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and the associative law of multiplication is
a × ( b × c )=( a × b ) × c
(3.2)
e.g. 1
3
Note that substraction is not associative:
×
(2
×
3) = (1
×
2)
×
a
( b
c )
=( a
b )
c
(3.3)
e.g. 1
(2
3)
=(1
2)
3
3.2.2 Commutative Law
The commutative law in algebra states that when two elements are linked
through some binary operation, the result is independent of the order of the
elements. The commutative law of addition is
a + b = b + a
(3.4)
e.g. 1 + 2 = 2 + 1
and the commutative law of multiplication is
a
×
b = b
×
a
(3.5)
e.g. 2
2
Note that subtraction is not commutative:
×
3=3
×
a − b = b − a
(3.6)
e.g. 2
3
=3
2
3.2.3 Distributive Law
The distributive law in algebra describes an operation which when performed
on a combination of elements is the same as performing the operation on
the individual elements. The distributive law does not work in all cases of
arithmetic. For example, multiplication over addition holds:
a
×
( b + c )= ab + ac
(3.7)
e.g. 3
5
whereas addition over multiplication does not:
×
(4 + 5) = 3
×
4+3
×
a +( b × c ) =( a + b ) × ( a + c )
(3.8)
e.g. 3 + (4
(3 + 5)
Although most of these laws seem to be natural for numbers, they do
not necessarily apply to all mathematical constructs. For instance, the vector
product , which multiplies two vectors together, is not commutative.
×
5)
=(3+4)
×
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