🏃🏽‍♀️‍➡️Intro to Mathematical Analysis Unit 1 – The Real Number System

Key Concepts and Definitions

• Real numbers encompass all rational and irrational numbers, forming a complete ordered field
• Rational numbers can be expressed as a ratio of two integers $\frac{p}{q}$, where $q \neq 0$ (fractions, terminating decimals, repeating decimals)
• Irrational numbers cannot be expressed as a ratio of two integers (non-terminating, non-repeating decimals like $\sqrt{2}$, $\pi$)
  • Irrational numbers have infinite decimal expansions without any repeating pattern
• Real numbers are dense, meaning between any two real numbers, there exists another real number
• Completeness property states that every non-empty set of real numbers that is bounded above has a least upper bound (supremum)
• Archimedean property asserts that for any positive real numbers $x$ and $y$, there exists a natural number $n$ such that $nx > y$

Properties of Real Numbers

• Commutative property for addition: $a + b = b + a$ for all real numbers $a$ and $b$
• Commutative property for multiplication: $ab = ba$ for all real numbers $a$ and $b$
• Associative property for addition: $(a + b) + c = a + (b + c)$ for all real numbers $a$, $b$, and $c$
• Associative property for multiplication: $(ab)c = a(bc)$ for all real numbers $a$, $b$, and $c$
• Distributive property: $a(b + c) = ab + ac$ for all real numbers $a$, $b$, and $c$
• Identity element for addition: $a + 0 = a$ for all real numbers $a$
• Identity element for multiplication: $a \cdot 1 = a$ for all real numbers $a$
• Inverse element for addition: For every real number $a$, there exists a unique real number $-a$ such that $a + (-a) = 0$

Number Sets and Subsets

• Natural numbers ($\mathbb{N}$): Positive integers starting from 1 ($1, 2, 3, \ldots$)
• Whole numbers ($\mathbb{W}$): Non-negative integers including 0 ($0, 1, 2, 3, \ldots$)
• Integers ($\mathbb{Z}$): Positive and negative whole numbers, including 0 ($\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$)
• Rational numbers ($\mathbb{Q}$): Numbers that can be expressed as a ratio of two integers ($\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$)
  • Includes terminating and repeating decimals
• Irrational numbers ($\mathbb{I}$): Real numbers that cannot be expressed as a ratio of two integers (non-terminating, non-repeating decimals)
• Real numbers ($\mathbb{R}$): Union of rational and irrational numbers ($\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$)
• Subset relationships: $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

Algebraic Operations on Real Numbers

• Addition of real numbers is always closed, meaning the sum of any two real numbers is also a real number
• Subtraction of real numbers is always closed, as subtracting a real number is equivalent to adding its additive inverse
• Multiplication of real numbers is always closed, meaning the product of any two real numbers is also a real number
• Division of real numbers is closed, except when dividing by zero, which is undefined
  • Dividing by a non-zero real number is equivalent to multiplying by its multiplicative inverse (reciprocal)
• Exponentiation of real numbers is closed when the base is a positive real number, or when the base is zero and the exponent is positive
  • Negative bases with rational exponents are closed if the denominator of the exponent is odd
• Roots of non-negative real numbers are closed (square roots, cube roots, etc.)
  • Even roots of negative numbers are not closed in the real number system

Order and Inequalities

• Real numbers are totally ordered, meaning for any two distinct real numbers $a$ and $b$, either $a < b$ or $b < a$
• Trichotomy law: For any real numbers $a$ and $b$, exactly one of the following holds: $a < b$, $a = b$, or $a > b$
• Transitive property of inequality: If $a < b$ and $b < c$, then $a < c$ for real numbers $a$, $b$, and $c$
• Addition property of inequality: If $a < b$, then $a + c < b + c$ for real numbers $a$, $b$, and $c$
• Multiplication property of inequality: If $a < b$ and $c > 0$, then $ac < bc$ for real numbers $a$, $b$, and $c$
  • If $a < b$ and $c < 0$, then $ac > bc$ for real numbers $a$, $b$, and $c$
• Solving linear inequalities involves applying the properties of inequality to isolate the variable
• Graphing inequalities on a number line uses open or closed circles to represent strict or inclusive inequalities, respectively

Absolute Value and Distance

• Absolute value of a real number $a$, denoted $|a|$, is the non-negative value of $a$ without regard to its sign
  • $|a| = a$ if $a \geq 0$, and $|a| = -a$ if $a < 0$
• Distance between two real numbers $a$ and $b$ on the real number line is given by $|a - b|$
• Triangle inequality: For any real numbers $a$ and $b$, $|a + b| \leq |a| + |b|$
• Absolute value equations can have zero, one, or two solutions depending on the structure of the equation
• Absolute value inequalities can be solved by considering the cases where the expression inside the absolute value is positive or negative
• Absolute value is used in various applications, such as finding the magnitude of a number, the distance between points, or the error in measurements

Real Number Line and Intervals

• Real number line is a visual representation of the set of real numbers, with each point on the line corresponding to a unique real number
• Positive numbers are located to the right of zero, while negative numbers are located to the left of zero
• Intervals are subsets of the real number line, representing a range of real numbers
• Notation for intervals:
  • $(a, b)$ represents an open interval, which does not include the endpoints $a$ and $b$
  • $[a, b]$ represents a closed interval, which includes the endpoints $a$ and $b$
  • $(a, b]$ and $[a, b)$ represent half-open intervals, which include one endpoint but not the other
• Unbounded intervals, such as $(a, \infty)$ or $(-\infty, b)$, represent all real numbers greater than $a$ or less than $b$, respectively
• Union and intersection of intervals follow the properties of set operations

Applications and Problem-Solving

• Real numbers are used in a wide range of mathematical and real-world applications
• Solving equations and inequalities involves applying the properties of real numbers to isolate the variable
  • Linear equations and inequalities (one variable, degree one)
  • Quadratic equations and inequalities (one variable, degree two)
  • Systems of linear equations (multiple variables, degree one)
• Modeling real-world situations using real numbers and algebraic expressions
  • Representing quantities, measurements, and relationships between variables
  • Formulating equations or inequalities to solve problems
• Analyzing functions and their properties using real numbers
  • Domain and range of functions as subsets of the real numbers
  • Continuity and differentiability of functions defined on the real numbers
• Approximating irrational numbers and using them in computations
  • Decimal approximations of irrational numbers ($\pi \approx 3.14159$, $\sqrt{2} \approx 1.41421$)
  • Rational approximations of irrational numbers (continued fractions, convergents)