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The 1 is usually understood, that is, not stated, and the opera-
tion is called canceling. Canceling leaves you with the fraction EXAMPLE A.5
1,000 g/1 kg, which is a conversion ratio that can be used to con- A service station sells gasoline by the liter, and you fill your tank with
vert from kilograms to grams. You simply multiply the conver- 72 liters. How many gallons is this? (Answer: 19 gal)
sion ratio by the numerical value and unit you wish to convert:
1,000 g
_

= 2.3 kg ×
1 kg
2.3 × 1,000 kg × g
_ _ A.5 SCIENTIFIC NOTATION

=

1 kg
Most of the properties of things that you might measure in
= 2,300 g your everyday world can be expressed with a small range of
The kilogram units cancel. Showing the whole operation with units numerical values together with some standard unit of measure.
only, you can see how you end up with the correct unit of grams: The range of numerical values for most everyday things can be
dealt with by using units (1s), tens (10s), hundreds (100s), or
kg⋅g
g
_ _ perhaps thousands (1,000s). But the actual universe contains

kg × = = g

kg kg some objects of incredibly large size that require some very
big numbers to describe. The Sun, for example, has a mass of
Since you did obtain the correct unit, you know that you used
about 1,970,000,000,000,000,000,000,000,000,000 kg. On the
the correct conversion ratio. If you had blundered and used an
other hand, very small numbers are needed to measure the
inverted conversion ratio, you would obtain
size and parts of an atom. The radius of a hydrogen atom, for
1 kg
kg
_ _ 2 example, is about 0.00000000005 m. Such extremely large and

2.3 kg × = .0023
1,000 g g small numbers are cumbersome and awkward since there are
so many zeros to keep track of, even if you are successful in
2
which yields the meaningless, incorrect units of kg /g.
carefully counting all the zeros. A method does exist to deal
Carrying out the mathematical operations on the numbers
with extremely large or small numbers in a more condensed
and the units will always tell you whether or not you used the
form. The method is called scientifi c notation, but it is also
correct conversion ratio.
sometimes called powers of ten or exponential notation, since it
is based on exponents of 10. Whatever it is called, the method
EXAMPLE A.4 is a compact way of dealing with numbers that not only helps
A distance is reported as 100.0 km, and you want to know how far this you keep track of zeros but provides a simplified way to make
is in miles. calculations as well.
In algebra, you save a lot of time (as well as paper) by writing
5
(a × a × a × a × a) as a . The small number written to the right
SOLUTION and above a letter or number is a superscript called an exponent.
First, you need to obtain a conversion factor from a textbook The exponent means that the letter or number is to be multiplied
5
or reference book, which usually lists the conversion factors by by itself that many times; for example, a means a multiplied by
properties in a table. Such a table will show two conversion factors for itself five times, or a × a × a × a × a. As you can see, it is much
kilometers and miles: (1) 1 km = 0.621 mi and (2) 1 mi = 1.609 km. easier to write the exponential form of this operation than it is to
You select the factor that is in the same form as your problem; for write it out in long form. Scientific notation uses an exponent to
example, your problem is 100.0 km = ? mi. The conversion factor in
this form is 1 km = 0.621 mi. indicate the power of the base 10. The exponent tells how many
Second, you convert this conversion factor into a conversion ratio times the base, 10, is multiplied by itself. For example,
by dividing the factor by what you wish to convert from: 10,000 = 10 4
conversion factor: 1 km = 0.621 1,000 = 10 3
divide factor by what you 1 km 0.621 mi 100 = 10 2
=
want to convert from: 1 km 1 km 1
10 = 10
_
0.621 mi
resulting conversion rate: 0

km 1 = 10
Note that if you had used the 1 mi = 1.609 km factor, the resulting 0.1 = 10 –1
units would be meaningless. The conversion ratio is now multiplied by –2
0.01 = 10
the numerical value and unit you wish to convert:
0.001 = 10 –3
_
0.621 mi
100.0 km × 0.0001 = 10 –4

km
_ This table could be extended indefinitely, but this somewhat
km . mi

(100.0)(0.621)
km shorter version will give you an idea of how the method works.
4
62.1 mi The symbol 10 is read as “ten to the fourth power” and means
A-5 APPENDIX A Mathematical Review 627