Introduction to scientific notation | Pre-Algebra | Khan Academy

November 22, 2019 0 By Kody Olson


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.