Real Numbers - Algebra II Honors - Algebra II - 2011-08-22

Properties of Real Numbers: Objectives

  • to graph and order real numbers
  • to identify and use properties of real numbers
  • to evaluate algebraic equations 
  • to simplify algebraic expressions

Subsets of Real Numbers:

natural numbers - numbers used for counting; starts with number 1 and goes up by 1's.   1,2,3,4....

whole numbers - natural numbers + zero.  0,1,2,3,4,...

integers - natural numbers (positive integers), zero, and the negative integers      ...-4,-3,-2,-1,0,1,2,3,4...

               - Each negative integer is the opposite, or additive inverse, of a

                  positive integer

               Z is the standard symbol for the set of integers.

rational numbers - all the numbers that can be written as quotients of integers.  Examples:  7/5, -3/2, -4/5, 0, 0.3, -1.2, 9

  •   Each quotient must have a nonzero denominator.
  •   Some rational numbers can be written as terminating decimals.  Ex. 1/8=0.125
  •   All other rational numbers can be written as repeating decimals.  Ex. 1/3  or 0.33.

Use Q for the quotient.


irrational numbers - all the numbers that cannot be written as quotients of integers.

  • their decimal representations neither terminate nor repeat.
  • if a positive rational number is not a perfect square such as 25 or 4/9, then its square root is irrational.


Properties of Real Numbers

Let a, b, and c represent real numbers.

Property      Addition      Multiplication
Closure      a + b is a real number.            ab is a real number    
Commutativ         a + b b + a              ab=ba
Associative   (a + b) +  c =   a + (b+ c)           (ab)c = a(bc)
Identity          a + 0= a,  0 + a = a        a · 1 = a, 1 · a = a
Inverse          a + (-a) = 0          a · 1/a = 1
Distributive          a (b + c) = ab + bc










absolute value - the difference a number is from zero.

coefficient - (for now) a number multiplied against something, typically a power of a variable.  For ex.: 3x (the coefficient is 3).

equation - two expressions connected by equality (=).  For example:  x2 + 2x - 1 = 5x - 2

term -product of a coefficient and one or more variables to various powers.  For example, 7x2, 3y, 2x3y.  A plain number can also be a term.

  -     like terms - terms that are identical with respect to variables (differ only in coefficients).  For ex.: 3x and 4x are like, 3y and 2x are not.

variable - a symbol, usually a lowercase letter, that represents one or more numbers.  Ex. x, y, t.


Properties for Simplifying Algebraic Expressions

Let  a, b, and c represent real numbers.

Definition of Subtraction                          a - b = a + ( - b )

Definition of Division                                 a ÷ b = a · 1/b,  b ± 0

Distributive Property for Subtraction          a (b - c) = ab - ac

Multiplication by 0                                      0 · a = 0

Multiplication by - 1                                  - 1 · a = - a

Opposite of a Sum                                    - (a + b) = - a + (- b) 

Opposite of a Difference                         - ( a - b) = b - a

Opposite of a Product                              - (ab) = - a · b = a · (- b )

Opposite of an Opposite                          - (- a) = a





Solving Equalities and Inequalities.  Solving Absolute Value Equations and Inequations



  • to solve equations
  • to solve and graph inequalities
  • to solve absolute value equations
  • to solve absolute value inequalities


Properties of Equality


Reflexive                                        a=a                                                                 
Symmetric   if a = b, then b = 
Transitive  if a = b and b = c, then a = c
Addition  if a = b, then a + c = b + c
subtraction   if a = b, then a - c = b - c
multiplication  if a = b, then ac = bc
division  if a = b and c≠0, then a/c = b/c

 if a = b, then b may be substituted for a in any expression to obtain an equivalent expression.






Properties of Inequalities

  let a, b, and c represent all real numbers

Transitive if a≤b and b≤c, then a≤c
addition if a≤b then a+c≤b+c
subtraction if a≤b then a-c≤b-c

if a≤b and c>0, then ac≤bc

if a≤b and c<0 then ac≥bc


if a≤b and c>o then a/c≤b/c

if a≤b and c<0 then a/c≥b/c



  • Compound inequality is a pair of inequalities joined by and or or
  • To solve a compound inequality joined by and, find all values of the variable that make both inequalities true
  • to solve a compound inequality joined by or, find all values of the variable that makes at least one of the inequalities true.


  • Algebraic definition of Absolute Value
  • if x>0, then lxl=x
  • if x<0 then lxl=-x
  • when solving an absolute value equation, remember that there are two solutions, a positive and a negative.
  • extraneous solution is a solution of an equation derived from an original equation that is not a solution of the original equation.
StudyUp Author: NLeonhardt
Honors Biology

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