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
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.
|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
|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.