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P.1 Real Numbers
Be prepared to take notes when the bell rings.
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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
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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:
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Real Numbers
Irrational Numbers
Rational Numbers
Non-Integer
Fractions
Integers
Negative Integers
Whole Numbers
Natural Numbers
Zero
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Real Number Line
Origin
0
Positive
Negative
Coordinate: • Every point on the real number line corresponds to exactly
one real number called its coordinate
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Ordering Real NumbersInequalities
Example:a b
a b
−143¿
√26¿
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Describe the subset of real numbers represented by each inequality. A.
B.
C.
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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
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Properties of Absolute Value
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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:
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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
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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.
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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.
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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)
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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
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Let a, b and c be real numbers, variables or algebraic expressions.
Properties of Negation and Equality
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Let a, b and c be real numbers, variables or algebraic expressions.
Properties of Zero
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Homework Problems Page 9 #’s 1-25 odd, 29, 33-39 odd, 43-47,
51-55, 59, 89-93, 99-107, 111-115