# Introduction to scientific notation | Pre-Algebra | Khan Academy

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.