Scientific Notation
Read through this lesson on scientific notation and then complete the
handout.
Keep this power point presentation on your laptop and use it when you
do the assignments. The assignment will be turned in and graded. (Make
sure you use pencil.)
Scientific notation will be included on the next test. This is an SOL topic
so make sure you have a good understanding of it. You will see questions
about scientific notation on the SOL test in May.
Scientists often work with very large or very small numbers.
6,020,000,000,000,000,000,000,000,000,000,000
is a very
large number.
0.000000000000000000624
is a very small number.
(The more zeros there are after the decimal the smaller the number.)
Using zeros to keep track of place value can be tedious with such large
numbers. Scientific notation uses powers of ten to make it easier to work
with such numbers. Look at each number written in scientific notation.
6,020,000,000,000,000,000,000,000,000,000,000 =
0.000000000000000000624 = 6.24 x 10-19
6.02 x 1033
6,020,000,000,000,000,000,000,000,000,000,000 =
Standard or decimal notation.
6.02 x 1033
Scientific notation.
Notice how the large number has a positive exponent and the *small number has a
negative exponent.
*smaller than one
0.000000000000000000624 = 6.24 x 10-19
Standard or decimal notation.
Scientific notation.
A number written in scientific notation is written in the following form:
n x 10p
This must be
a number
greater than
1 but less
than 10.
To get this number you
move the decimal to
where there is only 1
digit in front of it.
This exponent
is positive if the
number is large
and negative if
the number is
smaller than
one.
To get the exponent you count
how many places you had to
move the decimal.
Write the number 625,000,000 in scientific notation.
625,000,000
Put the decimal here to get a
number greater than 1 but less
than 10.
6 . 25000000
Now count how many
spaces you had to move the
decimal.
6.25000000
The decimal
started out here.
And I moved it
to here.
How many places did I move it?
6 . 2 5 0 0 0 0 0 0 How many places did I move it?
8 places
Since the decimal was moved 8 places and the number was a large number the
exponent will be 8.
625,000,000 = 6.25 x 10
8
Write 245,000,000,000 in scientific notation.
2 .4 5 0 0 0 0 0 0 0 0 0
Move the decimal
from the end to behind
the 2.
Now, drop all the zeros.
2.45
So far we have 2.45 x 10p
To figure out what the exponent is count how many places the
decimal was moved.
The decimal was moved 11 places so the exponent is 11. The
exponent is positive because we are dealing with a large number.
2.45 x 1011
You try it.
Write 714,000 in scientific notation.
Did you get
7.14 x 105
Let’s do a small number.
Write 0.00000000008834 in scientific notation.
0.00000000008834
Move the decimal to
here to get 8.834
Now count how many places the decimal was moved.
It was moved 11 places.
Since we are dealing with a small number the exponent will be
negative. So the exponent is -11
0.00000000008834 =
8.834 x 10-11
Your turn.
Write 0.0000025 in scientific notation.
You should have gotten
2.5 x 10-6
Now lets work backwards. Instead of writing a number in scientific notation you can take a
number that is already in scientific notation and write in standard or decimal notation.
Write 2.67 x 106 in standard notation.
To convert from scientific notation to standard notation you move the decimal
from its current position. You will move the decimal the number of places
according to the exponent.
Since the exponent here is 6 you will move the decimal 6 places.
Since the exponent is positive you will move the decimal to the right. (Remember
numbers to the right are positive.)
2. 6 7 _ _ _ _
The decimal will
have to go here.
Now lets work backwards. Instead of writing a number in scientific notation you can take a
number that is already in scientific notation and write in standard or decimal notation.
Write 2.67 x 106 in standard notation.
To convert from scientific notation to standard notation you move the decimal
from its current position. You will move the decimal the number of places
according to the exponent.
Since the exponent here is 6 you will move the decimal 6 places.
Since the exponent is positive you will move the decimal to the right. (Remember
numbers to the right are positive.)
2 6 7 _0 _0 0_ 0_
.
The decimal will
have to go here.
Now, fill the gaps in with zeros.
Now lets work backwards. Instead of writing a number in scientific notation you can take a
number that is already in scientific notation and write in standard or decimal notation.
Write 2.67 x 106 in standard notation.
To convert from scientific notation to standard notation you move the decimal
from its current position. You will move the decimal the number of places
according to the exponent.
Since the exponent here is 6 you will move the decimal 6 places.
Since the exponent is positive you will move the decimal to the right. (Remember
numbers to the right are positive.)
2 6 7 _0 _0 0_ 0_
.
Now, fill the gaps in with zeros.
So,
2.67 x 106 = 2, 670,000
Write 7.882 x 10-7 in standard notation.
Since the exponent is -7 we have to move the decimal 7 places to the left.
(Remember, negative numbers are to the left.)
______7.882
Move the
decimal
from here...
Write 7.882 x 10-7 in standard notation.
Since the exponent is -7 we have to move the decimal 7 places to the left.
(Remember, negative numbers are to the left.)
______7.882
… to
here.
Write 7.882 x 10-7 in standard notation.
Since the exponent is -7 we have to move the decimal 7 places to the left.
(Remember, negative numbers are to the left.)
._ __ __ _ 7
… to
here.
882
Write 7.882 x 10-7 in standard notation.
Since the exponent is -7 we have to move the decimal 7 places to the left.
(Remember, negative numbers are to the left.)
. _0 _0 _0_0 _0 _0 7
882
Now fill in the gaps with zeros.
So, 7.882 x 10-7 = .0000007822
Your turn.
Write
3.567 x 10-4 in standard notation.
You should have gotten .0003567.
If you didn’t get that answer go back and review the process again.
Try the next problems before you attempt the homework.
Write 550,000,000 in scientific notation.
5.5 x 108
Write 0.00000113 in scientific notation.
1.13 x 10-6
Write 3.09 x 105 in standard notation.
309,000
Write 9.6 x 10-3 in standard notation.
0.0096
Multiplying Numbers in Scientific Notation
You can multiply numbers that are in scientific notation.
For example:
(2.35 x 105)(1.6 x 103)
Since we are allowed to rearrange multiplication we can re-write this as:
(2.35 x 1.6)(105 x 103)
When you are multiplying
powers of 10 you can just
add the exponents.
3.76
x
105 + 3
3.76 x 108
Multiply (3.11 x 104)(2.42 x 103)
First rearrange:
(3.11 x 2.42)(104 x 103)
Now multiply the decimals: 7.5262
Now put it all together:
Now multiply the
powers of 10. Just add
the exponents.
7.5262 x 107
104 + 3 = 10
7
Your turn!
Multiply:
(1.02 x107 )(6 x 103)
multiply
add the exponents
The answer is: 6.12 x 1010
Now how about some division!
Dividing Numbers in Scientific Notation
Divide:
2.25  10 5
1.68  103
First break it apart:
2.25 10 5
 3
1.68 10
Next divide the decimals: 1.3398285714 Now divide the powers
105-3 = 102
of 10. To divide powers
of 10 you subtract the
bottom exponent from
the top exponent. In this
case 5 - 3
Now put them back together:
1.3398285714 x 102
Your turn!
Divide:
8.25  106
1.25  103
Did you get 6.6 x 103 ? If not compare your work to mine.
8.25 106
 3
1.25 10
6.6 x 103
6-3=3
BEWARE!!!
Sometimes when you multiply or divide numbers in scientific notation you may
end up with an answer that appears to be in scientific notation but really isn’t.
Check your product or quotient to be sure it is in scientific notation. If it is not
then make the adjustment to the decimal needed to put the number in scientific
notation.
Ex: You multiply two numbers together and get 34.7 x 104 . This is not in
scientific notation because 34.7 is not a number greater than 1 and less than 10.
Here you have to move the decimal one place to the left so you have 3.47. Since
you are moving the decimal you must change the exponent. Since 34.7 is bigger
than one the exponent is positive so you add one to the exponent. The new
answer is 3.47 x 105
Homework tip: Keep in mind that when you are multiplying or dividing numbers in
scientific notation and you add or subtract the exponents you could end up with a
negative exponent. It is ok to have a negative exponent.
Example:
3.6  10 2.625  10  3.6  2.62510
3
9
3
9.45
x
 10 9 
103 + (-9)
9.45 x 10-6
Homework tip: Keep in mind that when you are multiplying or dividing numbers in
scientific notation and you add or subtract the exponents you could end up with a
negative exponent. It is ok to have a negative exponent.
Example:
8.35  10 4 8.35 10 4
 11
11 
2  10
2
10
4.675
x 104 -
4.675 x 10-7
11
Now you need to complete the handout. Make sure you do all the
problems. This is a graded assignment.
See the next slide for some useful calculator information.
GOOD LUCK!!
Scientific Notation in the Calculator
When you use the Casio to multiply very large or very small numbers it may give
you the answer in scientific notation. Here is an example of what you would see in
the calculator:
3.13 x 1011 would be shown as 3.13E+11 The E indicates scientific notation. The +11
means that the exponents on 10 is positive 11.
5.441 x 10-13 would be shown as 5.441E - 13 The -13 means that the exponent on the
10 is negative 13.
If you see 5.42E+4 in the calculator it means 5.42 x 104
If you see 3.6E - 16 in the calculator it means 3.6 x 10-16
FYI: You can multiply and divide scientific notation in the calculator. To raise 10 to a
power such as 10 to the 5th power you put 10^5.
Come see me. I’ll be happy to show you how!