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Axioms for Multiplication
Properties for Multiplication
The CLOSURE property for Multiplication
The ASSOCIATIVE property for Multiplication
The COMMUTATIVE property for Multiplication
IDENTITY Property of Multiplication
For all real numbers a and b :
ab is a unique real number
a , b , ! c ;(ab  c)
For all real numbers a, b, and c:
ab c  a bc
b g bg
a, b, c ; b
abg
c  abg
bc
For all real numbers a and b:
ab  ba
a , b ; ab  ba
There is a unique real number 1 such that for
every real number a,
a 1  a and 1 a  a .
a ; (a  1  a)  1 a  a
b
Property of RECIPROCALS
g
For every non-zero real number a , there is a
1
unique real number
such that
a
1
1
a   1 and  a  1
a
a
1
1
1
a   a  0, ! ; (a   1)   a  0
a
a
a
F
G
H
Definition of Division
For all real numbers a and b , the quotient a  b is defined by
1
a  b  a  for b  0
b
to multiply by b, multiply by the reciprocal of b.
1
a , b , b  0; a  b  a 
b
Definition of Division
Rule for Multiplication by -1
Theorem: Multiplication by -1
Theorem: Negative times a positive
Theorem: Negative times a negative is a
positive
For all real numbers a:
1 a  a
a ;  1 a  a
1 1   1  1
bg
Rules for Multiplication and Addition
The DISTRIBUTIVE property
(of multiplication with respect to addition)
1
For all real numbers a, b, and c:
a b  c  ab  ac
b g
a, b, c ; ab
b  cg
 ab  ac
mlt_prop.doc
6/28/2017
IJ
K
Axioms for Multiplication
Sample Quiz on Axioms for Multiplication
1. Find a solution set for 4 x  8
4x  8
Given
bg
F
I
G
HJ
K
1
1
 4x  8
4
4
1
1
 4 x  8
4
4
1
1 x   8
4
1
x  8
4
x2
2. Find a solution set for x  3  9
x3 9
Given
1
x  9
3
1
x  3  93
3
F
I
G
HJ
K
F1 I
x  G 3J 9  3
H3 K
x 1  9  3
x  93
x  27
2
mlt_prop.doc
6/28/2017
Axioms for Multiplication
3. Find a solution set for 6  x  1 (Additive Solution)
6 x 1
Given
6  x  1
b6   xg x  1  x
6b
 x  xg
 1 x
6  0  1 x
6  1 x
1 x  6
x 1  6
bx  1g 1  6  1
x b
1  1g
 6  1
x  0  6  1
x  6  1
x 5
4. Find a solution set for 6  x  1 (Multiplicative Solution)
6 x 1
Given
6  x  1
b g
6  c
6b
1 xg
h 6  1
b6  6g 1 x  6  1
6  1 x  1
0  1 x  6  1
1 x  6  1
1 x  5
b g
b1 1g x  1 5
1 1 x  1 5
1 x  5
x 5
3
mlt_prop.doc
6/28/2017
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