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Introduction to scientific notation. An in-depth discussion about why and how scientific notation is used. Created by Sal Khan.
Video transcript
I don't think it's any secret that if one were to do any kind of science, they're going to be dealing with a lot of numbers. It doesn't matter whether you do biology, or chemistry, or physics, numbers are involved. And in many cases, the numbers are very large. They are very, very large numbers. Very large numbers. Or, they're very, small, very small numbers. Very small numbers. You could imagine some very large numbers. If I were to ask you, how many atoms are there in the human body? Or how cells are in the human body? Or the mass of the Earth, in kilograms, those are very large numbers. If I were to ask you the mass of an electron, that would be a very, very small number. So any kind of science, you're going to be dealing with these. And just as an example, let me show you one of the most common numbers you're going to see, in especially chemistry. It's called Avogadro's number. Avogadro's number. And if I were write it in just the standard way of writing a number, it would literally be written as-- do it in a new color. It would be 6022-- and then another 20 zeroes. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. And even I were to throw some commas in here, it's not going to really help the situation to make it more readable. Let me throw some commas in here. This is still a huge number. If I have to write this on a piece of paper or if I were to publish some paper on using Avogadro's number, it would take me forever to write this. And even more, it's hard to tell if I forgot to write a zero or if I maybe wrote too many zeroes. So there's a problem here. Is there a better way to write this? So is there a better way to write this than to write it all out like this? To write literally the 6 followed by the 23 digits, or the 6022 followed by the 20 zeroes there? And to answer that question-- and in case you're curious, Avogadro's number, if you had 12 grams of carbon, especially 12 grams of carbon-12, this is how many atoms you would have in that. And just so you know, 12 grams is like a 50th of a pound. So that just gives you an idea of how many atoms are laying around at any point in time. This is a huge number. The point of here isn't to teach you some chemistry. The point of here is to talk about an easier way to write this. And the easier way to write this we call scientific notation. Scientific notation. And take my word for it, although it might be a little unnatural for you at this video. It really is an easier way to write things like things like that. Before I show you how to do it, let me show you the underlying theory behind scientific notation. If I were to tell you, what is 10 to the 0 power? We know that's equal to 1. What is 10 to the 1 power? That's equal to 10. What's 10 squared? That's 10 times 10. That's 100. What is 10 to the third? 10 to the third is 10 times 10 times 10, which is equal to 1,000. I think you see a general pattern here. 10 to the 0 has no 0's. No 0's in it. 10 to the 1 has one 0. 10 to the second power-- I was going to say the two-th power. 10 to the second power has two 0's. Finally, 10 to the third has three 0's. Don't want to beat a dead horse here, but I think you get the idea. Three 0's. If I were to do 10 to the 100th power, what would that look like? I don't feel like writing it all out here, but it would be 1 followed by-- you could guess it-- a hundred 0's. So it would just be a bunch of 0's. And if we were to count up all of those 0's, you would have one hundred 0's right there. And actually, this might be interesting, just as an aside. You may or may not know what this number is called. This is called a googol. A googol. In the early '90s if someone said, hey, that's a googol, you wouldn't have thought of a search engine. You would have thought of the number 10 to the 100th power, which is a huge number. It's more than the number of atoms, or the estimated number of atoms, in the known universe. In the known universe. It raises the question of what else is there out there. But I was reading up on this not too long ago. And if I remember correctly, the known universe has the order of 10 to the 79th to 10 to the 81 atoms. And this is, of course, rough. No one can really count this. People are just kind of estimating it. Or even better, guesstimating this. But this is a huge number. What may be even more interesting to you is this number was the motivation behind the naming a very popular search engine-- Google. Google is essentially just a misspelling of the word "googol" with the O-L. And I don't know why they called it Google. Maybe they got the domain name. Maybe they want to hold this much information. Maybe that many bytes of information. Or, it's just a cool word. Whatever it is-- maybe it was the founder's favorite number. But it's a cool thing to know. But anyway, I'm digressing. This is a googol. It's just 1 with a hundred 0's. But I could equivalently have just written that as 10 to 100, which is clearly an easier way. This is an easier way to write this. This is easier. In fact, this is so hard to write that I didn't even take the trouble to write it. It would have taken me forever. This was just twenty 0's right here. A hundred 0's I would have filled up this screen and you have found it boring. So I didn't even write it. So clearly, this is easier to write. This is just good for powers of 10. But how can we write something that isn't a direct power of 10? How can we use the power of this simplicity? How can we use the power of the simplicity somehow? And to do that, you just need to make the realization. This number, we can write it as-- so this has how many digits in it? It has 1, 2, 3, and then twenty 0's. So it has 23 digits after the 6. 23 digits after the 6. So what happens if I use this-- if I try to get close to it with a power of 10? So what if I were to say 10 to the 23? Do it in this magenta. 10 to the 23rd power. That's equal to what? That equals 1 with 23 0's. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23. You get the idea, that's 10 to the 23rd. Now, can we somehow write this guy as some multiple of this guy? Well, we can. Because if we multiplied this guy by 6-- if we multiply 6 times 10 to the 23rd, what do we get? Well, we're just going to have a 6 with twenty three 0's. We're going to have a 6, and then you're going to have twenty three 0's. Let me write that. You're going to have twenty three 0's like that. Because all I did, if you take 6 times this. You know how to multiply. You'd have the 6 times this 1. You'd get a 6. And then all the 6 times the 0's will all be 0. So you'll have 6 followed by twenty three 0's. So that's pretty useful. But still, we're not getting quite to this number. I mean, this had some 2's in there. So how could we do it a little bit better? Well, what if we wrote it as a decimal? This number right here is identical to this number if these 2's were 0's. But if we want to put those 2's there, what can we do? We could put some decimals here. We could say that this is the same thing as 6.022 times 10 to the 23rd. And now, this number is identical to this number, but it's a much easier way to write it. And you could verify it, if you like. It will take you a long time. Maybe we should do it with a smaller number first. But if you multiply 6.022 times 10 to the 23rd, and you write it all out, you will get this number right there. You will get Avogadro's number. Avogadro's number. And although this is complicated or it looks a little bit unintuitive to you at first, this was just a number written out. This has a multiplication and then a 10 to a power. You might say, hey, that's not so simple. But it really is. Because you immediately know how many 0's there are. And it's obviously a much shorter way to write this number. Let's do a couple of more. I started with Avogadro's number because it really shows you the need for a scientific notation. So you don't have to write things like that over and over again. So let's do a couple of other numbers. And we'll just write them in scientific notation. So let's say I have the number 7,345. And I wanted to write it in scientific notation. So I guess the best way to think about it is, it's 7,345. So how can I represent a thousand? Well, I wrote it over here, 10 to the third is 1,000. So we know that 10 to the third is equal to 1,000. So that's essentially the largest power of 10 that I can fit into this. This is seven 1,000's. So if this is seven 1,000's, and then it's 0.3 1,00's, then it's 0.4 1,000's-- I don't know if that helps you, we can write this as 7.345 times 10 to the third because it's going to be seven 1,000's plus 0.3 1,000's. What's 0.3 times 1,000? 0.3 times 1,000 is 300. What's 0.04 times 1,000? That's 40. What's 0.05 times 1,000? That's a 5. So 7.345 times 1,000 is equal to 7,345. Let me multiply it out just to make it clear. So if I took 7.345 times 1,000. The way I do it is I ignore the 0's. I essentially multiply 1 times that guy up there. So I get 7, 3, 4, 5. Then I had three 0's here, so I put those on the end. And then I have three decimal places. 1, 2, 3. So 1, 2, 3. Put the decimal right there. And there you have it, 7.345 times 1,000 is indeed 7,345. Let's do a couple of them. Let's say we wanted to write the number 6 in scientific notation. Obviously, there's no need to write in scientific notation. But how would you do it? Well, what's the largest power of 10 that fits into 6? Well, the largest power of 10 that fits into 6 is just 1. So we could write it as something times 10 to the 0. This is just 1, right? That's just 1. So 6 is what times 1? Well, it's just 6. So 6 is equal to 6 times 10 to the 0. You wouldn't actually have to write it this way. This is much simpler, but it shows you that you really can express any number in scientific notation. Now, what if we wanted to represent something like this? I had started off the video saying in science you deal with very large and very small numbers. So let's say you had the number-- do it in this color. And you had 1, 2, 3, 4. And then, let's say five 0's. And then you have followed by a 7. Well, once again, this is not an easy number to deal with. But how can we deal with it as a power of 10? As a power of 10? So what's the largest power of 10 that fits into this number, that this number is divisible by? So let's think about it. All the powers of 10 we did before were going to positive or going to-- well, yeah, positive powers of 10. We could also do negative powers of 10. We know that 10 to the 0 is 1. Let's start there. 10 to the minus 1 is equal to 1/10, which is equal to 0.1. Let me switch colors. I'll do pink. 10 to the minus 2 is equal to 1 over 10 squared, which is equal to 1/100, which is equal to 0.01. And you I think you get the idea that the--, well, let me just do one more so that you can get the idea. 10 to the minus 3. 10 to the minus 3 is equal to 1 over 10 to the third, which is equal to 1/1,000, which is equal to the 0.001. So the general pattern here is 10 to the whatever negative power is however many places you're going to have behind the decimal point. So here, it's not the number of 0's. In here, 10 to the minus 3, you only have two 0's but you have three places behind the decimal point. So what is the largest power of 10 that goes into this? Well, how many places behind the decimal point do I have? I have 1, 2, 3, 4, 5, 6. So 10 to the minus 6 is going to be equal to 0.-- and we're going to have six places behind the decimal point. And the last place is going to be a 1. So you're going to have Five 0's and a 1. That's 10 to the minus 6. Now, this number right here is 7 times this number. If we multiply this times 7, we get 7 times 1. And then we have 1, 2, 3, 4, 5, 6 numbers behind the decimal point. So 1, 2, 3, 4, 5, 6. So this number times 7 is clearly equal to the number that we started off with. So we can rewrite this number. Instead of writing this number every time, we can write it as being equal to this number. Or, we could write it as 7. This is equal to 7 times this number. But this number is no better than that number. But this number is the same thing as 10 to the minus 6. 7 times 10 to the minus 6. So now you can imagine numbers like-- imagine the number-- what if we had a 7-- let me think of it this way. Let's say we had a 7, 3 over there. So what would we do? Well, we'd want to go to the first digit right here because this is kind of the largest power of 10 that could go into this thing right here. So if we wanted to represent that thing, let me do another decimal that's like that one. So let's say I did 0.0000516 and I wanted to represent this in scientific notation. I'd go to the first non-digit 0-- the first non-zero digit, not non-digit 0, which is there. And I'm like, OK, what's the largest power of 10 that will fit into that? So I'll go 1, 2, 3, 4, 5. So it's going to be equal to 5.16. So I take 5 there, then everything else is going to be behind the decimal point. Times 10. So this is going to be the largest power of 10 that fits into this first non-zero number. So it's 1, 2, 3, 4, 5. So 10 to the minus 5 power. Let me do another example. So the point I wanted to make is you just go to the first-- if you're starting at the left, the first non-zero number. That's what you get your power from. That's where i got my 10 to the minus 5 because I counted 1, 2, 3, 4, 5. You got to count that number just like we did over here. And then, everything else will be behind the decimal. Let me do another example. Let's say I had 0.-- and my wife always point out that I have to write a 0 in front of my decimal points because she's a doctor. And if people don't see the decimal point, someone might overdose on some medication. So let's write it her way, 0.0000000008192. Clearly, this is a super cumbersome number to write. And you might forget about a 0 or add too many 0's, which could be costly if you're doing some important scientific research. Or, maybe doing-- well, you wouldn't prescribe medicine at this small a dose. Or maybe you would, I don't want to get into that. But how would I write this in scientific notation? So I start off with the first non-zero number, if I'm starting from the left. So it's going to be 8.192. I just put a decimal and write 0.192 times-- times 10 to what? Well, I just count. Times 10 to the 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. I have to include that number, 10 to the minus 10. And I think you'll find it reasonably satisfactory that this number is easier to write than that number over there. Now, and this is another powerful thing about scientific notation. Let's say I have these two numbers and I want to multiply them. Let's say I want to multiply the number 0.005 times the number 0.0008. This is actually a fairly straightforward one to do, but sometimes it can get quite cumbersome. And especially if you're dealing with twenty or thirty 0's on either sides of the decimal point. Put a 0 here to make my wife happy. But when you do it in scientific notation, it will actually simplify it. This guy can be rewritten as 5 times 10 to the what? I have 1, 2, 3 spaces behind the decimal. 10 to the third. And then this is 8, so this is times 8 times 10 to the-- sorry, this is 5 times 10 to the minus 3. That's very important. 5 times 10 to the 3 would have been 5,000. Be very careful about that. Now, what is this guy equal to? This is 1, 2, 3, 4 places behind the decimal. So it's 8 times 10 to the minus 4. If we're multiplying these two things, this is the same thing as 5 times 10 to the minus 3 times 8 times 10 to the minus 4. There's nothing special about the scientific notation. It literally means what it's saying. So for multiplying, you could write it out like this. And multiplication, order doesn't matter. So I could rewrite this as 5 times 8 times 10 to the minus 3 times 10 to the minus 4. And then, what is 5 times 8? 5 times 8 we know is 40. So it's 40 times 10 to the minus 3 times 10 to the minus 4. And if you know your exponent rules, you know that when you multiply two numbers that have the same base, you can just add their exponents. So you just add the minus 3 and the minus 4. So it's equal to 40 times 10 to the minus 7. Let's do another example. Let's say we were to multiply Avogadro's number. So we know that's 6.022 times 10 to the 23rd. Now, let's say we multiply that times some really small number. So times, say, 7.23 times 10 to the minus 22. So this is some really small number. You're going to have a decimal, and then you're going to have twenty one 0's. Then you're going ti have a 7 and a 2 and a 3. So this is a really small number. But the multiplication, when you do it in scientific notation, is actually fairly straightforward. This is going to be equal to 6.0-- let me write it properly. 6.022 times 10 to the 23rd times 7.23 times 10 to the minus 22. We can change the order, so it's equal to 6.022 times 7.23. That's that part. So you can view it as these first parts of our scientific notation times 10 to the 23rd times 10 to the minus 22. And now, this is-- you're going to do some little decimal multiplication right here. It's going to be-- some number-- 40 something, I think. I can't do this one in my head. But this part is pretty easy to calculate. I'll just leave this the way it is. But this part right here, this will be times. 10 to the 23rd times 10 to the minus 22. You just add the exponents. You get times 10 to the first power. And then this number, whatever it's going to equal, I'll just leave it out here since I don't have a calculator. 0.23. Let's see, it will be 7.2. Let's see, 0.2 times-- it's like a fifth. It'll be like 41-something. So this is approximately 41 times 10 to the 1. Or, another way is approximately-- it's going to be 410-something. And to get it right, you just have to actually perform this multiplication. So hopefully you see that scientific notation is, one, really useful for super large and super small numbers. And not only is it more useful to kind of understand the numbers and to write the numbers, but it also simplifies operating on the numbers.