Fundamentals of Math Analysis Course Outline - Week 1 PDF

**Fundamentals of Math Analysis Course Outline - Week 1** Real Numbers ------------ ### Elementary Properties of Real Numbers 1. *Definition of Real Numbers*: Real numbers include (ℝ) all the numbers on the number line, encompassing rational and irrational numbers. - **Rational Nu...

**Fundamentals of Math Analysis Course Outline - Week 1** Real Numbers ------------ ### Elementary Properties of Real Numbers 1. *Definition of Real Numbers*: Real numbers include (ℝ) all the numbers on the number line, encompassing rational and irrational numbers. - **Rational Numbers**: Numbers that can be expressed as the quotient of two integers (e.g., , 3, −4, −0.75). - **Irrational Numbers**: Numbers that cannot be expressed as a simple fraction (e.g., 𝜋,  2. *Specific Examples of Real Numbers*: - 2 (an integer (ℤ), rational number (ℚ)) o −5 (an integer (ℤ), rational number (ℚ)) o 0 (an integer (ℤ), rational number (ℚ)) ~o~ (a rational number (ℚ)) o 0.5 (a rational number (ℚ)) o (an irrational number (ℝ\\ℚ)) - 𝜋 (an irrational number (ℝ\\ℚ)) - 𝑒 (an irrational number (ℝ\\ℚ)) 3. *Complex Numbers*: Complex numbers (ℂ) include all numbers of the form 𝑎 + 𝑏𝑖, where 𝑎 and 𝑏 are real numbers, and 𝑖 is the imaginary unit with the property that 𝑖^2^ = −1. 4. *Properties of Real Numbers*: - **Commutative Property**: The order in which two numbers are added or multiplied does not change their sum or product. - **Addition**: 𝑎 + 𝑏 = 𝑏 + 𝑎 - **Multiplication**: 𝑎 ⋅ 𝑏 = 𝑏 ⋅ 𝑎 - **Associative Property**: The way in which numbers are grouped in addition or multiplication does not change their sum or product. - **Addition**: (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐) - **Multiplication**: (𝑎 ⋅ 𝑏) ⋅ 𝑐 = 𝑎 ⋅ (𝑏 ⋅ 𝑐) - **Distributive Property**: The sum of two numbers times a third number is equal to the sum of each addend times the third number. - 𝑎 ⋅ (𝑏 + 𝑐) = (𝑎 ⋅ 𝑏) + (𝑎 ⋅ 𝑐) - **Identity Property**: Adding 0 or multiplying by 1 leaves a number unchanged. - **Addition**: 𝑎 + 0 = 𝑎 - **Multiplication**: 𝑎 ⋅ 1 = 𝑎 - **Inverse Property**: Every number has an additive inverse *(opposite)* and a multiplicative inverse *(reciprocal)* that result in the identity for the operation. - **Addition**: 𝑎 + (−𝑎) = 0 - **Multiplication**: for 𝑎 ≠ 0 𝑎 - **Closure Property**: Performing an operation (addition or multiplication) on any two real numbers results in another real number. - **Addition**: The sum of any two real numbers is a real number. - **Multiplication**: The product of any two real numbers is a real number. ### Order of Operations 1. *Definition:* Order of operations is the sequence in which operations should be performed in a mathematical expression to ensure consistent and correct results. 2. *PEMDAS Rule*: An acronym to remember the order of operations: Parentheses, Exponents, Multiplication, and Division (left-to-right), Addition and Subtraction (left-to-right). o **P**arentheses o **E**xponents (including roots, such as square roots) ~o~ **M**ultiplication and **D**ivision (left-to-right) ~o~ **A**ddition and **S**ubtraction (left-to-right) Inequalities 1. *Definition:* Inequalities are mathematical statements that compare two values or expressions using inequality symbols. 2. *Symbols*: Inequality symbols are used to show the relative size of two values. - Greater than (\>) o Less than (\ 𝑏 and 𝑏 \> 𝑐, then 𝑎 \> 𝑐. - **Addition Property**: If 𝑎 \> 𝑏, then 𝑎 + 𝑐 \> 𝑏 + 𝑐. o **Multiplication Property**: If 𝑎 \> 𝑏 and 𝑐 \> 0, then 𝑎 ⋅ 𝑐 \> 𝑏 ⋅ 𝑐. ### Absolute Value - \|𝑥\| = 𝑥 if 𝑥 ≥ 0 o \|𝑥\| = −𝑥 if 𝑥 \< 0 - \|𝑎𝑏\| = \|𝑎\| ⋅ \|𝑏\| 𝑎 \|𝑎\| - \| \| = for 𝑏 ≠ 0 𝑏 \|𝑏\| ### Intervals and Bounded Sets 1. *Definition of Intervals*: An interval is a set of real numbers between two endpoints. 2. *Types of Intervals*: - **Closed Interval**: \[𝑎, 𝑏\] includes all numbers between 𝑎 and 𝑏, including the endpoints. - **Open Interval**: (𝑎, 𝑏) includes all numbers between 𝑎 and 𝑏, excluding the endpoints. - **Half-Open Interval**: \[𝑎, 𝑏) or (𝑎, 𝑏\] includes all numbers between 𝑎 and 𝑏, including only one endpoint. 3. *Bounded Sets*: - A set is bounded if it has both upper and lower bounds. - **Upper Bound**: A number 𝑀 is an upper bound of a set 𝑆 if 𝑥 ≤ 𝑀 for all 𝑥 ∈ 𝑆. - **Lower Bound**: A number 𝑚 is a lower bound of a set 𝑆 if 𝑚 ≤ 𝑥 for all 𝑥 ∈ 𝑆. **Fundamentals of Math Analysis Course Outline - Week 2** Algebraic Expressions --------------------- ### Polynomials 1. *Definition:* A polynomial is an algebraic expression composed of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. 2. *Form:* 𝑃(𝑥) = 𝑎~𝑛~𝑥^𝑛^ + 𝑎~𝑛−1~𝑥^𝑛−1^ + ⋯ + 𝑎~1~𝑥 + 𝑎~0~ 3. *Degree:* The highest power of the variable in the polynomial. 4. *Types of Polynomials*: ### Operations on Polynomials 1. *Addition*: Combine like terms. - *Example:* (2𝑥^2^ + 3𝑥 + 4) + (𝑥^2^ + 2𝑥 + 1) = 3𝑥^2^ + 5𝑥 + 5 2. *Subtraction*: Distribute the negative sign and combine like terms. - *Example:* (2𝑥^2^ + 3𝑥 + 4) − (𝑥^2^ + 2𝑥 + 1) = 𝑥^2^ + 𝑥 + 3 3. *Multiplication*: Use distributive property. - *Example:* (2𝑥 + 3)(𝑥 + 1) = 2𝑥^2^ + 5𝑥 + 3 4. *Division:* Polynomial long division or synthetic division. ### Special Products ### Factoring Polynomials 1. *Factoring by Grouping*: Group terms to factor out common factors. 2. *Factoring Quadratic Polynomials*: 𝑎𝑥^2^ + 𝑏𝑥 + 𝑐 ~o~ Find two numbers that multiply to 𝑎𝑐 and add to 𝑏. o *Example:* 𝑥^2^ + 5𝑥 + 6 = (𝑥 + 2)(𝑥 + 3) 3. *Factoring Perfect Square Trinomials*: 𝑎^2^ + 2𝑎𝑏 + 𝑏^2^ = (𝑎 + 𝑏)^2^ 4. *Factoring Difference of Squares*: 𝑎^2^ − 𝑏^2^ = (𝑎 + 𝑏)(𝑎 − 𝑏) Rational Expressions -------------------- ### Rational Expressions ### Simplification of Rational Expressions ### Negative Integral Exponents ### Operations on Rational Expressions 1. *Addition and Subtraction*: Find a common denominator, combine the numerators, and then simplify. 2. *Multiplication and Division*: Multiply or divide the numerators and denominators, then simplify. 𝑏 𝑑 𝑏𝑑 𝑏 𝑑 𝑏 𝑐 𝑏𝑐 ### Complex Fractions 1. *Definition*: A complex fraction is a fraction where the numerator, the denominator, or both, contain fractions. 2. *Simplification*: Simplify by finding a common denominator for the fractions within the numerator and the denominator, then perform the division. 𝑏 𝑐 𝑏𝑐 Radical Expressions ------------------- ### Radical Expressions ### Fractional or Rational Exponents ### Rules Governing Fractional Exponents - 𝑎𝑛 = ^𝑛^√𝑎^𝑚^ 𝑚 𝑞 𝑚𝑝 - (𝑎𝑛) = 𝑎^𝑛𝑞^ 𝑚 𝑝 𝑚𝑞+𝑛𝑝 - 𝑎 𝑎 𝑛𝑞 ### Simplification of Radicals ### Operations on Radicals 1. *Addition and Subtraction*: Combine like radicals. - *Example:*. 2. *Multiplication*: If possible, use the distributive property and combine under the same radical. - *Example:* . 3. *Division*: Rationalize the denominator if necessary.