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### Course: Class 7 > Unit 5

Lesson 5: Large numbers in standard form# Scientific notation examples

More scientific notation examples. Created by Sal Khan.

## Want to join the conversation?

- can you upvote me i need badges.(24 votes)
- How did scienctific notation even come to be?(5 votes)
- The concept's actually pretty old, Archimedes invented the concept of multiplying super small and large numbers by powers of ten. It's just an easier way to write ridiculously long numbers.(4 votes)

- At5:56is that supposed to be negative 0.0000064?(3 votes)
- A negative exponent on the 10s indicates division by 10s.

10^(-1) = 1/10

10^(-2) = 1/10^2 = 1/100

etc.

0.0000064 = 64/10000000

This is division by a lot of 10s.

So, in scientific notation it becomes 6.4 * 10^(-6)

Hope this helps.(6 votes)

- Sal said that 7,012,000,000,000 had 12 places in it. There are actually 13!(3 votes)
- When he said 12 places,he meant 12 places after the 7. Because when you write that in scientific notation it would be 7.012 x 10^12(5 votes)

- would 200 as a scientific notation be:

2.00 x 10^-2 or 2.00 x 10^2(3 votes)- Sorry if this is late but 200 in scientific notation is 2*10^2

The reason it is not the first one is because having a negative exponent means we divide the number instead of multiplying. The simplified version of 2*10^-2 would be 0.02

Hope that helped! Have a great day.(4 votes)

- i use a simulation programm to calculate magnitude of displacement etc and i see notations like max displacement 4.0858e-02 (mm)

How does this translate?(2 votes)- e relates to 10^-2, so it would be .040858(3 votes)

- so the decimal always go's between the first and second number(3 votes)
- My teacher showed me something different,this is so confusing!!(3 votes)
- What would be the scientific notation for 437.5?(1 vote)
- Normal scientific notation involved having only one digit before the decimal point. We can write 437.5 as:

437.5 * 10^0, since any number to the 0th power = 1

From here, we keep increasing the power of ten (and dividing 437.5 by 10 accordingly) until we get our desired format):

43.75 * 10^1

4.375 * 10^2

We have our answer at 4.375 * 10^2. Since 10 squared is a hundred, this answer makes sense. Hope this helps!(4 votes)

- At12:00I thought you had to multiply or divide equaly on both sides of the equation. Here you multiply on one side and divide on the other.(2 votes)
- He is making two changes to the
*same*side of the equation. The only way that is legal is when the changes cancel each other.(1 vote)

## Video transcript

It always helps me to see a lot
of examples of something so I figured it wouldn't hurt to
do more scientific notation examples. So I'm just going to write a
bunch of numbers and then write them in scientific notation. And hopefully this'll cover
almost every case you'll ever see and then at the end of this
video, we'll actually do some computation with them to just
make sure that we can do computation with
scientific notation. Let me just write down
a bunch of numbers. 0.00852. That's my first number. My second number
is 7012000000000. I'm just arbitrarily
stopping the zeroes. The next number is 0.0000000
I'll just draw a couple more. If I keep saying 0, you
might find that annoying. 500 The next number --
right here, there's a decimal right there. The next number I'm going
to do is the number 723. The next number I'll do -- I'm
having a lot of 7's here. Let's do 0.6. And then let's just do one more
just for, just to make sure we've covered all of our bases. Let's say we do 823 and then
let's throw some -- an arbitrary number of 0's there. So this first one, right here,
what we do if we want to write in scientific notation, we want
to figure out the largest exponent of 10 that
fits into it. So we go to its first
non-zero term, which is that right there. We count how many positions to
the right of the decimal point we have including that term. So we have one, two, three. So it's going to
be equal to this. So it's going to be equal
to 8 -- that's that guy right there -- 0.52. So everything after that
first term is going to be behind the decimal. So 0.52 times 10 to the
number of terms we have. One, two, three. 10 to the minus 3. Another way to think of it:
this is a little bit more. This is like 8 1/2
thousands, right? Each of these is thousands. We have 8 1/2 of them. Let's do this one. Let's see how many 0's we have. We have 3, 6, 9, 12. So we want to do -- again,
we start with our largest term that we have. Our largest non-zero term. In this case, it's going
to be the term all the way to the left. That's our 7. So it's going to be 7.012. It's going to be equal to
7.012 times 10 to the what? Well it's going to be times 10
to the 1 with this many 0's. So how many things? We had a 1 here. Then we had 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12 0's. I want to be very clear. You're not just
counting the 0's. You're counting everything
after this first term right there. So it would be equivalent
to a 1 followed by 12 0's. So it's times 10
to the twelfth. Just like that. Not too difficult. Let's do this one right here. So we go behind our
decimal point. We find the first
non-zero number. That's our 5. It's going to be equal to 5. There's nothing to the right of
it, so it's 5.00 if we wanted to add some precision to it. But it's 5 times and then how
many numbers to the right, or behind to the right of the
decimal will do we have? We have 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, and we have to include this one, 14. 5 times 10 to the
minus 14th power. Now this number, it might be a
little overkill to write this in scientific notation, but it
never hurts to get the practice. So what's the largest 10
that goes into this? Well, 100 will go into this. And you could figure out 100 or
10 squared by saying, "OK, this is our largest term." And then
we have two 0's behind it because we can say 100
will go into 723. So this is going to be equal to
7.23 times, we could say times 100, but we want to stay in
scientific notation, so I'll write times 10 squared. Now we have this
character right here. What's our first non-zero term? It's that one right there, so
it's going to be 6 times and then how many terms do we have
to the right of the decimal? We have only one. So times 10 to the minus 1. That makes a lot of sense
because that's essentially equal to 6 divided by 10
because 10 to the minus 1 is 1/10 which is 0.6. One more. Let me throw some commas
here just to make this a little easier to look at. So let's take our largest
value right there. We have our 8. This is going to be 8.23 -- we
don't have to add the other stuff because everything else
is a 0 -- times 10 to the -- we just count how many
terms are after the 8. So we have 1, 2, 3, 4,
5, 6, 7, 8, 9, 10. 8.23 times 10 to the 10. I think you get the idea now. It's pretty straightforward. And more than just being able
to calculate this, which is a good skill by itself, I want
you to understand why this is the case. Hopefully that last
video explained it. And if it doesn't, just
multiply this out. Literally multiply 8.23
times 10 to the 10 and you will get this number. Maybe you could try it
with something smaller than 10 to the 10. Maybe 10 to the fifth. And well, you'll get a
different number but you'll end up with five
digits after the 8. But anyway, let me do a couple
more computation examples. Let's say we had the numbers
-- let me just make something really small -- 0.0000064. Let me make a large number. Let's say I have that number
and I want to multiply it. I want to multiply it by --
let's say I have a really large number -- 3 2 -- I'm just going
to throw a bunch of 0's here. I don't know when
I'm going to stop. Let's say I stop there. So this one, you
can multiply out. But it's a little difficult. But let's put it into
scientific notation. One, it'll be easier to
represent these numbers and then hopefully you'll see that
the multiplication actually gets simplified as well. So this top guy right here,
how can we write him in scientific notation? It would be 6.4 times
10 to the what? 1, 2, 3, 4, 5, 6. I have to include the 6. So times 10 to the minus 6. And what can this
one be written as? This one is going to be 3.2. And then you count how many
digits are after the 3. 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11. So 3.2 times 10 to the 11th. So if we multiply these two
things, this is equivalent to 6 -- let me do it in a different
color -- 6.4 times 10 to the minus 6 times 3.2
times 10 to the 11th. Which we saw in the last video
is equivalent to 6.4 times 3.2. I'm just changing the order
of our multiplication. Times 10 to the minus 6
times 10 to the 11th power. And now what will
this be equal to? Well, to do this, I don't
want to use a calculator. So let's just calculate it. So 6.4 times 3.2. Let's ignore the
decimals for a second. We'll worry about
that at the end. So 2 times 4 is 8,
2 times 6 is 12. Nowhere to carry the
1, so it's just 128. Put a 0 down there. 3 times 4 is 12, carry the 1. 3 times 6 is 18. You've got a 1
there, so it's 192. Right? Yeah. 192. You had them up and you
get 8, 4, 1 plus 9 is 10. Carry the 1. You get 2. Now, we just have to
count the numbers behind the decimal point. We have one number there, we
have another number there. We have two numbers behind
the decimal point, so you count 1, 2. So 6.4 times 3.2 is equal to
20.48 times 10 to the -- we have the same base here, so we
can just add the exponents. So what's minus 6 plus 11? So that's 10 to the
fifth power, right? Right. Minus 6 and 11. 10 to the fifth power. And so the next question,
you might say, "I'm done. I've done the computation."
And you have. And this is a valid answer. But the next question is is
this in scientific notation? And if you wanted to be a real
stickler about it, it's not in scientific notation because we
have something here that could maybe be simplified
a little bit. We could write this --
let me do it this way. Let me divide this by 10. So any number we can
multiply and divide by 10. So we could rewrite
it this way. We could write 1/10 on this
side and then we can multiply times 10 on that side, right? That shouldn't
change the number. You divide by 10 and
multiply it by 10. That's just like multiplying
by 1 or dividing by 1. So if you divide this side
by 10, you get 2.048. You multiply that side by
10 and you get times 10 to the -- times 10 is just
times 10 to the first. You can just add the exponents. Times 10 to the sixth. And now, if you're a stickler
about it, this is good scientific notation
right there. Now, I've done a lot
of multiplication. Let's do some division. Let's divide this
guy by that guy. So if we have 3.2 times 10 to
the eleventh power divided by 6.4 times 10 to the minus
six, what is this equal to? Well, this is equal
to 3.2 over 6.4. We can just separate them out
because it's associative. So, it's this times 10
to the 11th over 10 to the minus six, right? If you multiply these
two things, you'll get that right there. So 3.2 over 6.4. This is just equal
to 0.5, right? 32 is half of 64 or 3.2
is half of 6.4, so this is 0.5 right there. And what is this? This is 10 to the 11th
over 10 to the minus 6. So when you have something
in the denominator, you could write it this way. This is equivalent to 10 to the
11th over 10 to the minus 6. It's equal to 10 to the
11th times 10 to the minus 6 to the minus 1. Or this is equal to 10 to the
11th times 10 to the sixth. And what did I do just there? This is 1 over 10
to the minus 6. So 1 over something is just
that something to the negative 1 power. And then I multiplied
the exponents. You can think of it that way
and so this would be equal to 10 to the 17th power. Or another way to think about
it is if you have 1 -- you have the same bases, 10 in this
case, and you're dividing them, you just take the 1 the
numerator and you subtract the exponent in the denominator. So it's 11 minus minus
6, which is 11 plus 6, which is equal to 17. So this division problem
ended up being equal to 0.5 times 10 to the 17th. Which is the correct answer,
but if you wanted to be a stickler and put it into
scientific notation, we want something maybe greater
than 1 right here. So the way we can do
that, let's multiply it by 10 on this side. And divide by 10 on this
side or multiply by 1/10. Remember, we're not changing
the number if you multiply by 10 and divide by 10. We're just doing it to
different parts of the product. So this side is going to become
5 -- I'll do it in pink -- 10 times 0.5 is 5, times 10 to
the 17th divided by 10. That's the same thing as
10 to the 17th times 10 to the minus 1, right? That's 10 to the minus 1. So it's equal to 10
to the 16th power. Which is the answer when
you divide these two guys right there. So hopefully these examples
have filled in all of the gaps or the uncertain
scenarios dealing with scientific notation. If I haven't covered something,
feel free to write a comment on this video or pop me an e-mail.