
Natural numbers
 Includes numbers we use for counting:
 1,2,3,4,5,6,7,8,9,...

Whole numbers
 Includes natural numbers together with 0:
 0,1,2,3,4,5,6,7,8,9,...

Integers
 includes natural numbers, 0, and the negatives of the natural numbers:
 5,4,3,2,1,0,1,2,3,4,5,...

Prime numbers
 Are natural number greater than 1 that are divisible by 1 and themselves:
 2,3,5,7,11,13,17,19,23

Composite numbers
 Are natural numbers greater than 1 that are not prime:
 4,6,8,9,10,12,14,15,16,18,20,21

Even integers
 Are integers that are exactly divisible by 2:
 8,6,4,2,0,2,4,6,8,...

Odd integers
 Are integers that are not exactly divisible by 2:
 9,7,531,1,3,5,7,9,...

Rational numbers
 a a is an integer and b is a nonzero integer
 b

Irrational numbers
x x is a nonterminating, nonrepeating decimal

Real numbers
x x is a terminating decimal, a repeating decimal, or a nontermination nonrepeating decimal

Interval notation
 (5,8]
 parenthesis indicates that endpoints are not included, brackets indicates that the endpoints are included

Compound Inequalities
Expressions that involve more than one inequality and involves the word (or,and) so it can be written in interval notation with the union symbol (U, upside down for and)

Absolute value
is the distance on a number line between 0 and the point with the coordinate.

Sum
Result when two numbers are added

Difference
Result when one number is subtracted from another number

Product
Result when two numbers are multiplied

Quotient
 Result when two numbers are divided
 Division by 0 is undefined

Mean
The Sum of the values divided by the number of values

Median
 The middle value
 if odd, choose the middle
 if even, find the mean of the middle two values

Mode
Value that occurs mors often

Perimeter
 Distance around a figure
 Square P=4s
 Rectangle P=2l+2w
 Triangle P= a+b+c
 Trapezoid P=a+b+c+d

Circumfrence
 Distance around a circle
 C=pie * D
 C=2*pie*radius
 (pie is approximately)

Communitive Property
Multiplication and Addition

Associative Property
Multiplication and Addition
(a+b) + c= a + (b+c)

Distributive Property
Multiplication over addition
a(b+c) = ab + ac

The product Rule of exponents
x^{m} x^{n} = x^{m+n}

Power Rules of Exponents
 (x^{m})^{n} = x^{m*n}
 (xy)^{n} = x^{n}*y^{n}
 (x/y)^{n}= x^{n}/y^{n}


Negative Exponents
 x^{n} = 1/x^{n}
 and
 1/x^{n} = x^{n}

Quotient Rule
x^{m}/x^{n} = x^{mn}

Fractions to Negative Powers
(x/y)^{n} = (y/x)^{n} = y^{n}/x^{n}

Equation
A statement indicating that two quantities are equal

Conditonal equations
Equations have exactly one solution

Identiy
Equation that is satisfied by every number for which both sides of the the equation are defined. All real numbers

Contradiction
Equation that has no solution

Right angle
angle whose measure is 90 degree

Straight angle
Angle whose measure is 180 degree

Acute anle
Angle whose measure is greater than 0 degree but less than 90

Complementary angles
the sum of two angles equals 90 degrees

Supplementary anles
the sum of two angles equals 180 degrees

Coordinate System
 xaxishorizontal line
 yaxisvertical line
 originpoint where axis cross
 coordinate planetwo axis form this
 quadrantscoordinate plane divide into four regions
 xcoordinatepoint on x axis
 ycoordinatepoint on y axis
 ordered pairwhen the order of the coordinate is important

Linear equation
general form
 When the graph of an equation is a line
 Ax + By = C
 A,B,C are constants
 x,y are variables

Yintercept
Point of a line where the line intersects the yaxis

Xintercept
Point of a line where the line intersects the xaxis

Horizonatal and Vertical lines
 the graph of x=a is a vertical line with xintercept at (a,0)
 the graph of y=b is a horizoneal line with y intercept at (0,b)

Midpoint
the middle point of a line with ends at P(x1,y1) and Q(x2,y2) calculated:
(x1+x2/2 , y1=y2/2)

Slope of the line
Constant rate of change of line passing through points (x1,y1) and (x2,y2) calculated:
 m= Change in y/Change in x
 m= y2y1/ x2x1
 m+ rise/run

Slopes of Horizontal and Vertical lines
 all horizontal lines (lines with equations of the form y=b) have a slope of O
 all vertical lines (lines with equations of the form x=a) have no defined slope

Slope of parallel lines
 Nonvertical parallel lines have the same slope, and lines having the same slope are parallel
 Since vertical lines are parallel, lines with no defined slop are parallel

Negative reciprocals
Two real numbers a and b if ab=1

Slopes of perpendicular lines
If two nonvertical lines are perpendicular, their slopes are negative reciprocals.

Point slope form
 the point slope equation of a line passing through P(x1,y1) and with the slope M is:
 yy1=m(xx1)

SlopeIntercept Form
 The slopeintercept equation of a line with slope m and yintercept (0,b) is:
 y=mx+b

Slope and Yintercept from the general form
 Ax+By=C
 slope =  a/b
 yinetercept = (0, c/b)

Relations
Sets of ordered pairs

Domain of the relation
the set of all the first components in the relation

Range of the relation
The set of all the second components in the relation

Function
Is any set of ordered pairs (a relation) in which each first component determines exactly one second component

Vertical line test
Determines whether the graph of an equation represents a function. If every vertical line that intersects a graph does so exactly once, the graph represents a function.

Function Notation
The notation y= f(x) denotes that the variable is a function of x.

y is a Function of x
An equation, table, or graph in x and y in which each value of x (the input) determines exactly one value of y (the output) is a function of x.

Graph of a function
the graph of the ordered pairs (x,f(x)) that define the function

Linear function
 a function defined by an equation that can be written in the form:
 f(x)=mx+b
 or
 y=mx+b

Squaring function
 f(x)=x^2 (or y=x^2)
 parabola

Cubing function
f(x)=x^3 (or y=x^3)

Absolute value function
 f(x)=lxl (or y=lxl)
 V shaped graph

Horizontal Translations
 If f is a fuction and k is a positive number, then:
 the graphe of f(xk) is identical to the graph of f(x), except that it is translated k units to the right
 The graph of f(x+k) is identical to the graph of f(x), except that it is translated k units to the left.

Vertical translations
 If f is a function f and k is a positive number, then:
 The graph of f(x)+k is identical to the graph of f(x), except that it is translated k units upward
 The graph of f(x)k is identical to the graph f(x), except that it is translated k units downward

Reflection of a Graph
The graph of y=f(x) is the graph of f(x) reflected about the xaxis

Solving the system
The process of finding the ordered pair that satisfies both equations in the system

Consistent system
When the system has a solution

Inconsistent system
When the system has no solution, the solution set is 0 (put line thru the zero)

Dependent system
When the system has infinately many solutions, 2 equations give the same line.

Substitution method
 1. solve one equation for one of its variables
 2. substitute the result from Step 1 into the other equation and solve
 3. find the value of the other variable by substittuting the value from step 2 into one of the original equations.
 4. State the solution

Addition Medthod
 1. write both equations in general form
 2. if necessary, multiply the terms of one or both equations to cake the coefficients on one of the variables differ only in sign
 3. Add the equation and solve.
 4. substitute the value in step 3 into either of the original equations and solve
 5. state the solution

Solving a system of three linear equations in three variables
 1. pick any two equations and elimate a variable
 2. pick a different pair and elinate the same variable
 3. solve the resulting pair of two equations in two variables
 4. find the value of the third variable, substitute the values from step three into one of the original equations with three variables and solve the equation

