Scientific notation is the way that scientists easily
handle very large numbers or very small numbers. For example,
instead of writing 0.0000000056, we write 5.6 x 10^{9}.
So, how does this work?
We can think of 5.6 x 10^{9}
as the product of two numbers: 5.6 (the digit term) and 10^{9}
(the exponential term).
Here are some
examples of scientific notation.
10000 = 1 x 10^{4} 
24327 = 2.4327 x 10^{4} 
1000 = 1 x 10^{3} 
7354 = 7.354 x 10^{3} 
100 = 1 x 10^{2} 
482 = 4.82 x 10^{2} 
10 = 1 x 10^{1} 
89 =
8.9 x 10^{1} (not usually
done) 
1 = 10^{0} 

1/10 = 0.1 = 1 x 10^{1} 
0.32 = 3.2 x 10^{1}
(not usually done) 
1/100 = 0.01 = 1 x 10^{2} 
0.053 = 5.3 x 10^{2} 
1/1000 = 0.001 = 1 x 10^{3} 
0.0078 = 7.8 x 10^{3} 
1/10000 = 0.0001 = 1 x 10^{4} 
0.00044 = 4.4 x 10^{4} 
As you can see, the exponent of 10 is the number of places the
decimal point must be shifted to give the number in long form. A
positive
exponent shows that the decimal point is shifted that number of
places to the right. A negative
exponent shows that the decimal point is shifted that number of
places to the left.
In
scientific notation, the digit term indicates the number of
significant figures in the number. The exponential term only places
the decimal point. As an example,
46600000 = 4.66 x 10^{7}
This number only has 3
significant figures. The zeros are not significant; they are only
holding a place. As another example,
0.00053 = 5.3 x 10^{4}
This number has 2 significant
figures. The zeros are only place holders.
How to
do calculations:
On
your scientific calculator:
Make sure that the number
in scientific notation is put into your calculator correctly.
Read
the directions for your particular calculator. For inexpensive
scientific calculators:

Punch the number (the
digit number) into your calculator.

Push the EE or EXP button. Do
NOT
use the x (times) button!!

Enter the exponent
number. Use the +/ button to change its sign.

Voila! Treat this number
normally in all subsequent calculations.
To
check yourself, multiply 6.0 x 10^{5}
times 4.0 x 10^{3} on your calculator.
Your answer should be 2.4 x 10^{9}.
On
your cheap nonscientific calculator:
You
will need to be familiar with exponents since your calculator cannot
take care of them for you. For an introduction to rules concerning
exponents, see the section on Manipulation of Exponents.
Addition and Subtraction:

All numbers are converted to the same power of 10, and the digit
terms are added or subtracted.

Example: (4.215 x 10^{2})
+ (3.2 x 10^{4})
= (4.215 x 10^{2})
+ (0.032 x 10^{2})
= 4.247 x 10^{2}

Example: (8.97 x 10^{4})  (2.62 x 10^{3})
= (8.97 x 10^{4})  (0.262 x 10^{4})
= 8.71 x 10^{4}
Multiplication:

The digit terms are multiplied in the normal way and the
exponents are added. The end result is changed so that there is
only one nonzero digit to the left of the decimal.

Example: (3.4 x 10^{6})(4.2 x 10^{3})
= (3.4)(4.2) x 10^{(6+3)} = 14.28 x
10^{9} = 1.4 x 10^{10}
(to 2 significant figures)

Example: (6.73 x 10^{5})(2.91
x 10^{2}) = (6.73)(2.91) x 10^{(5+2)}
= 19.58 x 10^{3}
= 1.96 x 10^{2}
(to 3 significant figures)
Division:

The digit terms are divided in the normal way and the exponents
are subtracted. The quotient is changed (if necessary) so that
there is only one nonzero digit to the left of the decimal.

Example: (6.4 x 10^{6})/(8.9 x 10^{2})
= (6.4)/(8.9) x 10^{(62)} = 0.719 x
10^{4} = 7.2 x 10^{3}
(to 2 significant figures)

Example: (3.2 x 10^{3})/(5.7
x 10^{2})
= (3.2)/(5.7) x 10^{3(2)}
= 0.561 x 10^{5} = 5.6 x 10^{4}
(to 2 significant figures)
Powers of Exponentials:

The digit term is raised to the indicated power and the exponent
is multiplied by the number that indicates the power.

Example: (2.4 x 10^{4})^{3}
= (2.4)^{3} x 10^{(4x3)}
= 13.824 x 10^{12} = 1.4 x 10^{13}
(to 2 significant figures)

Example: (6.53 x 10^{3})^{2}
= (6.53)^{2} x 10^{(3)x2}
= 42.64 x 10^{6}
= 4.26 x 10^{5}
(to 3 significant figures)
