Exponential and Scientific Notation
A. Exponential Notation
Frequently in science we
must write very large or very small numbers. A convenient method of expressing
these types of numbers is by using exponential notation. This notation
expresses the numbers as powers of ten or tenths. The number 1,000 is the result of the
following multiplication: 10 x 10 x 10. Written in exponential terms, this is:
This would normally be
referred to as "ten to the third power." In science, the base is
normally 10.
How can you easily convert a
decimal number to an exponential number? Consider the following relationships.
Decimal 



Exponential 
10 
= 
10 
= 
10^{1} 
100 
= 
10 x 10 
= 
10^{2} 
1,000 
= 
10 x 10 x 10 
= 
10^{3} 
10,000 
= 
10 x 10 x 10 x
10 
= 
10^{4} 
Now, what about numbers that are smaller than one? The number 0.01 is the result of
multiplying 0.1 x 0.1. Written in exponential terms, this is:
10^{2}
Once again, consider the
relationship between decimal and exponential versions of numbers that are
smaller than one.
Decimal 



Exponential 
0.1 
= 
0.1 
= 
10^{1} 
0.01 
= 
0.1 x 0.1 
= 
10^{2} 
0.001 
= 
0.1 x 0.1 x 0.1 
= 
10^{3} 
0.0001 
= 
0.1 x 0.1 x 0.1 x 0.1 
= 
10^{4} 
Notice that again, the
number of zeros in the decimal number, including the one to the left of the
decimal point, determines the exponent (except it is always a negative value).
The number 1 written in exponential terms is 100.
B. Scientific Notation
Frequently numbers consist of digits other than ones and zeros. The decimal
form of numbers like these are frequently expressed in a form of exponential
notation, called scientific notation, in which a decimal number between 1 and
10 is followed by a power of ten. For example, the distance from the sun to the
earth is 93,000,000 miles. Written in scientific notation, this becomes:
Notice that in this case you
cannot simply count zeros. In order to determine the exponent, count the number
of places needed to move the decimal point (real or imaginary) until you obtain
a number between 1 and 10.
Numbers Larger than 10
Write 670,620 in scientific notation.
Step 1  Move the decimal point of 670,620 to the left to get a number between 1 and 10 times a power of ten.
670,620 = 6.7062 x 10^{exponent}
Step 2  Find the exponent by counting how many places you had to move the decimal point. This is the value of the exponent.
exponent = 5
Step 3  Write the number in scientific notation.
670,620 = 6.7062 x 10^{5}
Numbers smaller than 1
Write 0.00234 in scientific notation.
Step 1  Move the decimal point of 0.00234 to the right to get a number between 1 and 10 times a power of ten.
0.00234 = 2.34 x 10^{exponent}
Step 2  Find the exponent by counting how many places you had to move the decimal point. This is the value of the exponent.
exponent = 3
Step 3  Write the number in scientific notation.
0.00234 = 2.34 x 10^{3}
In some cases you may want
to convert a number from scientific notation to decimal notation. If the
exponent is positive, move the decimal point rightward the number of places
indicated by the exponent. If the exponent is negative, move the decimal point
leftward the number of places indicated by the exponent. In both cases it may
be necessary to add zeros.
Convert the following
scientific notation to decimal notation.
7.33 x 10^{4}
= 0.000733
This is the result of moving the decimal point four
places.
C. Multiplication
When exponential numbers are
multiplied, the exponents are added.
10^{a} x 10^{b}
= 10^{a + b
}
Thus, if 100 is multiplied by 10,000, in exponential terms this would be:
10^{2}
x 10^{4} = 10^{2} + 4 = 10^{6}
Be careful when working with
negative exponents. Consider the following two examples.
10^{5} x 10^{7} = 10^{5 + (7)} = 10^{5}
 7 = 10^{2}
10^{6} x 10^{8} = 10^{6 + (8)} = 10^{6 8} = 10^{14}
If there is a coefficient in
front of the exponentials, multiply them, and then multiply the exponential
component. Finally, put the answer in scientific notation if not already so,
and then round off to the appropriate number of significant figures. Let’s try
the following calculations.
(3.8930 x 10^{4})
(2.600 x 10^{8}) 
= 
10.1218 x 10^{4} 


= 
1.01218 x 10^{3} 
Since the coefficient has
become smaller by a factor of 10, the exponential portion becomes bigger by a
factor of 10. 

= 
1.012 x 10^{3} 





(4.658 x 10^{2})
(4.2 x 10^{5}) 
= 
19.5636 x 10^{7} 


= 
1.95636 x 10^{6} 


= 
2.0 x 10^{6} 

D. Division
When exponential numbers are
divided, the exponents are subtracted.
10^{a} 


 
= 
10^{a
 b} 
10^{b} 


Let’s try dividing 100,000 by
100 using exponential notation.




100,000 
/ 100 = 
10^{5} / 10^{2} 
= 10^{5  2} = 10^{3} 




As with the multiplication
of exponentials, be careful when working with negative exponents. Consider the
following two examples.




10^{8 }/ 10^{3 }= 
10^{8  (3)}
= 10^{8 + 3} = 10^{11} 

10^{5 }/10^{11
}= 
10^{5  (11)}
= 10^{5 +11} = 10^{6} 
If there is a coefficient in front of the
exponentials, divide them, and then divide the exponential component. Finally,
put the answer in scientific notation if not already so, and then round off to
the appropriate number of significant figures. Let’s try the following
calculations.



(6.750 x 10^{5}) 
/ (4.20 x 10^{3}) = 
1.607143 x 10^{2} = 1.61 x 10^{2} 




(5.99 x 10^{6}) ^{ }/
(30.01 x 10^{3})^{ }=

0.1996001 x 10^{3} = 1.996001 x 10^{4 }=^{ }2.00^{ }x 10^{4} 



The key to adding or subtracting numbers
in Scientific Notation is to make sure the exponents are the same. For example,
(2.0 x 10^{2}) + (3.0 x 10^{3})
can be rewritten as:
(0.2 x 10^{3}) + (3.0 x 10^{3})
Now you
just add 0.2 + 3 and keep the 10^{3} intact. Your answer is 3.2 x 10^{3},
or 3,200. We can check this by converting the numbers first to the more
familiar form.
So:
2 x 10^{2} + 3.0 x 10^{3 } = 200 + 3,000 =
3,200=
3.2 x 10^{3}
Let's try
a subtraction example.
(2.0 x 10^{7})  (6.3 x 10^{5})
The
problem needs to be rewritten so that the exponents are the same. So we can
write
(200 x 10^{5})  (6.3 x 10^{5})
=
193.7 x 10^{5}
which in Scientific Notation would be written 1.937 x 10^{7}.
Let's
check by working it another way:
2 x 10^{7}  6.3 x 10^{5}
= 20,000,000  630,000 =
19,370,000 =
1.937 x 10^{7}