Properties of Real Numbers
COMMUTATIVE: if you change the order of the numbers being added, the answer does not change. This also
works for multiplying. The following are examples of this property…
Add:
-3 + -5 = -5 + -3
½ + ¾ = ¾ + ½
Multiply: (5)(-10) = (-10)(5)
6(x + 3) = (x + 3)6
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ASSOCIATIVE: changing the grouping in an addition problem will not affect the answer. This also works for
multiplication. The following are examples…
Add: -3 + (-2 + 5) = (-3 + -2) + 5
(0.5 + 0.25) + 0.1 = 0.5 + (0.25 + 0.1)
Multiply: (3 · 5) · 2 = 3 · (5 · 2)
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IDENTITY: of Addition or Additive Identity…any number plus zero is itself (adding zero to a number doesn’t
change the number (identity)). -5 + 0 = -5; 0 + c = c
of Multiplication or Multiplicative Identity…any number multiplied by one is itself (multiplying by one doesn’t
change the number (identity)). (8)(1) = 8; 1 · y = y
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INVERSE: of Addition or Additive Inverse…a number added to its additive inverse (opposite) gives zero. -5 + 5 = 0;
½+-½=0
of Multiplication or Multiplicative Inverse…any number multiplied by its multiplicative inverse (reciprocal) gives
𝟑
𝟒
one. 2 · ½ = 1; − ∙ − = 𝟏
𝟒
𝟑
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DISTRIBUTIVE: the factor outside the parentheses is distributed (multiplied) to each of the terms inside the
parentheses. a(b + c) = ab + ac; 4(2x + 7) = 8x + 28