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# Introduction to scientific notation

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 you'll have to do with any kind of science, you'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. There 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 number. If I were to ask you, how
many atoms are there in the human body? Or how many 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, 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. Just as an example, let me show
you one of the most common numbers you're going to see,
especially chemistry, it's called Avogadro's number. Avogadro's number. And if I were to 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 6 0 2 2 and then another twenty 0's. One, two, three, four, five,
six, seven, eight, nine, ten, eleven, twelve, thirteen,
fourteen, fifteen, sixteen, seventeen, eighteen,
nineteen, twenty. And even if 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. It's still a huge number, and I
don't even, you know, if I had to write this on a piece of
paper, or if I were to publish some paper on, you know, 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 0, or if I maybe
wrote too many 0's. 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 6 0 2 2 followed
by the twenty 0's there. And to answer that question,
and in case you're curious, Avogadro's number, if you had
twelve grams of carbon, especially twelve grams of
carbon-12, this is how many atoms you would have in that. And so you know, twelve grams
is like 1/50th 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 here isn't to
teach you some chemistry. The point 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, even
though it might be a little unnatural for you, it really is
an easier way to write things like, things like that. So let me, 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,
what is 10 to the 0 power? We know that's equal to 1. What is 10 to the 1 power? Well 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, right? 10 to the 1 has 1 0. 10 to the second power, I was
going to say the "twoth" power. 10 to the second power has
two 2 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, if I were to
do 10 to the 100th power, 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, 100 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 100 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 nineties if
someone said, hey that's, you know, 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. If you actually, it's more than
the number of atoms, or the estimated number of atoms
in the known universe. In the known universe. I mean, you know, raises
the question of what else is it that's out there? But you know, I was reading up
on this not too long ago, and if I remember correctly, the
known universe as 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 guestimating it. But this is a huge number. But maybe even more interesting
to you, this number was the motivation behind the naming of
a very popular search engine Google. googol with the "ol". 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, you know, was the founder's favorite number. But it's a cool
thing, you know. But anyway, I'm digressing. This is a googol, just
1 with a hundred 0's. But I could equivalently just
written that as 10 to 100. Which is, you know, 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 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 would have found it boring,
so I didn't even write it. So clearly this is
easier to write. And so, you might very well
think, right, this is just good for powers of ten, right? But how can we write something
that isn't a direct power of 10? How can we use the power
of the simplicity? How can we use the power of
the simplicity somehow? And to do that, you just need
to make the realization. This number, this number,
we would write it as. So this has how many
digits in It has 1, 2, 3, and then twenty 0's. So that's twenty-three
digits after the 6, right? Twenty-three digits
after the 6. What happens if I have, 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? With this magenta. 10 to the 23rd power. That's equal to what? That equals 1 with
twenty-three 0's. So one, two, three, four, five,
six, seven, eight, nine, ten, eleven, twelve, thirteen,
fourteen, fifteen, sixteen, seventeen, eighteen, nineteen,
twenty, twenty-one, twenty-two, twenty-three. You get the idea. That's 10 to the 23rd. Now can we somehow
write this guy as some Because if we multiplied this
guy by 6, so what is 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 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 6 times
this 1, and you'd get a 6, and all the 6 times the
0'x will all be 0's. So we'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, 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, this number right
here, is identical to this number is if these
2's were 0's. But if we have, we want
to put those 2's there, what could we do? Well we could put
some decimals here. We could say that it's
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'll take a long time. Maybe we should do it with
a smaller number first. But if you multiply 6.022 times
10 to the twenty-third, 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. You know, this was just
a number written out. This has a multiplication,
and then a 10 to a power. You 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 more. This was, 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, let's say I have the number 7,345. And I want to write it
in scientific notation. So the, I guess the best
way to think about it is, 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 essentially the largest
power of 10 that I can fit into this. This is 7 thousand. So this is 7 thousands, and
then it's 0.3 thousands, then it's 0.04 thousands. 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 7
thousands plus 0.3 thousands. What's 0.3 times 1,000? 0.3 times 1,000 is 300. What's 0.04 times 1,000? It's 40. What's 0.005 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. One, two, three. So one, two, three, 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 the scientific notation, but how would you do it? 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. Now, what if we wanted to
represent something like this? I started off the video saying,
science, you deal with very large and very small numbers. So let's say you had the
number, do it in this color. Say you have the number 1, and
you have 1, 2, 3, 4, and then let's say five 0's, and you
have, followed by a 7. Once again, this is not an
easy number to deal with. What's the largest power of 10
that fits into this number? That this number's
divisible by? Let's think about it. All the powers of 10 we
did before were going to positive, or going to, yeah,
positive powers of 10. We could also do
negative powers of 10. You know that 10 to the 0 is 1. Let's start there. 10 to the minus 1 is equal to 1
over 10, which is equal to 0.1. Switch colors, I'll do pink. 10 to the minus 2 is equal to 1
over 10 squared, which is equal to 1 over 100, which is equal
to 0.01 And I think you get the idea. 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 over 1,000,
which is equal to 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'x. 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 decimal,
places behind the decimal point do I have? I have one, two, three,
four, five, six. So 10 to the minus 6 is going
to be equal to point, and we're going to have six places behind
the decimal point, and the last point is going to be, or 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, right? If we multiply this times 7,
times 7, we get 7 times 1, and then we have one, two, three,
four, five, six numbers behind the decimal point. One, two, three,
four, five, six. So this number times 7 is
clearly equal to the number that we started off with. So we can rewrite this number. We can rewrite, 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's no better
than that number, but this number's the same thing
as 10 to the minus 6th. 7 times 10 to the minus 6th. So now you can imagine. You know, numbers like, imagine
the number, what if we had a 7, let me think of it this way. What if we had a 7, 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 can be, that can go into this
thing right here. So if we want to represent that
thing, let me write, let me do another decimal that's
like that one. So let's see, I did 0.0000516. 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 one, two, three,
four, five, so it's going to be equal to 5.16, so I take 5
there, and 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 one, two, three,
four, five, 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, you go to the first, 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 one, two, three, four, five. You've got to count
that number, just like we did over here. And then everything else
would be behind the decimal. Let me do another example. If you had, and my wife always
points 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 points, someone might overdose on some medications. So let's write it her way. 0.0000000008192. Clearly this is a super
cumbersome number, right? And you know 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, or you wouldn't prescribe medicine in 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 of the what? eight, nine, ten, I have
to include that number. 10 to the minus 10. 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, times the
number 0.0008. This is actually a fairly
straightforward one to do. But, sometimes they can get
quite cumbersome, 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. Well 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 one, two, three
spaces behind the decimal, 10 to the third. And 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've been 5,000. Be very careful about that. And what is this guy equal to? This is one, two, three, four
places behind the decimal. So it's 8 times 10
to the minus 4. So if we're multiplying
numbers, so this is, if we're multiplying these two things,
it's 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 can 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. You can just add the
minus 3 and the minus 4. So it becomes equal to 10
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. Let's say we were
multiplying that times some really small number. So times, let's say it's 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 to 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 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 do it as these
first parts of our scientific notation. Times, times, 10 to the
twenty-third times 10 to the minus twenty-two. And now, this is, you know,
you're going to have to do some little decimal
multiplication right here. It's going to be some number,
forty something, I think. I can't do this one in my head. 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 times 10. And then this number, whatever
it's going to equal, I'll just leave it out here since I
don't have a calculator. 7.23, let's see
what it will be. 7.2, let's see, 0.2 times,
it's like 1/5, it'll be like 41 something. So this is approximately,
approximately, 41 times 10 to the 1. Or another way, it's
approximately, it's a little, 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.