to the minus 7. Now, I did this in my head just because I could, but you're certainly welcome to do that in your calculator. In your calculator, you type in 1.873 times 10 to the 2, which you'd use that E notation, whatever your calculator likes. Remember, 10 to the 6 I can write as 1E6, and 10 to the 3rd I can write as 1E3. And hopefully, your calculator gives you 1.873 times 10 to the minus 7th. It might tell you 0.00000-- I don't know, six 0s, and 18 and 7, and maybe they'll show you that 3 there, maybe it won't. So be careful that sometimes your calculator will chop off the digits that you need. Here, we have four sig figs. We better have four sig figs that we start with. Gives me four sig figs in my answer. If your calculator-- let me go ahead write that out. If your calculator gives you-- 5, 6, 7, 8, 9-- a lot of TI calculators will write the answer that way. That's not right. That's not the right answer. Just because your calculator only gives you those digits doesn't mean those digits are the right digits to have. You need to have four sig figs. It's really useful on your calculator to understand how to convert your calculator to tell you all the digits that you need. Most Casio calculators, other brands of calculators, they do this just fine. They know how to convert and when to convert to scientific notation. TI doesn't quite get it right. They will keep things in decimal notation until they only have one significant figure, and then when they're forced to, they will put things into scientific notation. I don't have my camera set up right now to show you the calculator, so I'm just going to hold it up to the screen here. But let's say I had that answer in front of me. I had 1, 2, 3, 4, 5, 6, 1, 8, 7, 3, and my calculator was just showing the 187, but I'm like, I know there should be more significant figures in there. One thing you can do is tell your calculator, can you please show this number to me in scientific notation? And your calculator can do that. How you do that on the TI calculator is if you look on top of the-- there's a 2nd key on the top left, and right above that-- or right next to that, there's a key that says DRG, and above that in blue, it says SCI/ENG. And so I'm going to put that up here. And here's my 2nd key, and here next to it is the DRG key. It says SCI/ENG. What I'm going to do is I'm going to press the 2nd. I'm going to press SCI/ENG key. And now it gives me a list of FLO SCI/ENG. I'm going to use the arrow keys, and I'm going to tell it SCI, which says, put this in scientific notation, please. And when I hit Enter, it actually gives me the answer in scientific notation, which is 1.873 times 10 to the minus 7th. And so I can actually get those digits back that I know I'm supposed to have. So be aware of that, especially if you're using the TI calculators, that they really play loose with these small numbers, and you sometimes need to convert it. Now, the problem is, now your calculator's in scientific mode. So I typed the number 23 in there, and it's telling me, oh yes, that's 2.3 times 10 to the 1st, and that's annoying, because when you're just trying 23, you want the calculator to say 23. So what you do is you do the same thing, but we're going to switch back out of SCI mode. So we hit the 2nd, DRG, and we're going to convert back to floating point mode, so F-L-O, which is regular mode, and now it tells me 23. So good thing to know how to do it on your calculator if you've got one of the TI calculators. Like I said, most other brands do that a little bit better. There we go. I'm like, it's not willing to go to the next page. So if you want or need more practice, and most of you will, these conversions with unit prefixes usually take people a while to grasp. I have a website. It's professorclements.com, and in the introductory chemistry section, there's an online practice part that will give you single step and two step metric conversions until the cows come home. It makes up a brand new one each time. It's not just a list of them, and it makes up different numbers of significant figures and different conversions, and so you can literally do this all day long until you get comfortable with it. What I recommend to people is literally go and do 40 of them. The first 10 will take you a long time. The second 10 will take you a little bit less time. The third 10 won't take you much time at all, and the fourth 10, you'll be like [SNAPS FINGERS] just going through them, not quite that fast, but really fast because you'll have the pattern down. But if you do 40 of them, you do 50 of them, and you can get to where you're getting all of them right, that will be really helpful as we move forward. What it looks like, it will give you a problem. It will ask for the answer. If you need to put in scientific notation, you can use that E notation. So let's say you needed to put in 4 times 10 to the 5th. You can write 4E5. You can also write 4 times 10 carat 5. It will accept either way. This way is a little more compact. You can also write it in decimal form if you want to as well. It'll check your answer and then it will also tell you the answer, and if you wanted to, there's a little button that says Show The Solution. It'll actually show you how to set it up. So here we are converting teraliters to liters, and it says you need 1 times 10 to the 12 liters on top. You need a teraliter on the bottom, and that gives you the answer of 2.72 times 10 to the 14th liters. So I really encourage you, go here, practice those single step ones first. Again, like 40 of them, 50 of them. Then move on to the two-step ones. Those ones, maybe not 40 or 50 because they take a while, but maybe like 20 or something like that, until you get to the point where you're really comfortable with them. The computer will let you do it all day long. It'll never tell you you're a bad person for doing too many of them, and it won't get impatient with you. So good resource. Encourage you to take advantage of it. Sorry about this extraordinarily long video. I think it's the longest one of the entire semester. I try to break things down a little bit more, but this was one kind of big topic about these metric prefixes and how to use them. Thanks so much for listening. Bye bye.