Calculating with Scientific Notation
Scientific notation is simply a method for expressing, and working with, very large or very
small numbers. It is a short hand method for writing numbers, and an easy method for
calculations. Numbers in scientific notation are made up of three parts: the coefficient, the
base and the exponent. Observe the example below:
5.67 x 105
This is the scientific notation for the standard number, 567,000. Now look at the number
again, with the three parts labeled.
5.67 x 105
coefficient
base
exponent
For a number to be in correct scientific notation, the following conditions must be true:
1. The coefficient must be greater than or equal to 1 and less than 10.
2. The base must be 10.
3. The exponent shows the number of decimal places that the decimal needs to be moved to
change the number to standard notation. A negative exponent means that the decimal is
moved to the left when changing to standard notation.
IMPORTANT REMINDER: 100 = 1
Any number to the zero power equals 1 (except 0)
X0 = 1 (provided that X does not equal 0)
Changing numbers from scientific notation to standard
notation
Change 6.03 x 107 to standard notation.
Remember, 107 = 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10,000,000
Therefore, 6.03 x 107 = 6.03 x 10,000,000 = 60,300,000
Answer = 60,300,000
Instead of finding the value of the base, we can simply move the decimal seven places to the
right because the exponent is 7.
Therefore, 6.03 x 107 = 60,300,000
Now let us try one with a negative exponent.
Change 5.3 x 10-4 to standard notation.
The exponent tells us to move the decimal four places to the left.
Therefore, 5.3 x 10-4 = .00053
Changing numbers from standard notation to scientific
notation
Change 56,760,000,000 to scientific notation
Remember, the decimal is at the end of the final zero.
The decimal must be moved behind the five to ensure that the coefficient is less than 10, but
greater than or equal to one.
The coefficient will then read 5.676
The decimal will move 10 places to the left, making the exponent equal to 10.
Answer equals 5.676 x 1010
Now we try a number that is very small.
Change .000000902 to scientific notation
The decimal must be moved behind the 9 to ensure a proper coefficient.
The coefficient will be 9.02
The decimal moves seven spaces to the right, making the exponent -7
Answer equals 9.02 x 10-7
IMPORTANT NOTE: When converting a number from standard notation, if the
decimal place is moved to make the number smaller, the exponent will become
larger. If the decimal place is moved to make the number bigger, the exponent will
become smaller.
Example: 31 becomes 3.1 x 101 but .31 becomes 3.1 x 10-1
Notice how both numbers become 3.1 as the coefficient in scientific notation, but the
exponent is greater in the number we made smaller, and smaller in the number we
made greater
Calculating with Scientific Notation
Scientific notation is simply a method for __________, and working with, very __________
or very __________ numbers. It is a short hand method for writing numbers, and an easy
method for calculations. Numbers in scientific notation are made up of three parts: the
__________, the __________ and the __________. Observe the example below:
5.67 x 105
This is the scientific notation for the __________number, 567,000. Now look at the number
again, with the three parts labeled.
5.67 x 105
__________
__________
__________
For a number to be in correct scientific notation, the following conditions must be true:
1. The __________ must be greater than or equal to 1 and less than 10.
2. The __________ must be 10.
3. The __________ shows the number of decimal places that the decimal needs to be moved to
change the number to standard notation. A negative exponent means that the decimal is
moved to the left when changing to standard notation.
IMPORTANT REMINDER: 100 = __________
Any number to the zero power equals 1 (except 0)
X0 = 1 (provided that X does not equal 0)
Changing numbers from scientific notation to standard
notation
Change 6.03 x 107 to standard notation.
Remember, 107 = 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10,000,000
Therefore, 6.03 x 107 = 6.03 x 10,000,000 = 60,300,000
Answer = 60,300,000
Instead of finding the value of the base, we can simply move the decimal __________ places
to the __________ because the __________ is 7.
Therefore, 6.03 x 107 = 60,300,000
Now let us try one with a negative exponent.
Change 5.3 x 10-4 to standard notation.
The exponent tells us to move the decimal four places to the __________.
Therefore, 5.3 x 10-4 = .00053
Changing numbers from standard notation to scientific
notation
Change 56,760,000,000 to scientific notation
Remember, the __________ is at the end of the final zero.
The decimal must be moved behind the five to ensure that the __________ is less than 10, but
greater than or equal to one.
The coefficient will then read 5.676
The decimal will move __________ places to the left, making the exponent equal to 10.
Answer equals 5.676 x 1010
Now we try a number that is very small.
Change .000000902 to scientific notation
The decimal must be moved behind the __________ to ensure a proper coefficient.
The coefficient will be __________
The decimal moves __________ spaces to the right, making the exponent -7
Answer equals 9.02 x 10-7
IMPORTANT NOTE: When converting a number from standard notation, if the
decimal place is moved to make the number smaller, the exponent will become
larger. If the decimal place is moved to make the number bigger, the exponent will
become smaller.
Example: 31 becomes 3.1 x 101 but .31 becomes 3.1 x 10-1
Notice how both numbers become 3.1 as the coefficient in scientific notation, but the
exponent is greater in the number we made smaller, and smaller in the number we
made greater
Name______________________________
Part I. Change the following numbers to proper scientific notation
1) 295
2) 841
3) .923
4) .0443
5) 5,310
6) .00710
7) 24,100
8) 6
9) 176,300
10) .000911
Name______________________________
Part I. Change the following numbers to proper scientific notation
1) 5,300,000
2) 73,000,000
3) 647.85
4) 246.0
5) 169,845,869
6) 0.64278
7) 5
8) .00572
9) .0000006243
10) 0.78500
Honors
Name______________________________
Part I. Change the following numbers to standard notation
1) 2.8 x 101
2) 4.67 x 102
3) 9.8 x 103
4) 6.382 x 10-1
5) 3.5 x 10-2
6) 7.8 x 10-3
7) 5.51 x 104
8) 1.9 x 105
9) 8.2 x 10-4
10) 9.8 x 100
Name______________________________
Part I. Change the following numbers to standard notation
1) 4.39 x 105
2) 1.139 x 106
3) 2.24 x 10-5
4) 1 x 106
5) 1 x 100
6) 2.681535 x 10-1
7) 4.56 x 10-6
8) 1.23456789 x 108
9) 1.03050709 x 10-4
10) 0 x 1010
Honors
Name______________________________
Part I. Change the following numbers to proper scientific notation
1)
65
2)
.5
3)
2,450
Part II. Change the following numbers to standard notation
4)
8.85 x 102
5)
1.847 x 103
6)
3.4 x 10-1
Name______________________________
Part I. Change the following numbers to proper scientific notation
1)
0.0803
2)
678.3
3)
3,450,000
Part II. Change the following numbers to standard notation
4)
6.5 x 10-2
5)
9.7 x 103
6)
1.4012 x 10-4
Honors