Sections 1.1 - 1.5 Flashcards by Caden Hickey

Complex Numbers

Have the form a+bi in which a and b are real numbers and i = the square root of -1

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Real Numbers

Numbers that have points on the number line

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Imaginary Numbers

Square roots of negative numbers, which have no points on the number line.

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Negative Numbers

Numbers less than 0

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Zero

Neither positive nor negative

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Positive Numbers

Numbers greater than 0

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Rational Numbers

Can be expressed exactly as a ratio of 2 integers

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Irrational Numbers

Cannot be expressed exactly as a ratio of 2 integers, but are real numbers

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Integers

Whole numbers and their opposites

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Nonintegers

Fractions, or numbers between the integers

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Radicals

Involve square roots, cube roots, etc. of integers

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Transcendental Numbers

Cannot be expressed as roots of integers (pi)

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Odd Numbers

Numbers not divisible by 2

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Even Numbers

Numbers divisible by 2

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Digits

Numbers from which the numerals are made (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

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Natural / Counting Numbers

Positive integers or counting numbers

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Field Axioms

The properties of algebra

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Definition of Subtraction

a - b = a + (-b)

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Definition of Division

a/b = a times 1/b

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Closure

The sum / product of any two members of a set must belong to the given set.

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Closure under addition

a + b is always a unique, real number

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Closure under multiplication

ab is always a unique, real number

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Commutativity

Changes the order of the operation

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Commutativity for addition

a + b = b + a

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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

The Identity Property of Multiplication

(a)(1) = a

The Identity Element of Multiplication

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

Sections 1.1 - 1.5 Flashcards

(75 cards)