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May 19th

May 19th

# Lecture Notes for Real Numbers

• Natural #’s: N = {1, 2, 3, 4, ...}
• Whole #’s: W = {0, 1, 2, 3, 4, ...}
• Integers: I = {...,−3,−2,−1, 0, 1, 2, 3, ...}
• Rational #’s:
• Irrational #’s: non-repeating, non- terminating decimal numbers

ALL of these sets, together, make up the set of Real Numbers ,

To change repeating decimals into rational form:
multiply by 10n where n is the number of digits that repeat
• ie: Suppose x =
• Write x in fraction form .

PROPERTIES OF REAL NUMBERS:

1. The Commutative Properties
• addition: a + b = b + a
• multiplication: ab = ba

2. The Associative Properties
• addition: a + (b + c) = (a + b) + c
• multiplication: a(bc) = (ab)c

3. The Distributive Property
• a(b + c) = ab + ac

4. Identities - an identity is a unique number where the answer for a particular operation
is the original number

is the original number
• for addition the identity element is 0: a + 0 = 0 + a = a
• for multiplication the identity element is 1: a · 1 = 1 · a = a

5. Inverses - an inverse is a number which when acted with the original number, results
in the identity

• Each real number a has a unique additive inverse represented by −a:

a + (−a) = (−a) + a = 0

• Each non zero real number a has a unique multiplicative inverse represented
by 1/a:

Working With Fractions

• Multiplying Fractions
Dividing Fractions
• Adding fractions if denominators are different :
Reducing Fractions
• Cross Products
• Using the LCD to add fractions

The Real Line

• Always label units on the graph.

Inequalities For a , b ∈ R, one and only one of the following holds:

a < b, a > b, or a = b

Sets and Intervals: A set is a well defined collection of objects. The objects in the set
are called elements. Sets are described by one of the following methods:

• roster method
• set builder notation
• graph

Symbols
• ∈: is an element of
is not an element of
• Union:

A ∪ B is the set which contains all elements in A or B or both.

• Intersection:
A∩B is the set which contains all elements in common , elements in both A and B.

• The empty set,
contains no elements.

Example: Given Q = {a, b, c, d, e, f} and R = {a, e, i, o, u}

• find: Q ∪ R

• find: Q ∩ R

Interval Notation: the following notation represents different intervals

 Interval Notation Inequality Notation Graph open interval (a, b) a < x < b closed interval [a, b] a ≤ x ≤ b half-open interval (a, b] a < x ≤ b undbounded interval (a,∞) x > a u

Example: Graph each of the following given A = (−2, 8] , B = (−∞,−1), and C = [5,∞)

• graph A

• graph B
• graph C
• find A ∩ B
• find A ∪ B
• find A ∩ C
• find A ∪ C
• find B ∩ C
• find B ∪ C

Absolute Value: the distance from a number to the origin; always positive or 0. Mathe -
matically,

• Find the absolute value of each of the following:

Distance between two points on a number line
• The distance between a and b is given by | a − b | or | b − a |
d(a, b) =| b − a |=| a − b |

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