Sections 1.1 - 1.5 Flashcards
Complex Numbers
Have the form a+bi in which a and b are real numbers and i = the square root of -1
Real Numbers
Numbers that have points on the number line
Imaginary Numbers
Square roots of negative numbers, which have no points on the number line.
Negative Numbers
Numbers less than 0
Zero
Neither positive nor negative
Positive Numbers
Numbers greater than 0
Rational Numbers
Can be expressed exactly as a ratio of 2 integers
Irrational Numbers
Cannot be expressed exactly as a ratio of 2 integers, but are real numbers
Integers
Whole numbers and their opposites
Nonintegers
Fractions, or numbers between the integers
Radicals
Involve square roots, cube roots, etc. of integers
Transcendental Numbers
Cannot be expressed as roots of integers (pi)
Odd Numbers
Numbers not divisible by 2
Even Numbers
Numbers divisible by 2
Digits
Numbers from which the numerals are made (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
Natural / Counting Numbers
Positive integers or counting numbers
Field Axioms
The properties of algebra
Definition of Subtraction
a - b = a + (-b)
Definition of Division
a/b = a times 1/b
Closure
The sum / product of any two members of a set must belong to the given set.
Closure under addition
a + b is always a unique, real number
Closure under multiplication
ab is always a unique, real number
Commutativity
Changes the order of the operation
Commutativity for addition
a + b = b + a
Commutativity for multiplication
ab = ba
Associativity
Changes the groupings and not the order
Associativity for addition
(a + b) + c = a + (b + c)
Associativity for multiplication
(ab)c = a(bc)
Inverses
The set must contain the opposite and the reciprocal to be a field.
Inverse of Addition
a + (-a) = 0
The Additive Inverse
Also known as the opposite, it is (-a)
Inverse of Multiplication
a(1/a) = 1
The Multiplicative Inverse
Also known as the reciprocal, it is 1/a
Vinculum
The “division / fraction line”
Identity
Any field must contain 0 and 1
The Identity Property of Addition
a + 0 = a
The Identity Element of Addition
0
The Identity Property of Multiplication
(a)(1) = a
The Identity Element of Multiplication
1
Distributitivity
a (b + c) = ab + ac
In order for a set of numbers to be a field . . .
The set must fit the requirements of closure, commutativity, associativity, inverses, identity, and distributivity
Variable
Any symbol that represents an unknown value
Expression
A collection of variables, constants, operations, and / or grouping symbols that can be simplified and / or evaluated
Exponentiation
The exponent power determines the amount of bases to multiply together
Order of Operations to Simplify
Groupings, exponentiation, multiplication (division), addition (subtraction)
Order of Operations to Solve
Addition (subtraction), multiplication (division), exponentiation, groupings
Polynomial
An algebraic expression that involves only the operations of addition, subtraction, and multiplication of variables
Reasons not to be a Polynomial
Variable expression in the denominator, variable expression under radical signs, and variable expression inside absolute value
Equation
A statement that sets two expressions equal to each other
Solution(s)
Value(s) that can be substituted for (a) variable(s) and make the statement true
Addition Property of Equality
If a = b, the a + c = b + c
Multiplication Property of Equality
If a = b, then ac = bc
Reasons for Extraneous Solutions
The solution is not a member of the domain, multiplying the equation by an unknown value (0) adds solutions, and dividing the equation by an unknown value (0).
Irreversible Steps
Multiplying or dividing both sides of an equation by an unknown value.
Zero Product Rule
If ab = 0, then a, b, or both must equal 0
Absolute Value
lxl = p
If P in Absolute Value is Positive . . .
x = p or x = -p
If P in Absolute Value is 0 . . .
x = 0
If P in Absolute Value is Negative . . .
No solution
Addition property of order
If a > b, then a + c > b + c
Multiplication Property of Order
If a > b, then . . .
If c is positive, ac > ab
If c = 0, then ac = bc
If c is negative, then ac < bc
Interval Notation
(x, y]
Set - Builder Notation
{x l e real numbers, x > 2}
Or statement
l x l > p
Negative or situation
x < -p
Positive Or Situation
x > p
And Statement
l x l < p
Negative and situation
x > -p
Positive and situation
x < p
Solution to l x l > -p
All real numbers
Solution to l x l < -p
No solution
Reflexive Property
If x is from a set of real numbers, then x = x
Symmetry property
If x = y, then y = x
Transitive property
If x = y and y = z, then x = z
Trichotomy
If x and y are from a set of real numbers, then one of the following is true:
x > y
x < y
x = y
