Introduction to Algebra

• Math includes algebra, geometry, and statistics
• These fields solve problems using logic, reasoning, and quantitative methods

Algebra

• Algebra uses symbols to represent numbers and quantities in formulas and equations
• It helps problem-solve in science, engineering, economics, and computer science
• Algebraic expressions use variables (x, y) and constants (3, 2), like 3x + 2y
• Equations show equality between two expressions like 3x + 2 = 7
• Solving equations means finding variable values that make the equation true
• Linear equations have the form ax + b = 0, where a and b are constants
• Quadratic equations have the form ax^2 + bx + c = 0, where a, b, and c are constants
• Factoring, completing the square, or using the quadratic formula solves quadratic equations
• Systems of equations have multiple equations with multiple variables
• Solutions to systems of equations satisfy all equations at the same time
• Substitution, elimination, and matrix methods are ways to solve systems of equations
• Inequalities use symbols like , ≤, and ≥ to compare expressions
• Solving inequalities means finding the range of values for the variable(s) that satisfy the inequality
• Absolute value is the distance of a number from zero
• Absolute value equations and inequalities need different methods to handle different cases
• Polynomials have variables and coefficients, using addition, subtraction, multiplication, and non-negative integer exponents such as x^2 + 3x + 2 and 5x^4 - 2x + 1
• Factoring polynomials simplifies them into smaller expressions
• Rational expressions are fractions with polynomials in the numerator and denominator
• Simplifying rational expressions involves canceling common factors
• Exponents show repeated multiplication (e.g., 2^3 = 2 * 2 * 2 = 8)
• Exponential functions are in the form f(x) = a^x, where a is a constant
• Logarithms are the opposite of exponential functions, finding the exponent to get a value (e.g., log_2(8) = 3, because 2^3 = 8)
• Logarithmic functions are in the form f(x) = log_a(x), where a is the base of the logarithm

Geometry

• Geometry studies the properties and relationships of points, lines, surfaces, solids, and higher dimensions
• Euclidean geometry is based on axioms and postulates, including the parallel postulate
• Points are basic, dimensionless objects showing locations in space
• Lines are straight, one-dimensional, and extend infinitely in both directions
• Line segments are parts of a line between two endpoints
• Rays are parts of a line that extend infinitely in one direction from an endpoint
• Angles form when two rays share a common endpoint (vertex)
• Angles are measured in degrees or radians
• Angle types are acute (less than 90°), right (90°), obtuse (greater than 90°), straight (180°), and reflex (greater than 180°)
• Parallel lines don't intersect on a plane
• Perpendicular lines intersect at a right angle (90°)
• Triangles are polygons with three sides and three angles
• The angles in a triangle always add up to 180°
• Triangle types are equilateral (all sides equal), isosceles (two sides equal), scalene (no sides equal), right (one angle is 90°), acute (all angles less than 90°), and obtuse (one angle is greater than 90°)
• The Pythagorean theorem says that in a right triangle, a^2 + b^2 = c^2
• Quadrilaterals are polygons with four sides and four angles
• Quadrilateral types are squares, rectangles, parallelograms, trapezoids, and rhombuses
• Polygons are closed shapes formed by line segments
• Regular polygons have equal sides and angles
• Circles are points at the same distance from a central point
• The radius is the distance from the center to any point on the circle
• The diameter is twice the radius (d = 2r)
• The circumference is the distance around the circle (C = 2πr)
• The area of a circle is πr^2
• Solid geometry studies three-dimensional objects
• Common solids are cubes, spheres, cylinders, cones, and pyramids
• The volume of a solid is the amount of space it takes up
• The surface area of a solid is the total area of its surfaces
• Geometric transformations include translations (sliding), rotations (turning), reflections (flipping), and dilations (scaling)
• Congruent figures have the same shape and size
• Similar figures have the same shape but different sizes
• Trigonometry studies the relationships between the sides and angles of triangles
• Trigonometric functions include sine (sin), cosine (cos), and tangent (tan)

Statistics

• Statistics is the science of collecting, organizing, analyzing, interpreting, and presenting data
• Descriptive statistics summarizes and presents data
• Inferential statistics makes inferences and generalizations about a population based on a sample
• Population is the entire group of individuals or items being studied
• Sample is a subset of the population selected for analysis
• Variables are characteristics that can take on different values
• Variable types are categorical (qualitative) and numerical (quantitative)
• Categorical variables represent categories or labels (e.g., color, gender)
• Numerical variables represent quantities (e.g., height, temperature)
• Discrete variables can only take on specific values (e.g., number of children)
• Continuous variables can take on any value within a range (e.g., height, temperature)
• Measures of central tendency include mean (average), median (middle value), and mode (most frequent value)
• Measures of dispersion include range (difference between maximum and minimum values), variance (average squared deviation from the mean), and standard deviation (square root of the variance)
• Probability measures the likelihood that an event will occur
• Probability is a number between 0 and 1
• Random variables are variables whose values are determined by chance
• Probability distributions describe the probabilities of different values of a random variable
• Normal distribution is a bell-shaped distribution that is commonly found in nature
• Hypothesis testing evaluates evidence for or against a claim about a population
• Statistical significance means study results probably didn't happen by chance
• Correlation measures the strength and direction of the linear relationship between two variables
• Regression analysis models the relationship between a dependent variable and one or more independent variables
• Data visualization uses graphs and charts to represent data
• Common data visualization techniques are histograms, bar charts, scatter plots, and pie charts
• Statistical software packages perform statistical analysis and data visualization