Scientific Notation:
The advantage of Scientific Notation is that you can
add, subtract, multiply, and divide large numbers or extremely small
numbers quickly with or without a calculator.
However you must know some basic things about algebra
and scientific notation numbers.
Before we start let us establish what is actually scientific notation.
Most students have learned that numbers are not written in scientific
notation unless they have one and only one digit to the left of the decimal.
Thus for the number 540000:
The correct scientific notation would only be 5.4
X 105
However scientific notation is the expression of a
number as a product of a number and 10 raised to a power. Thus 54 X
104 is a correct scientific notation form
of the number 540000.
What is the difference between 5.4 X 105
and 54 X 104?
The number 5.4 X 105 is
the Standard Form of Scientific Notation
(SFSN).
Thus we can add, subtract, multiply, and divide all
types of scientific notation numbers and worry about whether or not
a number is in the Standard Form of Scientific
Notation when we come to the end of the problem.
Let us add the following scientific notation numbers.
8.3 x 106 + 9.7
X 104 + 31 X 108
| 1. |
In order to add numbers with exponents all
of the exponents must be the same. (like terms).
So you must decide how to make all the exponents the same.
Obviously you are going to have to move the decimal points in
two of the three numbers so that they will match the third (thus
all be alike).
|
| 2. |
Which number you choose to change is up to
you. With practice you eventually see the advantages of change
certain numbers and not changing others. I will not insist that
you do things a certain way.
I am going to choose to make all the exponents
equal to 6.
8.3 x 106
will not change
9.7 X 104
becomes .097 X 106
31 X 108 becomes 3100
X 106
|
| 3. |
Now I can add the numbers together because
the exponents are alike.
8.3
X 106
.097 X
106
3100. X
106
3108.397 X 106
|
| 4. |
Of course
if we want to express the answer in Standard Form we would move
the decimal point to the left and the answer would be 3.10897
X 109 |
| 5. |
The
same procedure applies to subtraction. |
Multiplication and division of Scientific Notation numbers
is easier than addition and subtraction. But you must observe the
law of exponents.
-
When you multiply numbers
with exponents you must add the exponents.
-
When you divide numbers
with exponents you subtract the exponent in the denominator from
the exponent in the numerator.
Let us multiply the following numbers.
(3.1 X 104)(2 X 10-6)(21
X 108)
| 1. |
Since the exponents
do not have to be the same for multiplication we can use our calculator
and quickly finish this problem. |
| |
(3.1)(2)(21) = 130.2
( 104)(10-6)(108)
= 106
The final answer is 130.2 X 106
Or in Standard Form 1.302
X 108
|
Let us divide the following:
7.56 X 10 24
35.5 X 1028
| 1. |
Since the exponents
do not have to be the same for division we can use our calculator
and quickly finish this problem as well. |
| |
7.56/35.5 = .213
10 24 =
10-4 Law
of exponents
1028
The final answer is .213 X 10-4
Or in Standard Form 2.13
X 10-5
|
You will find Examples of addition,
subtraction, multiplication, and division in the Example
Problems Page of this Web Site.
You may use the side bar link, as well: Go to
the Example Page. All Links to Scientific Notation Examples
are listed in a Table.
Page
8 begins the study of the application of the conversion of metric
units and scientific notation to density.
Forward to Page 8
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