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
In order 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 must show 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.
Changing numbers from scientific notation to standard notation.
Ex.1 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
so,
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.
So, 6.03 x 107 = 60 300 000
Now let us try one with a negative exponent.
Ex.2 Change 5.3 x 10-4 to standard notation.
The exponent tells us to move the decimal four places to the left.
so, 5.3 x 10-4 = 0.00053
Changing numbers from standard notation to scientific notation
Ex.1 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.
Ex.2 Change 0.000000902 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
Calculating with Scientific Notation
Not only does scientific notation give us a way of writing very large and very small
numbers, it allows us to easily do calculations as well. Calculators are very helpful tools,
but unless you can do these calculations without them, you can never check to see if your
answers make sense. Any calculation should be checked using your logic, so don't just
assume an answer is correct. This page will explain the rules for calculating with
scientific notation.
Rule for Multiplication - When you multiply numbers with scientific notation, multiply
the coefficients together and add the exponents. The base will remain 10.
Ex 1. Multiply (3.45 x 107) x (6.25 x 105)
first rewrite the problem as: (3.45 x 6.25) x (107 x 105)
Then multiply the coefficients and add the exponents:
21.5625 x 1012
Then change to correct scientific notation and round to correct significant digits:
2.16 x 1013
NOTE - we add one to the exponent because we moved the decimal one place to the
left.
Remember that correct scientific notation has a coefficient that is less than 10, but greater
than or equal to one.
Ex. 2. Multiply (2.33 x 10-6) x (8.19 x 103)
rewrite the problem as: (2.33 x 8.19) x (10-6 x 103)
Then multiply the coefficients and add the exponents: 19.0827 x 10-3
Then change to correct scientific notation and round to correct significant digits
1.91 x 10-2
Remember that -3 + 1 = -2
Rule for Division - When dividing with scientific notation, divide the coefficients and
subtract the exponents. The base will remain 10.
Ex. 1 Divide 3.5 x 108 by 6.6 x 104
rewrite the problem as:
3.5 x 108
--------6.6 x 104
Divide the coefficients and subtract the exponents to get:
0.530303 x 104
Change to correct scientific notation and round to correct significant digits to
get: 5.3 x 103
Note - We subtract one from the exponent because we moved the decimal one place
to the right.
Rule for Addition and Subtraction - when adding or subtracting in scientific notation,
you must express the numbers as the same power of 10. This will often involve changing
the decimal place of the coefficient.
Ex. 1 Add 3.76 x 104 and 5.5 x 102
move the decimal to change 5.5 x 102 to 0.055 x 104
add the coefficients and leave the base and exponent the same: 3.76 + 0.055 = 3.815
x 104
following the rules for rounding, our final answer is 3.815 x 104
Rounding is a little bit different because each digit shown in the original problem must be
considered significant, regardless of where it ends up in the answer.
Ex. 2 Subtract (4.8 x 105) - (9.7 x 104)
move the decimal to change 9.7 x 104 to 0.97 x 105
subtract the coefficients and leave the base and exponent the same: 4.8 - 0.97 = 3.83
x 105
round to correct number of significant digits: 3.83 x 105