Overview
This lecture introduces the foundations of algebraic expressions, mathematical models, the set of real numbers, and set notation. It covers variables, exponents, order of operations, evaluating expressions, equations, formulas, mathematical modeling, and the classification and operations of sets and numbers.
Variables and Exponential Notation
- A variable is a letter (commonly x or y, but any letter can be used) that represents different numbers.
- Exponential notation: ( B^N )
- B is the base; N is the exponent (or power).
- ( B^N ) means multiplying B by itself N times.
- Read as "B to the Nth power" or "B to the N."
- Special cases:
- ( B^0 = 1 ) for any B ≠0.
- ( B^1 = B ).
- ( B^2 ) is "B squared."
- ( B^3 ) is "B cubed."
- Examples:
- ( 7^0 = 1 )
- ( 9^1 = 9 )
- ( 8^2 = 64 ) ("8 squared")
- ( 5^3 = 125 ) ("5 cubed")
- ( 2^4 = 16 )
Algebraic Expressions and Order of Operations
- An algebraic expression is a mathematical phrase with numbers, variables, and operations (addition, subtraction, multiplication, division, exponents, roots) but no equal sign.
- Examples: ( 3x + 6 ), ( x^2 - 3 )
- To evaluate an expression, substitute values for variables and use the correct order of operations.
- Order of operations (PEMDAS):
- Parentheses (and brackets)
- Exponents
- Multiplication/Division (left to right)
- Addition/Subtraction (left to right)
- Multiplication and division are performed in the order they appear from left to right, as are addition and subtraction.
Evaluating Expressions with Exponents
- Substitute the given value for the variable, then follow PEMDAS.
- Negative numbers and exponents:
- A negative number raised to an even exponent gives a positive result.
- A negative number raised to an odd exponent gives a negative result.
- Parentheses are essential: ( (-2)^2 = 4 ), but ( -2^2 = -4 ).
- Examples:
- ( x^2 ) for ( x = -2 ): ( (-2)^2 = 4 )
- ( x^3 ) for ( x = -3 ): ( (-3)^3 = -27 )
- ( x^5 ) for ( x = -1 ): ( (-1)^5 = -1 )
- ( x^4 ) for ( x = -2 ): ( (-2)^4 = 16 )
- ( x + 4^2 - 3 ) for ( x = -5 ): ( (-5 + 4)^2 - 3 = (-1)^2 - 3 = 1 - 3 = -2 )
Equations and Mathematical Models
- An equation is formed by placing an equal sign between two algebraic expressions (e.g., ( 2x - 3 = 4x^3 )).
- A formula is an equation that uses variables to represent relationships between quantities.
- Example: ( F = \frac{9}{5}C + 32 ) (converts Celsius to Fahrenheit)
- Can be rearranged to solve for any variable.
- Mathematical modeling uses formulas to describe real-world situations.
- Example: ( T = 4x^2 + 330x + 3310 ) models average tuition cost, where x is years after 2000.
- To find tuition for 2010: plug in ( x = 10 ).
Sets and Set Notation
- A set is a collection of distinct objects, called elements, written inside braces { }.
- Roster method: lists all elements (e.g., ( {1, 2, 3, 4} )).
- Set-builder notation: describes elements (e.g., ( {x \mid x ) is a counting number less than 6}).
- Intersection (( A \cap B )): elements common to both sets.
- Example: ( {7, 8, 9, 10, 11} \cap {6, 8, 10, 12} = {8, 10} )
- Union (( A \cup B )): all elements in either set, without repetition.
- Example: ( {7, 8, 9, 10, 11} \cup {6, 8, 10, 12} = {6, 7, 8, 9, 10, 11, 12} )
- Empty set (( \emptyset )): a set with no elements.
Types of Numbers
- Natural numbers (( \mathbb{N} )): ( {1, 2, 3, 4, ...} ) (counting numbers)
- Whole numbers (( \mathbb{W} )): ( {0, 1, 2, 3, ...} ) (natural numbers plus zero)
- Integers (( \mathbb{Z} )): ( {..., -3, -2, -1, 0, 1, 2, 3, ...} ) (positive and negative whole numbers)
- Rational numbers (( \mathbb{Q} )): numbers that can be written as a ratio of integers (( \frac{a}{b} ), where a and b are integers, ( b \neq 0 ))
- Includes terminating and repeating decimals.
- Examples: ( -5 = \frac{-5}{1} ), ( 2/3 = 0.6\overline{6} ), ( 0.4 = 2/5 )
- Irrational numbers: decimals that are non-terminating and non-repeating; cannot be written as a ratio of integers.
- Examples: ( \sqrt{2} \approx 1.4142... ), ( \pi \approx 3.1415... )
- Real numbers (( \mathbb{R} )): all rational and irrational numbers.
Real Number Line and Inequalities
- The real number line is a graphical representation of real numbers, with zero in the center, positive numbers to the right, and negative numbers to the left.
- Inequality symbols:
- ( < ): less than
- ( \leq ): less than or equal to
- ( > ): greater than
- ( \geq ): greater than or equal to
- The inequality sign opens toward the larger number.
- Examples:
- ( -4 < 2 )
- ( -3 \geq -7 )
- ( -5 \leq 0 )
- ( 9 \geq 9 )
Absolute Value
- Absolute value (( |x| )) is the distance from zero on the number line; always non-negative.
- ( |a| = a ) if ( a \geq 0 )
- ( |a| = -a ) if ( a < 0 )
- Think of absolute value as the magnitude or distance, regardless of direction.
- Examples:
- ( |3| = 3 )
- ( |-2| = 2 )
- ( -|5| = -5 )
- ( -| -7 | = -7 )
- ( |17 - 23| = | -6 | = 6 )
- ( |2 - \pi| = \pi - 2 ) (since ( 2 - \pi ) is negative)
- ( |\sqrt{3} - 1| = \sqrt{3} - 1 ) (since ( \sqrt{3} - 1 ) is positive)
Key Terms & Definitions
- Variable: a symbol representing a number in expressions or equations.
- Exponent: indicates how many times to multiply the base by itself.
- Algebraic Expression: a mathematical phrase with numbers, variables, and operations, but no equal sign.
- Equation: a statement showing equality between two expressions.
- Formula: an equation expressing a relationship among variables.
- Set: a collection of distinct elements.
- Intersection: elements common to two sets.
- Union: all elements from both sets, without repetition.
- Empty Set: a set containing no elements.
- Natural Numbers: counting numbers starting from 1.
- Whole Numbers: natural numbers plus zero.
- Integers: whole numbers and their negatives.
- Rational Numbers: numbers that can be written as a fraction of integers.
- Irrational Numbers: numbers not expressible as a fraction; non-repeating, non-terminating decimals.
- Real Numbers: all rational and irrational numbers.
- Absolute Value: the distance from zero on the number line.
Action Items / Next Steps
- Practice evaluating expressions using PEMDAS and substitution for variables.
- Convert between roster and set-builder notation for sets.
- Identify and classify numbers as natural, whole, integer, rational, or irrational.
- Solve problems involving set intersection and union, and recognize the empty set.
- Review and memorize the hierarchy and symbols for number sets.
- Practice using the real number line and inequalities to compare values.
- Work with absolute value in expressions and equations, including simplifying and evaluating.