The speed of light is 299,792,458 m/s … which is a very long number to write.
It takes up a lot of paper, and it takes me a long time to type it into my calculator.
For most calculations, I don’t really need this degree of precision.
It’s rare for me to do a calculation with more than 3 or 4 significant figures.
If I wanted to round this value to 4 significant figures, it would be 299,800,000 m/s.
That’s still quite long to write out.
And now it’s really easy to make a mistake about the number of zeros to write.
Scientific Notation is a much quicker way to write very large numbers.
In Scientific Notation, the speed of light written to 4 significant figures is
2.998 x 10^8 m/s.
If I need to write it with 3 significant figures, it’s 3.00 x 10^8 m/s.
This notation is much more compact and easier to read.
We can also use Scientific Notation to abbreviate very small numbers.
For example, the mass of a proton is 0.000000000000000000000000001672626219 kilograms.
Imagine if you had to write this out every time you did a problem involving atomic mass.
You’d waste so much paper.
It’s so much faster and easier to write this number in Scientific Notation.
With 3 sig figs, it’s 1.67 x 10^-27 kilograms.
Notice that for very large numbers, the exponent on the 10 is positive, and for very small
numbers, the exponent on the 10 is negative.
How do we convert these very long numbers into the much shorter Scientific Notation?
Let’s start with the speed of light example.
The first step is to find where the decimal point is.
When a number is written as an integer like this - there is an understood decimal
at the end.
So let’s go ahead and put that in.
The next step is to move the decimal point until there is only one non-zero digit to
the left of the decimal.
Each time you move a decimal point, you are changing the number by a factor of 10, so
we need to keep track of the number of moves.
Let’s count.
1..2..3..4..5..6..7..8
We moved the decimal to the left 8 times.
That’s equivalent to dividing the number by 10^8. To make sure we don’t actually
change the value of the number, we have to multiply by 10^8.
Here’s another way to think of it.
Let’s use a smaller number like 45000 so we can see the pattern more easily.
We’ll put in the decimal at the end of 45000.
45000 is the same as 45000 x 10^0.
10^0 = 1, so you can multiply any number by 10^0 and you won’t change the number.
I’m putting that in so you can see it’s not a big deal to have that x 10 at the end.
X 10^0 is a way to say the decimal hasn’t changed position at all.
If we move the decimal one place to the left, we now have 4500.0 That’s a different number.
4500 is much less than 45000.
10 times less.
By moving the decimal point one step to the left, we divided 45000 by 10.
So to cancel the effect of moving the decimal, we need to multiply by 10
So it’s like 4500.0 x 10^0 x 10^1 = 4500.0 x 10^1
Let’s move the decimal again.
450.00 x 10^1 x 10^1 = 450.00 x 10^2
Each time we move the decimal, we need to multiply by another 10.
Count up all the 10s and you’ll see that to write 45000 in Scientific Notation, we
moved the decimal point 4 times, so we needed to multiply by 10, 4 times.
We get 4.5 x 10^4
We do something very similar when we use Scientific Notation for very small numbers.
Let’s use the human red blood cell as an example.
It’s 8 micrometers in diameter.
That’s 0.000008 m.
Once again, we really don’t want to have to write down all those zeros.
Let’s re-write it in Scientific Notation.
Step 1 - find the decimal point.
This one is here already.
Step 2 - move the decimal point until there is one non-zero digit to the left of the decimal.
Don’t forget to count how many steps you move the decimal point.
1..2..3..4..5..6.
Notice we’re moving the decimal point to the right.
Each time we move the decimal point to the right, that’s the same as
multiplying the number by 10.
That means we’re going to have to DIVIDE the final number by 10 for every step that
decimal point takes to the right.
Step 3 - write the number and multiply by
10 raised to the negative of the number of steps.
So in this case, we get 8 x 10^-6.
Are you already familiar with negative powers of 10?
Let’s review this idea.
When we move the decimal to the right, we need to undo that by dividing by 10.
We can write that as 10^-1, which is the same as 1/10.
If we move the decimal point twice, that means we need to divide by 10 twice.
We can think of that as 1/10 x 1/10, which is the same as 1/10^2,
which is the same as 10^-2.
Let’s continue in this way until we get to 10-6.
How about if we want to go in the other direction?
What if we started with a number in scientific notation, and we want to expand it?
For example, the approximate age of the Earth is 4.543 x 10^9 years.
To “get rid” of that 10^9, we need to move the decimal point to the right.
Remember, moving the decimal point to the right is the same as multiplying by 10.
So each time you move the decimal to the right, take away one of those 10s on the right.
45.43 x 10^8
454.3 x 10^7
4543 x 10^6
45430 x 10^5
454300 x 10^4
4543000 x 10^3
45430000 x 10^2
454300000 x 10^1
4543000000.
A quick word about vocabulary: We say that in Scientific Notation, a number is written as
a product between a coefficient and a power of 10.
Here you can see on the left is the coefficient.
It’s always a number between 1 and 10.
The coefficient is multiplied by a power of 10 - and it can either be a positive or a
negative power of 10.
10 is the base, and this is the exponent.
Recognizing each of these parts is important when it comes to adding and subtracting numbers
in Scientific Notation.
Now let’s turn to calculations using Scientific Notation.
First, we’ll look at the rules for adding and subtracting, and then we’ll look at
examples for multiplying and dividing.
Add the mass of a proton and the mass of a neutron.
1.673 x 10^-27 kg + 1.675 x 10^-27 kg.
These numbers have the same base and exponent, so all we need to do is add together the coefficients,
so we get 3.348 x 10^-27 kg.
The coefficient is still between 1 and 10, so there is no need to change the exponent
on the right.
Let’s look at an example where the exponent would change.
Find the mass of 10 protons: We can add 1.673 x 10-27 10 times - that’s
the same as multiplying the coefficient by 10.
We get 16.73 x 10-27 We’ll move the decimal point one step to the left, and then we can
rewrite this as 1.673 x 10-26
What do we do when the exponents are different?
We’ll need to manipulate the numbers so they have the same exponents.
This involves moving the decimal point.
For example, if we do this calculation: 8.43 x 10^3 + 9.765 x 10^6 we need to shift
the decimal point on one of these numbers so the exponents match.
Let’s move the decimal point on 9.765 x 10^6 so we have 103.
To do that, we move the decimal to the right 3 places.
9.765 x 10^6 is the same as 97.65 x 10^5, which is the same as 976.5 x 10^4, which is
the same as 9765 x 10^3.
Now that these two numbers have the same base and exponents, we can just add their coefficients.
8.43 + 9765 = 9773.43 and don’t forget the x 103.
Move the decimal point to get back to scientific notation: 9.77343 x 10^6 Notice that there
were only two decimal places in 8.43, so we’ll round to two decimal places in our final answer:
9.77 x 10^6 If you need a refresher on significant figures, check out our video on that subject.
The same rules apply if you are subtracting.
As long as the base and exponent of the two numbers are the same, you can just do the
operation on the coefficients.
For example, 3.1415 x 1064 - 2.71828 x 10^4 = 0.42322 x
10^4 Rewrite in scientific notation by moving the decimal point one place to the right.
4.2322 x 103 - and we already have the correct number of sig figs.
To multiply two numbers in Scientific Notation, you multiply their coefficients and add their
exponents.
Keep the same number of significant digits as the number with the fewest significant
digits.
For example: (5.56 x 10^3 ) x (2.124 x 10^5)
= 11.80944 x 10^8 Round to 3 sig figs: 11.8 x 10^8, Again if you need a refresher on sig
figs, we have a video all about those rules.
To divide two numbers in Scientific Notation, you divide the coefficients and subtract the
exponent.
Round your final answer to the number of significant figures in the coefficient with the smallest
number of significant figures.
(4.5 x 10^7) / (7.82 x 10^9) = 0.57544757 x 10^-2
Round to 2 sig figs: 5.8 x 10^-3
If you’re working with Scientific Notation on a calculator, make sure you know how to
enter in these values correctly.
It depends on the calculator.
There may be a 10x button, or there may be a button that says E or EE.
These all mean the same thing.
If you want to enter in 5 x 10^3, you would type in 5, then the multiply button, then
the 10x button, then 3.
Or you would type in 5, then the E button, then 3.
Either way, you’ll get 5000.