# Chapter 1: Preliminary Information

Complex numbers

Have the form a + bi, where a and b are real and i = √-1ex) 2 - 3i, 7, root 3, and root of -5

Real numbers

Have points on the number line. (Complex)ex) 3, -17.4, root 5, pi, 4/3

Imaginary numbers

Square roots of negative numbers. Have no points on the number line. (Complex)ex) root -3, 5 root -2, - of root -9

Positive numbers

Numbers that are greater than zero. (Complex, real)ex) 0.001, 37, root 15

Zero

Neither positive nor negative. (Complex, real)ex) 0

Negative numbers

Numbers less than 0. (Complex, real)ex) -7.5, -58, - of root 3

Rational numbers

Can be expressed exactly as a ratio of two integers. (Complex, real)ex) 4/5, -2/3, 7.31, -5

Irrational numbers

Cannot be expressed exactly as a ratio of two integers, but are real numbers. (Complex, real)ex) root 5, - of 3 root 11, pi

Involve square root, cube root, etc. of integers (Complex, real, irrational)ex) root 5, - of 3 root 11

Transcendental numbers

Cannot be expressed as roots of integers. (Complex, real, irrational)ex) pi

Integers

Whole numbers and their opposites. (Complex, real, rational)ex) 2, -17, 2001, 0)

Nonintegers

Fractions: numbers between the integers. (Complex, real, rational)ex) 3/4, 1 2/7, -4.63

Natural numbers

Positive integers or counting numbers. (Complex, real, rational, integers)ex) 2, 2001

Digits

numbers from which the numerals are made (Complex, real, rational, integers)ex) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Even numbers

Integers divisible by 2. (Complex, real, rational, integers)ex) 2046, -38, 562, 0

Odd numbers

integers not divisible by 2. (Complex, real, rational, integers)ex) -49, 15, 2001, 1

axioms

What are some properties that are accepted without proof and are used as starting points for working numbers?

closure

What is it called when real numbers are closed under addition and multiplication? ex. 0x1=1 not closed 1+1=2

commutativity

What is it called when x + y = y + x or xy = yx?

associative

What is it called when if (x+y)+z = x+ (y+z)?

Distributive

What is it called when x+ (y+z) = xy+xz?

identity elements

What is it called when x+0=x or y x 1 = y?

inverse

What is it called when you have x + (-x)=0 or when y x 1/y = 1?

Variable

A letter which stands for an unspecified number from a given set.

expression

What is a collection of variables and constants connected by operation signs, which stands for a number?

Polynomial

Algebraic expressions that involve only the operations of addition, subtraction, and multiplication of variables. (No absolute values, division, or square roots)

monomial

What is only one number of terms in the equation?

binomial

What is a polynomial with two terms called?

trinomail

What is a polynomial with three terms called?

constant

What is a variable with no degree called?

linear

What is a variable with one degree?

What is a variable squared called?

cubic

What is a variable that is cubed called?

Quartic

What is a variable that is raised to the fourth power called?

Quintic

What is a variable that is raised to the fifth power called?

6th degree

Which degree doesn't have a special name?

extraneous solution

a number which satisfies a transformed equation, but not the original equation

Reflexive Property

if x is a real number, x=x

Symmetry Property

If x=y, then y=x

Transitive Property

For equality, if x=y and y=z, then x=zFor order, if x<y and y<z, then x<z

Trichotomy Property

If x and y are two real numbers, then exactly one of the following must be true.y<xy>xy=x