Introduction to Algebra

• Math encompasses a broad range of topics including algebra, geometry, calculus, trigonometry, and statistics.
• These areas share common principles but focus on distinct types of problems and methodologies.

Algebra

• Algebra deals with symbols and the rules for manipulating these symbols.
• It's a generalization of arithmetic, where letters and symbols represent numbers or quantities.
• Key concepts include variables, expressions, equations, and inequalities.
• Variables are symbols (usually letters) that represent unknown or changing values.
• Expressions are combinations of variables, numbers, and operations (like addition, subtraction, multiplication, and division).
• Equations are statements that two expressions are equal, and solving equations involves finding the value(s) of the variable(s) that make the equation true.
• Inequalities compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
• Linear equations are a fundamental topic, represented as equations of the form ax + b = c, where a, b, and c are constants, and x is the variable.
• Solving linear equations involves isolating the variable on one side of the equation.
• Quadratic equations are of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.
• Systems of equations involve two or more equations with the same variables.
• Solving a system of equations means finding values for the variables that satisfy all equations simultaneously.
• Common methods for solving systems of equations include substitution, elimination, and graphing.
• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, and multiplication, and non-negative integer exponents.
• Factoring polynomials involves breaking them down into simpler expressions that, when multiplied together, give the original polynomial.
• Rational expressions are fractions where the numerator and/or denominator are polynomials.
• Simplifying rational expressions involves factoring and canceling common factors.

Geometry

• Geometry is the study of shapes, sizes, positions, and properties of space.
• It encompasses both two-dimensional (plane) geometry and three-dimensional (solid) geometry.
• Key concepts include points, lines, angles, surfaces, and solids.
• Points are fundamental, representing a location in space.
• Lines are straight paths extending infinitely in both directions.
• Angles are formed by two rays (half-lines) sharing a common endpoint (vertex).
• Triangles are fundamental geometric shapes with three sides and three angles.
• The sum of the angles in a triangle is always 180 degrees.
• Common types of triangles include equilateral (all sides equal), isosceles (two sides equal), and right triangles (one angle is 90 degrees).
• The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (a² + b² = c²).
• Circles are sets of points equidistant from a central point.
• Key features of a circle include the radius (distance from the center to any point on the circle), diameter (distance across the circle through the center), circumference (distance around the circle), and area.
• Quadrilaterals are four-sided polygons.
• Common types of quadrilaterals include squares, rectangles, parallelograms, trapezoids, and rhombuses.
• Area is the measure of the two-dimensional space inside a shape, and volume is the measure of the three-dimensional space inside a solid.
• Coordinate geometry combines algebra and geometry by using a coordinate system to represent geometric shapes and solve geometric problems.
• Transformations involve changing the position, size, or orientation of geometric shapes.
• Common types of transformations include translations (sliding), rotations (turning), reflections (flipping), and dilations (scaling).

Calculus

• Calculus is the study of continuous change, and is divided into two main branches: differential calculus and integral calculus.
• Differential calculus deals with rates of change and slopes of curves.
• Key concepts include limits, derivatives, and differentiation rules.
• Limits describe the value that a function approaches as the input approaches some value.
• The derivative of a function measures the instantaneous rate of change of the function with respect to its input.
• Differentiation is the process of finding the derivative of a function.
• Integral calculus deals with accumulation of quantities and the areas under and between curves.
• Key concepts include antiderivatives, definite integrals, and indefinite integrals.
• The antiderivative of a function is a function whose derivative is the original function.
• Integration is the process of finding the antiderivative of a function.
• The definite integral of a function over an interval represents the net signed area between the function's graph and the x-axis.
• The fundamental theorem of calculus links differentiation and integration.
• It states that differentiation and integration are inverse operations.
• Applications of calculus include optimization (finding maximum or minimum values of functions), related rates (analyzing how rates of change of different variables are related), and finding areas and volumes.

Trigonometry

• Trigonometry is the study of the relationships between angles and sides of triangles.
• It is primarily concerned with right triangles and the trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
• Sine (sin) of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.
• Cosine (cos) of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.
• Tangent (tan) of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
• The reciprocal trigonometric functions are: cosecant (csc = 1/sin), secant (sec = 1/cos), and cotangent (cot = 1/tan).
• The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane, used to define trigonometric functions for all real numbers.
• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which the functions are defined.
• The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles.
• The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
• Applications of trigonometry include solving triangles (finding unknown sides and angles), modeling periodic phenomena (like sound waves and oscillations), and navigation.

Statistics

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• It involves methods for summarizing and describing data (descriptive statistics) and for making inferences and generalizations about populations based on samples (inferential statistics).
• Descriptive statistics involve methods for organizing, summarizing, and presenting data in a meaningful way.
• Key concepts include measures of central tendency (mean, median, mode), measures of dispersion (range, variance, standard deviation), and graphical representations (histograms, bar charts, pie charts, scatter plots).
• The mean is the average of a set of numbers.
• The median is the middle value in a sorted set of numbers.
• The mode is the value that appears most frequently in a set of numbers.
• The range is the difference between the largest and smallest values in a set of numbers.
• Variance and standard deviation measure the spread or variability of data around the mean.
• Inferential statistics involve methods for making inferences and generalizations about populations based on samples.
• Key concepts include probability, hypothesis testing, confidence intervals, and regression analysis.
• Probability is the measure of the likelihood that an event will occur.
• Hypothesis testing is a procedure for determining whether there is enough statistical evidence to reject a null hypothesis.
• Confidence intervals provide a range of values that is likely to contain the true population parameter.
• Regression analysis is a method for modeling the relationship between a dependent variable and one or more independent variables.
• Common types of regression include linear regression and multiple regression.
• Random variables are variables whose values are numerical outcomes of a random phenomenon.
• Probability distributions describe the probability of each possible value of a random variable.
• Common probability distributions include the normal distribution, the binomial distribution, and the Poisson distribution.