1.1 Real Numbers and Sets and Intervals - College Algebra (Math 1150)

1. Classifying Real Numbers

Real Numbers Organization
- Natural numbers: \,N={1,2,3,\ldots}\;
- Whole numbers: W=\{0,1,2,3,\ldots}\;
- Integers: Z=\{\ldots,-3,-2,-1,0,1,2,3,\ldots}\;
- Rational numbers: \mathbb{Q}={\frac{p}{q}\;|\, p\in\mathbb{Z},\; q\in\mathbb{N},\; q\neq 0}\;
- Irrational numbers: cannot be written as a ratio of two integers. Examples: \sqrt{2},\;\sqrt{5},\;\pi
- Real numbers: \mathbb{R}=\mathbb{Q}\cup{\text{irrational numbers}}\;
Decimal representations of rational numbers
- Terminating decimals: examples include
- \tfrac{1}{2}=0.5
- \tfrac{3}{4}=0.75
- 6.55=\tfrac{655}{100}
- Non-terminating repeating decimals: examples include
- 0.285714\overline{285714} = 0.\overline{285714}=\tfrac{2}{7}
Examples and clarifications
- Rational numbers can yield terminating or repeating decimals.
- Irrational numbers have non-terminating, non-repeating decimals.
The real number line and coordinates
- One-to-one correspondence: each real number corresponds to a unique point on the line.
- Origin is the point for coordinate 0.
- Positive real numbers lie to the right of the origin; negative numbers to the left.
- The real number line is a geometric representation of (\mathbb{R}).

2. The Real Numbers: The Real Number Line and Key Checks

True/False quick checks (from the transcript)
- Is (-1.5) rational? True
- Is (4) an integer, a rational number, and a real number? True
- Is (81) irrational? False ((81=9^2) is rational)
- Is (-\sqrt{12}) irrational? True (since (\sqrt{12}=2\sqrt{3}) is irrational)

2. The Real Number Line and Graphs

The real number line associates every real number with exactly one point:
- The coordinate of the point is the real number itself.
- The origin corresponds to coordinate 0.
Graph and coordinate language
- Real number: the coordinate; Graph shows position on the line.

3. Sets and Intervals

Sets
- A set is a collection of objects (elements).
- Notation examples:
- A=\\\{1,2,3,4,5,6}\n
- A={x\;|\; x\in \mathbb{N}\;\text{and}\; x<7}\n
Set operations
- Union: A\cup B contains all elements in A or B (without duplicates).
- Intersection: A\cap B contains elements common to A and B.
Example
- Let A={-2,-1,0,1,2}, B={-4,-2,0,2,4}, C={-3,3}.
- A\cap B={-2,0,2}
- A\cup B={-4,-2,-1,0,1,2,4}
- A\cap C=\emptyset\n
Intervals
- Open interval: ((a,b) = {x\;|\; a<x<b}) (excluding endpoints)
- Closed interval: ([a,b] = {x\;|\; a\le x\le b}) (including endpoints)
- Unbounded intervals:\n - ((a,\infty) = {x\;|\; x>a})\n - ((-
  \infty,b) = {x\;|\; x<b})\n - ((-
  \infty,\infty) = {x\in \mathbb{R}}\n- Notation vs. Graph
- Interval notation uses parentheses or brackets to indicate endpoint inclusion and whether the interval is open/closed.
- Graphical representations use open circles for open endpoints and closed circles for closed endpoints.
Union and intersection of intervals (example)
- If I=(-3,4) and J=[2,6], then
- I\cup J = (-3,6]
- I\cap J = [2,4)

3. Checks for interval operations

Example true/false checks
- Given I=(-x,5) and J=[-2,\infty), then
- (I\cup J = (-\infty, -2]) is False
- (I\cup J = (-\infty, \infty)) is True
- (I\cap J = [-2,5)) is True
A note on endpoint inclusion: -2 belongs to both I and J; 5 belongs to J but not I.

4. Absolute Value and Distance

Absolute value
- Definition: for real number (a),
  |a|=\begin{cases}a,& a\ge 0\-a,& a<0\end{cases}
- Examples:
- |4|=4
- |{-4}|=4
- |0|=0
- |(-3)+1|=| -2|=2
- |2(-3)+7|=|-6+7|=|1|=1
Geometric interpretation
- Absolute value is the distance of a real number from the origin on the number line.
- Distance from the origin for coordinate (a) is |a|\;.
Distance between two points on a real number line
- If the coordinates are (a) and (b), then
  d(a,b)=|a-b|
- Examples:
- d(-3,4)=|(-3)-4|=| -7|=7
- d(4,-3)=|4-(-3)|=|7|=7
Quick checks
- Intuition: distance is always nonnegative; absolute value yields a nonnegative result.

4. Evaluations and Quick Checks

Sample evaluations (illustrative, aligned with the style of the notes):
- Evaluate: 7 (as a standalone value) → 7
- Evaluate: 74-10 → 64
- Find distance: d(-7,2)=|(-7)-2|=| -9|=9

5. Mathematical Expressions

Vocabulary
- A term: a constant, a variable, a parenthetical group, or a product of these.
- Examples of terms: 6,\, 5x^2,\, x y (x+2),\, 3\sqrt{7} (illustrative examples; formatting may vary in notes)
- A mathematical expression consists of terms separated by + and − signs.
- Examples of expressions: 3.25+4,\, 2x^2-xy+4,\, xy(x+2),\, 7+\sqrt{7}
- The value of the expression is the result of performing the arithmetic operations or substituting numbers for the variables.
Order of Operations (PEMDAS)
- P: Parentheses, brackets, braces, fraction bars, absolute value bars, radicals.
- E: Exponents (including roots).
- MD: Multiplication and Division from left to right.
- AS: Addition and Subtraction from left to right.
Examples (step-by-step)
- Example 1: 3.25+4
- Value: 3.25+4=7.25
- Example 2: (-3)\cdot 5+20
- Step: (-3)\cdot 5=-15; then -15+20=5
- Example 3: 5-12\div 6\cdot 2
- Division/Multiplication left to right: 12\div 6=2; 2\cdot 2=4
- Then 5-4=1
- Example 4: -3+ (6-4)^2
- Inside parentheses: 6-4=2; square: 2^2=4; then -3+4=1
Evaluating algebraic expressions (variable substitution)
- Example: Evaluate expression \frac{(9+x)}{7} for (x=5)
- Substitution: (9+5=14); then value is \frac{14}{7}=2\;.

5. Properties of Real Numbers

Fundamental properties (for real numbers (a,b,c))
- Addition (Commutative):
  a+b=b+a
- Addition (Associative):
  a+(b+c)=(a+b)+c
- Multiplication (Commutative):
  ab=ba
- Multiplication (Associative):
  (ab)c=a(bc)
- Distributive:
  a(b+c)=ab+ac
Identity and Inverse
- Additive identity: there exists a unique real number 0 such that a+0=a\;\forall a\in\mathbb{R}
- Multiplicative identity: there exists a unique real number 1 such that a\cdot 1=a\;\forall a\in\mathbb{R}
- Additive inverse: every real number a has an opposite (-a) with a+(-a)=0
- Multiplicative inverse (reciprocal): every nonzero real number a has an inverse such that a\cdot a^{-1}=1\; (a\neq 0)
Examples illustrating the properties
- Commutative: (-2)+7=7+(-2) and (-11)(-4)=(-4)(-11)
- Associative: (15+(-9))+23=15+((-9)+23) and (3\cdot 4)\cdot 5 = 3\cdot(4\cdot 5) = 60
- Distributive: 4(12-7)=4\cdot12+4\cdot(-7)
- Identity: -6+0=-6\;\text{ and }\;23\cdot1=23
- Inverse: 3\cdot \tfrac{1}{3}=1

Connections, implications, and practice

Connections to foundations
- Classifications (N, W, Z, Q, irrationals) underpin number theory and algebraic structure.
- Real number line enables geometry of inequalities, intervals, and function domains.
- Set operations and interval notation are essential for solving inequalities and describing solution sets.
Practical implications
- Understanding decimal representations helps in identifying rational vs irrational numbers.
- Absolute value and distance are fundamental in measuring magnitude and in defining metrics.
- Order of operations ensures unambiguous evaluation of expressions across disciplines.
Ethical/philosophical notes (brief)
- Clear notation avoids misinterpretation in mathematical communication.
- Precise definitions (e.g., what counts as a number, what is an interval) underpin rigorous reasoning.

Key formulas and notations to remember

Real number classifications: \mathbb{N},\;\mathbb{W},\;\mathbb{Z},\;\mathbb{Q},\;\mathbb{R} with definitions above.
Rational number set: \mathbb{Q}={\tfrac{p}{q}\;|\;p\in\mathbb{Z},\; q\in\mathbb{N},\; q\neq 0}
Decimal types: terminating vs repeating decimals, with example relationships to fractions.
Absolute value: |a|=\begin{cases}a,& a\ge 0 \\ -a,& a<0\end{cases}
Distance on the number line: d(a,b)=|a-b|
Interval notations: open (a,b) = {x\in\mathbb{R}\;|\;a<x<b} , closed [a,b] = {x\in\mathbb{R}\;|\;a\le x\le b} , unbounded (a,\infty),\ (-\infty,b),\ (-\infty,\infty)
Set operations: A\cup B,\; A\cap B
Order of Operations: PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)
Basic properties: Commutative, Associative, Distributive, Identity, Inverse

Title: 1.1 Real Numbers: Algebra Essentials — Comprehensive Study Notes