Download Rules that apply to both addition and multiplication

ADDITION vs. MULTIPLICATION
With very few exceptions (see the last table on the reverse of this sheet for these), the rules governing the
application of addition (and subtraction) in arithmetic, algebra, and calculus are different from the rules for the
application of multiplication (and division). These rules, and invalid “rules” with which they are frequently
confused, are compared/contrasted below. Rules marked with a star (*) relate to advanced algebra. Rules
marked with a double star (**) relate to calculus, and are correct only to the extent that the stated limits,
derivatives, or integrals exist and are finite-valued. Rules marked with a dagger (†) relate to linear algebra.
Unless stated otherwise, each correct rule applies to all real values of the variables, except for those
combinations of values giving rise to zero denominators or indeterminate expressions such as 00 , which may
not be explicitly noted in order to conserve space. Correct rules marked with
also work for complex values
of the variables.
denotes the set of strictly positive integers (i.e., the set of natural or counting numbers).
When possible, the incorrect rules are marked with correct alternatives (in parentheses).
Rules that apply to multiplication/division, but not to addition/subtraction:
Correct rules for
Common errors resulting from misapplication
Properties
multiplication/division
of the “rule” to addition/subtraction
M1 Multiplicative
a 1  a
a 1  a (0 is the identity for addition)
identity
a
M2 Multiplicative
a   a   1 (0 is the identity for addition)
1
inverse
a
a c a c
a c a d ad
a c ac
 
   
 
;
M3 Combining
b d bd
b d b c bc
b d bd
fractions
(common denominator is undesirable
(common denominator is required for
for multiplication/division of fractions)
addition/subtraction of unlike fractions)
a b a b b
ab b
a b b
ba b
M4 Reducing


 ;
 ;

fractions
ac a c c
ac c
ac c
ca c
xa
M5 Combining
x a  xb  x a b ;
 x a b
x a  x b  x a  b ; x a  x b  x a b
b
exponents
x
M6 Combining
bases
M7 Combining
radicals
M8 Absolute value
and modulus
(stripes cannot be
removed from these
expressions safely
without knowledge
of the relative sizes
and/or signs of a
and b)
M9 Logarithms*
a  b
n
n
 a n  bn ;
an
a

 
bn
b
a na

a b  a  b ;
b nb
(restrictions: n ; n  2 ; a, b  0 )
n
n
n
n
a
a

a b  a  b ;
b
b
Other rules that are safe and reliable:
a  a ; a  b  b  a ;
a b  ba
logb  m  n   logb m  logb n ;
m
log b    log b m  log b n
n
(restrictions: b, m, n  0 ; b  1)
VC DEPARTMENT OF MATHEMATICS
( a  b) n  a n  b n
(Use FOIL method or binomial theorem)
ab  n a  n b
(Use binomial theorem on (a  b)1/ n )
n
a b  a  b
Other unreliable “rules” to avoid:
a   a ; a  a ; a  b   b  a ;
a b  a b ;
a b  a b;
a  b  a  b ;
a  b  a  b ;
a b  a b
logb  m  n   logb m  logb n ;
(There is no alternative rule for logb  m  n  )
REVISED SUMMER 2013
Rules that apply to addition/subtraction, but not to multiplication/division:
Common errors resulting from
Correct rules for
misapplication of the “rule” to
Properties
addition/subtraction
multiplication/division
A1 Additive
a0  a
a  0  a (1 is the identity for multiplication)
identity
a
A2 Additive
a   a   0
 0 (1 is the identity for multiplication)
inverse
a
a c a d bc a d  bc
a c a  d b c a b c  d
 


 


A3 Combining
b d bd bd
bd
b d bd bd
bd
fractions
(common denominator is required for
(common denominator is undesirable for
addition/subtraction of unlike fractions)
multiplication/division of fractions)
ax  bx   a  b  x
A4 Distributive
ax  bx   a  b  x
property
(correct simplification is ax  bx  abx 2 )
 n
  n 
a

b

a
 k k    k     bk  ;

k 1
 k 1   k 1 
 n

ak 


n
 ak   k 1 
  n



k 1  bk 
  bk 
 k 1 
 f ( x)  g ( x)   f ( x)   g ( x) ;
n
n
a
A5 Convergent
sums*
k 1
k

 

 bk     ak     bk 
 k 1   k 1 
n
n
(restriction:
 ak and
k 1
n
n
b
k 1
k
must converge)
 f ( x)   f ( x) 
A6 Derivatives**
 g ( x)   g ( x)


 
(do not confuse with L’Hospital’s rule; use
product or quotient rules instead)
 f ( x)  g ( x) dx    f ( x) dx     g ( x) dx 
A7 Integrals**
  f ( x)  g ( x) dx   f ( x) dx    g ( x) dx  
(use u-substitution or integration by parts)
A8 Linear trans- T  a  f    b  g     a  T  f    b  T  g  T  a  f  b  g   a  T f  b  T g
         

formations†
(restriction: a, b are constants)
 f ( x)  g ( x)   f ( x)   g ( x)
Rules that apply to both addition and multiplication:
Correct rules for addition/subtraction
Properties
a b  ba;
B1 Commutative &
anticommutative
a  b  (b  a )
properties
(a  b)  c  a  (b  c)
B2 Associative
property
(this does not apply to subtraction)
B3 Selected
properties of
limits**
Correct rules for multiplication/division
a b  b a
(these do not apply to division)
(a  b)  c  a  (b  c)
(this does not apply to division)
lim  f ( x)  g ( x)  lim f ( x)   lim g ( x)  ;
x a
 xa
  xa


lim  f ( x)  g ( x)  lim f ( x)   lim g ( x) 
f ( x) 
 f ( x)  lim
x a
xa
 x a
  xa


lim 

xa g ( x) 

 lim g ( x) 
 xa

VC DEPARTMENT OF MATHEMATICS
REVISED SUMMER 2013
VC DEPARTMENT OF MATHEMATICS
REVISED SUMMER 2013