Algebra Basics

Algebra

• Equations and Inequalities:
  • Linear equations: ax + by = c, where a, b, and c are constants
  • Quadratic equations: ax^2 + bx + c = 0, where a, b, and c are constants
  • Systems of linear equations: solving multiple equations simultaneously
• Functions:
  • Domain and range: input and output values of a function
  • Composition of functions: combining two or more functions
  • Inverse functions: reversing the input and output values
• Graphing:
  • Cartesian coordinates: plotting points on a grid
  • Graphing linear and quadratic functions

Geometry

• Points, Lines, and Planes:
  • Midpoint and distance formulas
  • Slope and intercept of a line
  • Angle relationships: complementary, supplementary, and corresponding angles
• Triangles:
  • Angle sum property: sum of interior angles is 180°
  • Pythagorean theorem: a^2 + b^2 = c^2 for right triangles
  • Congruent and similar triangles
• Polygons and Circles:
  • Properties of regular polygons: number of sides, interior and exterior angles
  • Circumference and area of circles

Number Theory

• Divisibility:
  • Factors and multiples: prime and composite numbers
  • Greatest common divisor (GCD) and least common multiple (LCM)
  • Divisibility rules: testing for divisibility by 2, 3, 4, etc.
• Prime Numbers:
  • Properties of prime numbers: uniqueness, distribution, and testing
  • Prime factorization: expressing a number as a product of prime factors
• Modular Arithmetic:
  • Congruences: equivalence relations modulo n
  • Properties of modular arithmetic: addition, multiplication, and inverses

Combinatorics

• Counting Principles:
  • Fundamental counting principle: product rule for counting
  • Permutations: arranging objects in a specific order
  • Combinations: selecting objects without regard to order
• Recurrence Relations:
  • Defining and solving recurrence relations
  • Applications to counting and probability
• Generating Functions:
  • Defining and applying generating functions
  • Extracting coefficients and solving recurrence relations

Vector

• Operations:
  • Vector addition and scalar multiplication
  • Dot product and cross product
• Properties:
  • Magnitude and direction of vectors
  • Unit vectors and normalization
• Applications:
  • Vector projections and decompositions
  • Work and energy: force and displacement vectors

Algebra

• Equations and Inequalities:
  • Linear equations have the form ax + by = c, where a, b, and c are constants.
  • Quadratic equations have the form ax^2 + bx + c = 0, where a, b, and c are constants.
  • Systems of linear equations involve solving multiple equations simultaneously.
• Functions:
  • Domain and range refer to the input and output values of a function.
  • Composition of functions involves combining two or more functions.
  • Inverse functions reverse the input and output values.

Geometry

• Points, Lines, and Planes:
  • Midpoint formula calculates the midpoint of a line segment.
  • Distance formula calculates the distance between two points.
  • Slope and intercept of a line can be used to graph it.
  • Angle relationships include complementary, supplementary, and corresponding angles.
• Triangles:
  • Angle sum property states that the sum of interior angles is 180°.
  • Pythagorean theorem states that a^2 + b^2 = c^2 for right triangles.
  • Congruent triangles have equal sides and angles, while similar triangles have proportional sides.
• Polygons and Circles:
  • Regular polygons have a fixed number of sides and interior and exterior angles.
  • Circumference and area of circles can be calculated using formulas.

Number Theory

• Divisibility:
  • Factors and multiples are prime or composite numbers.
  • Greatest common divisor (GCD) and least common multiple (LCM) can be calculated.
  • Divisibility rules test for divisibility by 2, 3, 4, etc.
• Prime Numbers:
  • Prime numbers have unique properties, such as distribution and testing.
  • Prime factorization expresses a number as a product of prime factors.
• Modular Arithmetic:
  • Congruences are equivalence relations modulo n.
  • Modular arithmetic has properties, such as addition, multiplication, and inverses.

Combinatorics

• Counting Principles:
  • Fundamental counting principle uses the product rule for counting.
  • Permutations arrange objects in a specific order.
  • Combinations select objects without regard to order.
• Recurrence Relations:
  • Defining and solving recurrence relations is used in counting and probability.
  • Applications include counting and probability.
• Generating Functions:
  • Defining and applying generating functions is used in counting and probability.
  • Extracting coefficients and solving recurrence relations is possible.

Vector

• Operations:
  • Vector addition and scalar multiplication are used to combine vectors.
  • Dot product and cross product are used to calculate vector quantities.
• Properties:
  • Magnitude and direction of vectors are important properties.
  • Unit vectors and normalization are used to simplify vectors.
• Applications:
  • Vector projections and decompositions are used in physics and engineering.
  • Work and energy involve force and displacement vectors.