For example to multiply by we multiply 4 by 5 and by

For example, to multiply by we multiply 4 by 5 and by
as follows:
To divide these numbers, we proceed likewise:
4 10
10
5 10
12
4
5
10
10
10
12
0.8 10
10 12
0.8 10
2
8 10
3
20 10
10 12
20 10
22
2 10
23
(4 10
10
) (5 10
12
) (4 5) (10
10
10
12
)
10
12
, 10
10
5 10
12
, 4 10
10
( a
1 > n
)
m
a
m > n
( a
1 > n
)
n
a
n > n
a
a
1 > n
,
a
1 > 2
1 a
1 a : a
1 > 2
a
1 > n
(2 3)
3
2
3
3
3
( a
#
b )
n
a
n
#
b
n
(3
2
)
3
3
2 3
3
6
3
2
3
3
3
1
1
3
3
2
3
3
3
5
( a
n
)
m
a
nm
a
n
a
m
a
n m
a
n
#
a
m
a
m n
a
0
1
A-2 APPENDIX 2 Mathematics Review

APPENDIX 2 Mathematics Review A-3
When performing additions or subtractions of numbers in scientific notation, we must
be careful to begin by expressing the numbers with the same power of 10. For exam-
ple, the sum of and is
A 2.4 Algebra
An equation is a mathematical statement that tells us that one quantity or a combination
of quantities is equal to another quantity or combination. We often have to solve for
one of the quantities in the equation in terms of the other quantities. For instance, we
may have to solve the equation
for x in terms of a and b . Here a and b are numerical constants or mathematical expres-
sions which are regarded as known, and x is regarded as unknown.
The rules of algebra instruct us how to manipulate equations and accomplish their
solution. The three most important rules are:
1. Any equation remains valid if equal terms are added or subtracted from its left
side and its right side.
This rule is useful for solving the equation We simply subtract a from both
sides of this equation and find
that is,
To see how this works in a concrete numerical example, consider the equation
Subtracting 7 from both sides, we obtain
or
Note that given an equation of the form we may want to solve for a
in terms of x and b , if x is already known from some other information but a is a math-
ematical quantity that is not yet known. If so, we must subtract x from both sides of
the equation, and we obtain
Most equations in physics contain several mathematical quantities which sometimes
play the role of known quantities, sometimes the role of unknown quantities, depend-
ing on circumstances. Correspondingly, we will sometimes want to solve the equation
for one quantity (such as x ), sometimes for another (such as a ).
2. Any equation remains valid if the left and the right sides are multiplied or divided
by the same factor.
This rule is useful for solving
ax b
a b x
x a b ,
x 2
x 5 7
x 7 5
x b a
x a a b a
x a b .
x a b
1.5 10
9
3 10
8
1.5 10
9
0.3 10
9
1.8 10
9
3 10
8
1.5 10
9

A-4 APPENDIX 2 Mathematics Review
We simply divide both sides by a , which yields
or
Often it will be necessary to combine both of the above rules. For instance, to solve
the equation
we begin by subtracting 10 from both sides, obtaining
or
and then we divide both sides by 2, with the result
or
3. Any equation remains valid if both sides are raised to the same power.
This rule permits us to solve the equation
Raising both sides to the power we find
or
As a final example, let us consider the equation
(as established in Chapter 2, this equation describes the vertical position of a particle
that starts at a height and falls for a time t ; but the meaning of the equation need not
concern us here). Suppose that we want to solve for t in terms of the other quantities in
the equation. This will require the use of all our rules of algebra. First, subtract x from
both sides and then add to both sides. This leads to
and then to
Next, multiply both sides by 2 and divide both sides by g ; this yields
t
2
2
g
( x
0
x )
1
2
gt
2
x
0
x
0
1
2
gt
2
x
0
x
1
2
gt
2
x
0
x
1
2
gt
2
x
0
x b
1 > 3
( x
3
)
1 > 3
b
1 > 3
1
3
,
x
3
b
x 3
x
6
2
2 x 6
2 x 16 10
2 x 10 16
x
b
a
ax
a
b
a