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The following
table lists the defining properties
of the real numbers (technically
called the field axioms).
These laws define how the
things we call numbers should
behave.
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Commutative
For
all real a, b
a
+ b = b
+ a |
Commutative
For
all real a, b
ab
= ba |
Associative
For
all real a, b,
c
a
+ (b + c)
= (a + b)
+ c |
Associative
For
all real a, b,
c
(ab)c
= a(bc) |
Identity
There
exists a real number
0 such that for every
real a
a
+ 0 = a |
Identity
There
exists a real number
1 such that for every
real a
a
´
1 = a |
Additive Inverse
(Opposite)
For
every real number a
there exist a real number,
denoted (-a), such that
a
+ (-a)
= 0 |
Multiplicative Inverse (Reciprocal)
For
every real number a
except 0 there exist
a real number, denoted
 ,
such that
a
´
 =
1 |
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For
all real a, b,
c
a(b + c) = ab + ac, and
(a + b)c
= ac + bc |
The commutative and associative
laws do not hold for subtraction
or division:
a
– b is not equal
to b – a
a
¸
b is not equal to b
¸
a
a
– (b – c) is
not equal to (a – b)
– c
a
¸
(b ¸ c) is not equal to (a ¸
b) ¸
c
Try
some examples with numbers
and you will see that they
do not work.
What these laws mean is that
order and grouping don't matter
for addition and multiplication,
but they certainly do matter
for subtraction and division.
In this way, addition and
multiplication are “cleaner”
than subtraction and division.
This will become important
when we start talking about
algebraic expressions. Often
what we will want to do with
an algebraic expression will
involve rearranging it somehow.
If the operations are all
addition and multiplication,
we don't have to worry so
much that we might be changing
the value of an expression
by rearranging its terms or
factors. Fortunately, we can
always think of subtraction
as an addition problem (adding
the opposite), and we can
always think of division as
a multiplication (multiplying
by the reciprocal).
You may have noticed that
the commutative and associative
laws read exactly the same
way for addition and multiplication,
as if there was no difference
between them other than notation.
The law that makes them behave
differently is the distributive
law, because multiplication
distributes over addition,
not vice-versa.. The distributive
law is extremely important,
and it is impossible to understand
algebra without being thoroughly
familiar with this law.
Example:
2(3 + 4)
According to the order of
operations rules, we should
evaluate this expression by
first doing the addition inside
the parentheses, giving us
2(3
+ 4) = 2(7) = 14
But we can also look at this
problem with the distributive
law, and of course still get
the same answer. The distributive
law says that
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