Download Properties of Real Numbers

Properties of Real Numbers
Name
Closure
Property
a + b is a unique real number
ab is a unique real number
Example
Commutative
a+b=b+a
ab = ba
2+3=3+2
(-6)(7) = (7)(-6)
Associative
(a + b) + c = a + (b + c)
(ab)c = a(bc)
(-1 + 4) + 9 = -1 + (4 + 9)
[(5)(8)]2 = 5[(8)(2)]
Distributive
a(b + c) = ab + ac
7(3 + 1) = (7)(3) + (7)(1)
Identity
a+0=0+a=a
a(1) = 1(a) = a
-3 + 0 = 0 + -3 = -3
24(1) = 1(24) = 24
Inverse
a + (-a) = (-a) + a = 0
a (1/a) = (1/a)a = 1
6 + (-6) = (-6) + 6 = 0
5 (1/5) = (1/5)5 = 1
Substitution
Replace an expression with
an equivalent expression
(4+5)(8) = (9)(8)
Definition of Subtraction
a – b = a + (-b)
8 – 2 = 8 + (-2)
Definition of Division
a ÷ b = a(1/b) when b 0
10 ÷ 5 = 10(1/5)
To add two numbers with the same sign:
• Add their absolute values
• Use their common sign
To add two numbers with different signs:
• Subtract their absolute values
• Use the sign of the number whose absolute value is larger
To subtract two numbers:
• Use the definition of subtraction to change to an addition problem
(i.e., add the opposite of the second number)
To multiply two numbers:
(+) · (+) = (+)
(+) · () = ()
() · () = (+)
() · (+) = ()
To divide two numbers:
(+) ÷ (+) = (+)
(+) ÷ () = ()
() ÷ () = (+)
() ÷ (+) = ()