or 
What are these numbers? How do you say
them? You might be surprised but these are realistic
numbers that come up frequently. They represent the
distance from Neptune to the Sun and the diameter of an
atom respectively. What is 7,500,000,000,000,000 /
4,500,000,000? Now imagine you left out one of the
zero’s in: 4,500,000,000. 450,000,000 is over four
billion less than 4,500,000,000. Quite a big difference.
Scientist have begun writing long numbers like these in a
notation that reduces the tendency of making a simple
mistake such as leaving out a zero which could have a
catastrophic effect. This notation is called scientific
notation. It is based on the properties of the
powers of 10.
The distance from Neptune to the Sun can more easily
be represented as:
and the diameter of an atom as:
.
This form of writing numbers is based on the properties
of exponents. This is how it
works.
109
= 10x10x10x10x10x10x10x10x10 = 1,000,000,000
4.50 x 109 =
4.50 x 1,000,000,000 = 4,500,000,000
It is important
to notice that the exponent, 9 in this case, is just the
number of places the decimal point is moved to the right
to obtain the fully written out number:
. When two
numbers are multiplied or divided, you first multiply or
divide the simple parts and then the powers of 10. Thus,
2.0 x 103 multiplied by 5.5 x 104
equals (2.0 x 5.5) x (103 x 104) =
11 x 107 where we used the rule for
multiplying exponents with the same base.
For number less than 1, e.g. .01, the exponent or
power of 10 is written with a negative sign: .01 = 1/100
= 1/102 = 1 x 10-2. Again the
decimal point has been moved the number of places
expressed in the exponent. A negative indicates moving
the decimal "X" units to the right while a
positive exponent indicates moving the decimal
"X" units to the left. The negative exponent
allows calculations to come out correctly.
2.0 x 10-2
x 5.5 x 104
(2.0 x 5.5) x (10-2
x 104)
11 x 102
The accepted method of writing scientific notation is to
make the mantisa number between 1 and 10 (1 can be used
but 10 should not be used). So while 45 x 107
is correct, 4.5 x 108 is preferred. Both
numbers are equal to 450,000,000.