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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
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Commutative
For all real a, b
ab = ba
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Associative
For all real a, b,
c
a + (b + c) = (a + b) + c
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Associative
For all real a, b,
c
(ab)c
= a(bc)
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Identity
There exists a real number 0
such that for every real a
a + 0 = a
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Identity
There exists a real number 1
such that for every real a
a ´
1 = a
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Additive Inverse
(Opposite)
For every real number a there exist a real number, denoted (-a),
such that
a + (-a) = 0
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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
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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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