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N
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Properties
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Comments
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1
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If
a = b
then
b = a. |
The property states equal rights for both sides of an equality.
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2
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If a = c
and b = c then
a = b .
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If two numbers are equal to the same number, then they are equal
to each other.
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3
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If a = b then
a + c = b + c . |
The property allows to add any number to both sides of the equality.
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4
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If a = b then
a c = b c .
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An equality holds true, if to multiply its both sides by any non-zero
number.
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5
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If a = b then
f
(a)
= f (b).
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If some operation with respect to a number a
gives a unique result f (a),
then a = b
implies f (a) =
f (b) .
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6
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a + b = b + a ,
a b = b a .
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Elements of a sum can be added in any order.
Elements of a product can be multiplied in any order.
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7
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a + ( b + c ) =
= ( a + b ) +
c ,
a ( b c ) =
( a b ) c .
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Elements of a sum can be combibed in groups.
Factors can be combined in groups.
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8
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a + (– a ) = a
– a = 0
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For any real number a, there exists
the additive inverse number (– a) such that by which the given number is added to produce zero.
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9
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a (1 / a ) =
a / a = 1
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For any non-real number a,
there exists the reciprocal (or multiplicative inverse) number
(1 / a)
such that by which the given number is multiplied to produce unity.
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10
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If a b = 0
then a = 0 ,
or b = 0 ,
or a = b = 0 .
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If a factored expression equals zero, then at least one of the
factors is equal to zero.
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11
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If a > b
and c > 0 ,
then a c > b c .
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The property allows to multiply both sides of an inequality by
any positive number.
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12
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If a > b
and c < 0 ,
then a c < b c .
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If the sides of an inequality are multiplied by a negative number,
then the sign of inequality must be reversed.
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Types of Numbers
