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Math Analysis Notes
Section 1.1
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Section 1.1: Real Numbers
Big Idea: ALGEBRA IS A SYSTEM FOR SIMPLIFYING AND UNDOING MULTI-STEP
ARITHMETIC CALCULATIONS!
This section is a review of how to represent the properties of real numbers, negative numbers, fractions, sets,
and absolute values in the (terse) language of mathematics.
Big Skill: You should be able to represent intervals of real numbers using set-builder notation, interval
notation, and graphical methods.
Subsets of the real number system:
Natural numbers:  1, 2,3, 4, 
Integers:
  , 3, 3, 1, ,1, 2,3,


m

 r r  , where m, n   Some rational numbers have a finite number of digits in
n


1

their decimal representation (½ = 0.5), while others have an infinite number of repeating digits   0.142857  .
7

Rational numbers:
Practice: Convert 0.87 to a fractional form.
Irrational numbers: Any number that can not be written as a rational number. An irrational number has an
infinite number of non-repeating digits in its decimal representation. We can only represent an irrational number
exactly by its description, such as 2  the positive solution to the equation x2 = 2, or  = the ratio of the
circumference to the diameter of any circle.
Real numbers:
= the union of the sets of all rational and irrational numbers.
Properties of Real Numbers:
Commutative Property of Addition: a  b  b  a
Commutative Property of Multiplication: ab  ba
Associative Property of Addition:  a  b   c  a  b  c 
Associative Property of Multiplication:  ab  c  a  bc 
Distributive Property: a  b  c   ab  ac ,  b  c  a  ba  ca
I state these properties to remind you that any algebraic expression, no matter how complicated, boils down to
doing repeated arithmetic operations on a pair of numbers, and because you are going to encounter
mathematical objects in higher mathematics classes where these properties don’t always apply.
Math Analysis Notes
Section 1.1
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Practice: State the properties of real numbers used in the following simplification:
2  x  5  3x   2 x  2  5  3x
  2 x  10   3 x
 10  2 x   3 x
 10   2 x  3 x 
 10    2  3 x 
 10  5 x
More Properties of Addition:
Additive Identity: The number 0; 0 leaves a number unchanged after addition: a + 0 = a.
Additive Inverse: The negative of a number; a number plus its additive inverse equals the additive identity:
a + -a = 0.
Inverse of the Addition Operation: Subtraction “undoes” addition. Subtraction is defined as the addition of
the negative of a number: a – b = a + (-b).
Properties of Negatives: These follow from the properties of addition.
1. (-1)a = -a
2. –(-a) = a
3. (-a)b = a(-b) = -(ab)
4. –(a + b) = -a – b
5. –(a – b) = b – a
More Properties of Multiplication:
Multiplicative Identity: The number 1; 1 leaves a number unchanged after multiplication: a·1 = a.
Multiplicative Inverse: The reciprocal of a number; a number times its multiplicative inverse equals the
multiplicative identity: a·(1/a) = 1.
Inverse of the Multiplication Operation: Division “undoes” multiplication. Division is defined as
multiplying by the reciprocal of a number: a  b = a·(1/b).
Properties of Fractions: These follow from the properties of multiplication.
a c ac
 
1.
b d bd
a c a d
  
2.
b d b c
a b ab
 
3.
c c
c
a c ad  bc
 
4.
b d
bd
ac a

5.
bc b
a c
6. If  , then ad  bc
b d
Math Analysis Notes
Section 1.1
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Ways to represent sets of real numbers:
1. Listing e.g. The set A of all integers whose absolute value is less than 2 can be written as A = {-1, 0, 1}
2. Set-builder notation (same example): A = {x | x is an integer and |x| < 2}, or A = {x | x  and |x| < 2}
3. Interval notation (combined with set builder notation) A = {x | x  (-2, 2) and x  Z}
4. Interval notation for real numbers:
a. [ or ] means the endpoint is in the set,
b. ( or ) means the endpoint is not in the set
c. [1, 5) means all numbers between 1 and five, including 1 but not including 5
d. (-1, ) means all numbers greater than -1
5. Combining sets using Union (  ): Forms a new set by combining all elements from both sets.
{1, 2, 3, 4, 5}  {4, 5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}
Practice: [-1, 1)  (0, 2] =
6. Combining sets using Intersection (  ): Forms a new set by combining only common elements from
both sets.
{1, 2, 3, 4, 5}  {4, 5, 6, 7, 8} = {4, 5}
Practice: [-1, 1)  (0, 2] =
Absolute value:
 a if a  0
Definition: The absolute value of a number a is a  
.
a if a  0
Properties of the Absolute Value:
1. a  0
2.
a  a
3.
ab  a b
4.
a
a

b
b
Linear Distance between points:
If a and b are points on the real line, then the distance between them is d  a, b   b  a .
Technology Practice:
Use Microsoft Excel to create a filled-in table for problem #80 from section 1.1