THE KINETIC ATMOSPHERE
Exponential Notation
EXPONENTIAL NOTATION
A. Exponential Notation. In cases where a number is multiplied by itself one or
more times, a shorthand method of writing the number has been universally
adopted. It is called exponential notation. For instance, the operation of 10 x 10 x
10 is written 103. The expression 103 is called, “ten to the third power” or more
usually “ten to the third”.
In this case, the number 10 is called the base, and the number 3 is called the
exponent or power. The exponent tells how many times the base appears in the
multiplication. Thus, 10 x 10 x 10 x 10 = 104. Similarly, 10 x10 = 102, and 10 =
101. Finally, for reasons of mathematical consistency, 100 = 1.
As examples,
100 = 1
101 = 10
102 = 100
103 = 1,000
104 = 10,000
105 = 100,000
106 = 1,000,000
20 = 1
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
3.1416 x 100 = 3.1416
3.1416 x 101 = 31.416
3.1416 x 102 = 314.16
3.1416 x 103 = 3,141.6
3.1416 x 104 = 31,416
3.1416 x 105 = 314,160
3.1416 x 106 = 3,141,600
and so on and so forth.
B. Negative Exponents. In the section above, all of the exponents are either
positive numbers or zero, with the value of the expression becoming smaller
as the exponent decreases. The value of the expression finally becomes
equal to 1 when the exponent becomes equal to 0. When the exponent
becomes a negative number, the value continues to diminish in a logical
fashion. The general rule is
Copyright 2008 by Patrick J. Tyson
Last revised in July of 2014
www.climates.com
THE KINETIC ATMOSPHERE
Exponential Notation
an 
1
, as long as a does not equal zero.
an
Therefore,
10-1 = 0.1
10-2 = 0.01
10-3 = 0.001
10-4 = 0.0001
10-5 = 0.00001
10-6 = 0.000001
2-1 = 1/2 = 0.5
2-2 = 1/4 = 0.25
2-3 = 1/8 = 0.125
2-4 = 1/16 = 0.0625
2-5 = 1/32 = 0.03125
2-6 = 1/64 = 0.015625
3.1416 x 10-1 = 0.31416
3.1416 x 10-2 = 0.031416
3.1416 x 10-3 = 0.0031416
3.1416 x 10-4 = 0.00031416
3.1416 x 10-5 = 0.000031416
3.1416 x 10-6 = 0.0000031416
And so on and so forth.
When the negative exponent is used with words or symbols other than
numbers, it means per. Thus, sec-1 means per second, yr-1 means per year, °K-1
means per degree Kelvin, cm-2 means per square centimeter, sec-2 means per
second per second (the standard acceleration of gravity is 980 centimeters per
second per second, or 9.8 x 102 cm sec-2), cm-3 means per cubic centimeter,
etcetera.
C. Fractional Exponents. Although not usually used with powers of ten,
fractional exponents are common in scientific literature. It is simply an easy way
to express the various roots of a number without drawing or printing the root sign
  . Thus,
1
n 2  n  the square root of n
1
n 3  3 n  the cube root of n
1
n 4  4 n  the fourth root of n
1
n m  m n  the m th root of n
and so on.
Copyright 2008 by Patrick J. Tyson
Last revised in July of 2014
www.climates.com
THE KINETIC ATMOSPHERE
Exponential Notation
D. Scientific Notation. Many scientific computations involve very large numbers
(the number of air molecules in a cubic centimeter of air at NTP is approximately
26,872,000,000,000,000,000), or very small numbers (each air molecule can be
approximated by a sphere with a radius of roughly 0.000000015 centimeter). To
eliminate the tedious writing out of zeros, the convention of using powers of ten
has been universally adopted in scientific literature. Thus, we would say that there
are 2.6872 x 1019 molecules per cm3, each having an average radius of 1.5 x 10-8
cm.
In these two examples, the numbers 26872 and 15 respectively are called the
significant figures. In scientific notation, the decimal point is placed after the first
significant figure unless there is some compelling reason to do otherwise. The
powers of ten (19 and -8 respectively) are called the orders of magnitude of the
expressions. Thus, in the first example, the significant figures are 26872 and the
order of magnitude is 19. In the second example, the significant figures are 15 and
the order of magnitude is -8. The two numbers may be said to differ from one
another by 27 orders of magnitude (19 minus -8 equals 27).
E. Operating Rules. In mathematical operations involving exponents, the
following rules should be observed:
m
n
m n
1. a  a  a
m
m
a

b
  ab 
2.
a 
m
3.
n
m
 a mn
am
 a m  n , but only where m  n and a  0
n
4. a
am
1

, but only where m  n and a  0
n
n m
a
5. a
Copyright 2008 by Patrick J. Tyson
Last revised in July of 2014
www.climates.com
THE KINETIC ATMOSPHERE
Exponential Notation
m
am
a
 b   b m , but only where b  0
6.  
In all honesty, if you aren’t comfortable with exponential notation, you are
unlikely to get very far in understanding atmospheric physics.
Copyright 2008 by Patrick J. Tyson
Last revised in July of 2014
www.climates.com