Introduction to Algebra

• Math is the abstract science of number, quantity, and space.
• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
• Trigonometry is the branch of mathematics dealing with the relationships between the sides and angles of triangles.
• Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Statistics is the science of collecting, analyzing, presenting, and interpreting data.
• Calculus is the branch of mathematics dealing with continuous change.

Algebra

• Algebra uses letters or symbols to represent numbers and quantities in formulas and equations.
• Algebraic expressions consist of numbers, variables, and operations.
• Equations in algebra are statements that two expressions are equal.
• Solving equations involves finding the value(s) of the variable(s) that make the equation true.
• Linear equations are equations in which the highest power of the variable is 1.
• Quadratic equations are equations in which the highest power of the variable is 2.
• Systems of equations involve two or more equations with the same variables.
• Inequalities are mathematical statements that compare two expressions using symbols like <, >, ≤, or ≥.
• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Factoring is the process of breaking down a polynomial into simpler terms or expressions that, when multiplied together, give the original polynomial.

Trigonometry

• Trigonometry focuses on the relationships between angles and sides of triangles.
• Right triangles have one angle that measures 90 degrees.
• The sides of a right triangle are referred to as the hypotenuse, opposite, and adjacent sides, relative to a given angle.
• Trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) relate the angles of a right triangle to the ratios of its sides.
• Sine (sin) is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) is the ratio of the opposite side to the adjacent side.
• Trigonometric identities are equations that are true for all values of the variables.
• The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).
• The unit circle is a circle with a radius of 1, used to define trigonometric functions for all real numbers.
• Trigonometric functions can be graphed and analyzed, showing periodic behavior.

Geometry

• Geometry deals with the properties and relationships of points, lines, angles, surfaces, and solids.
• Euclidean geometry is based on a set of axioms and postulates defined by Euclid.
• Points are locations in space that have no dimension.
• Lines are straight paths that extend infinitely in both directions.
• Line segments are parts of a line with two endpoints.
• Rays are parts of a line with one endpoint, extending infinitely in one direction.
• Angles are formed by two rays sharing a common endpoint (vertex).
• Triangles are polygons with three sides and three angles.
• Quadrilaterals are polygons with four sides and four angles.
• Circles are sets of points equidistant from a center point.
• Area is the measure of the two-dimensional space inside a closed figure.
• Volume is the measure of the three-dimensional space occupied by a solid.
• Congruence refers to figures that have the same shape and size.
• Similarity refers to figures that have the same shape but different sizes.

Statistics

• Statistics involves collecting, organizing, analyzing, interpreting, and presenting data.
• Descriptive statistics summarize and describe the characteristics of a data set.
• Inferential statistics use sample data to draw conclusions about a larger population.
• Populations are the entire group of individuals or items being studied.
• Samples are subsets of the population selected for analysis.
• Variables are characteristics or attributes that can be measured or observed.
• Data can be qualitative (categorical) or quantitative (numerical).
• Mean is the average of a set of numbers.
• Median is the middle value in a sorted set of numbers.
• Mode is the value that appears most frequently in a set of numbers.
• Standard deviation measures the spread or dispersion of data around the mean.
• Probability is the measure of the likelihood that an event will occur.
• Statistical hypothesis testing is a method for making decisions based on data.
• Regression analysis is a statistical technique for modeling the relationship between variables.

Calculus

• Calculus is the study of continuous change, covering limits, derivatives, integrals, and infinite series.
• Differential calculus deals with the rate of change of functions.
• Integral calculus deals with the accumulation of quantities.
• Limits describe the behavior of a function as the input approaches a certain value.
• Derivatives measure the instantaneous rate of change of a function.
• Integrals calculate the area under a curve.
• The fundamental theorem of calculus relates differentiation and integration, showing they are inverse operations.
• Applications of calculus include optimization, related rates, and modeling physical phenomena.
• Sequences are ordered lists of numbers.
• Series are the sum of the terms in a sequence.
• Infinite series are series with an infinite number of terms.