Properties of Real Numbers

The Commutative Property states that when adding or multiplying two or more numbers or terms, order is not important. That is:

• $2+7=7+2$
  

• $3\times5=5\times3$
  

• $7-2=2-7$

• $50\div10=10\div50$

The Associative Property states that when adding or multiplying three or more numbers or terms together, grouping is not important. That is:

• $\left(5+2\right)+6=5+\left(2+6\right)$

• $\left(5\times2\right)\times6=5\times\left(2\times6\right)$

• $\left(5-2\right)-3=5-\left(2-3\right)$

• $\left(20\div4\right)\div2=20\div\left(4\div2\right)$

The Distributive Property states that for any three terms a,b, and c: $a\left(b+c\right)=ab+ac$ 
Select the answer below that models this

•  $4\left(3+2\right)=\left(4\times3\right)+\left(4\times2\right)$  

•  $5\left(6+7\right)=56+57$  

•  $4\left(3+2\right)=\left(4+3\right)\times\left(4+2\right)$  

•  $5\left(6+7\right)=5\times6\times7$  

The Identity Property of Addition states that adding 0 to any expression leaves the expression unchanged.

• 3 + 0 = 3

• 5 + 0 = 0

• 7 + 0 = 70

• 3 + 0 = 3

The Identity Property of Multiplication states that multiplying any expression by 1 leaves the expression unchanged.

• 8(1) = 8

• 10(1) = 11

• 5(1) = 51

• 6(1) = 6

The Additive Inverse Property states that for every number a there is a number −a such that a + (−a) = 0.

• 5 + (-5) = 0

• 6 + (-6) = -12

• 2 + (-2) = 0

• 8 + (-7) = 0

The Multiplicative Inverse Property states that for every nonzero number a there is a number  $\frac{1}{a}$  such that  $a\times\frac{1}{a}=1$  

A common name used for the multiplicative inverse is the reciprocal

•  $8\ \times\frac{1}{8}=1$  

•  $5\times\frac{1}{5}=0$  

•  $6\times\frac{1}{12}=1$  

•  $3\times\frac{1}{3}=1$  

The Additive Property of Equality states that equality is maintained if the same amount is added to both sides of an equation. That is, if a = b, then a + c = b + c

• 5x = 10 then 5x + 2 = 12

• 3x = 9 then 3x + 3 = 9

• 4x = 8 then 4x - 4 = 4

• 6x = 12 then 6x - 3 = 6

The Multiplicative Property of Equality states that equality is maintained if both sides of an equation are multiplied by the same amount. That is, if a = b, then a · c = b · c

• y = 3 then 4y = 4(3)

• x = 2 then 2x = 22

• y = 6 then 3y = 3

• x = 4 then 2x = 8

A symbol used to represent one or more numbers. Letters of the English alphabet are used as _________. For example, in the expression 3x – (8.6xy + z), the _________ are x, y, and z.

• Variables

• Expressions

• Order of Operations

• Combine like terms