P.1 Real Numbers

Be prepared to take notes when the bell rings.

Real Numbers Set of numbers formed by joining the set of

rational numbers and the set of irrational numbers

Subsets: (all members of the subset are also included in the set) {1, 2, 3, 4, …} natural numbers {0, 1, 2, 3, …} whole numbers {…-3, -2, -1, 0, 1, 2, 3, …} integers

Rational and Irrational Numbers

Rational Number Irrational Number

A real number that can be written as the ratio of two integers, where q

Example:

Repeats = 0.125

Terminates

Repeats

A real number that cannot be written as the ratio of two integers

*infinite non-repeating decimals

Example:

Real Numbers

Irrational Numbers

Rational Numbers

Non-Integer

Fractions

Integers

Negative Integers

Whole Numbers

Natural Numbers

Zero

Real Number Line

Origin

0

Positive

Negative

Coordinate: • Every point on the real number line corresponds to exactly

one real number called its coordinate

Ordering Real NumbersInequalities

Example:a b

a b

−143¿

√26¿

Describe the subset of real numbers represented by each inequality. A.

B.

C.

Interval: subsets of real numbers used to describe inequalities

Notation

[𝑎 ,𝑏 ](𝑎 ,𝑏)[𝑎 ,𝑏 )

(𝑎 ,𝑏 ]

Interval TypeClosed

Open

Inequality

𝑎≤𝑥 ≤𝑏𝑎<𝑥<𝑏𝑎≤𝑥<𝑏𝑎<𝑥≤𝑏

*Unbounded intervals using infinity can be seen on page 4

Properties of Absolute Value

Absolute value is used to define the distance (magnitude) between two points on the real number line Let a and b be real numbers. The distance

between a and b is:

The distance between -3 and 4 is:

Algebraic Expressions Variables:

letter that represents an unknown quantity

Constant: Real number term in an

algebraic expression Algebraic Expression:

Combination of variables and real numbers (constants) combined using the operations of addition, subtraction, multiplication and division

Examples of algebraic expressions:

Terms: Parts of an algebraic

expression separated by addition

i.e.

Coefficient: Numerical factor of a

variable term Evaluate:

Substitute numerical values for each variable to solve an algebraic expression

Examples of Evaluation

Expression

−3 𝑥+53 𝑥2+2𝑥−1

Value of Variable

𝑥=3𝑥=−1

Substitute

−3 (3)+5

3 (−1)2+2(−1)−1

Value of Expressi

on

−9+5=−4

3−2−1=0Used Substitution Principle: If a=b, then a can be replaced by b in any expression involving a.

Basic Rules of Algebra 4 Arithmetic operations:

Addition, + Subtraction, - Division, / Multiplication,

Addition and Multiplication are the primary operations. Subtraction is the inverse of Addition and Division is the inverse of Multiplication.

Basic Rules of Algebra

Subtraction: add the opposite of b

Division: multiply by the reciprocal of b; if b0, then

is called the additive inverse (opposite of a real number)

is called the multiplicative inverse (reciprocal of a real number)

Let a, b and c be real numbers, variables or algebraic expressions.

Commutative Property of Addition

Commutative Property of Multiplication

Associative Property of Addition

Associative Property of Multiplication

Distributive Property

Additive Identity Property

Multiplicative Identity Property

Additive Inverse Property

Multiplicative Inverse Property

Let a, b and c be real numbers, variables or algebraic expressions.

Properties of Negation and Equality

Let a, b and c be real numbers, variables or algebraic expressions.

Properties of Zero

Homework Problems Page 9 #’s 1-25 odd, 29, 33-39 odd, 43-47,

51-55, 59, 89-93, 99-107, 111-115