Math: Arithmetic and Algebra

• Math is the science and study of quantity, structure, space, and change
• Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions

Arithmetic

• Arithmetic is the oldest and most elementary branch of mathematics
• It deals with basic operations on numbers: addition, subtraction, multiplication, and division
• The natural numbers (1, 2, 3, ...) form the basis of arithmetic
• Integers include natural numbers, zero, and negative numbers
• Rational numbers are numbers that can be expressed as a fraction a/b, where a and b are integers and b ≠ 0
• Real numbers include all rational numbers and irrational numbers (numbers that cannot be expressed as a fraction, like √2 or π)
• Complex numbers extend real numbers by including the imaginary unit i, defined as the square root of -1

Algebra

• Algebra is a branch of mathematics that generalizes arithmetic, using variables to represent numbers and express relationships
• Elementary algebra involves solving equations and inequalities with these variables
• Linear equations are algebraic equations where the highest power of the variable is 1
• Quadratic equations are algebraic equations where the highest power of the variable is 2
• Abstract algebra studies algebraic structures such as groups, rings, and fields
• Boolean algebra deals with logical operations and is fundamental to computer science

Geometry

• Geometry is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs
• Euclidean geometry is based on a set of axioms defined by Euclid, including concepts like points, lines, angles, and shapes
• Analytic geometry uses algebra to describe geometric shapes using a coordinate system
• Trigonometry studies the relationships between the angles and sides of triangles
• Differential geometry uses calculus to study the properties of curves and surfaces

Calculus

• Calculus is the study of continuous change
• Differential calculus deals with the rate of change of a function (its derivative)
• Integral calculus is concerned with the accumulation of quantities (its integral) and the areas under or between curves
• The fundamental theorem of calculus connects differentiation and integration
• Limits and continuity are foundational concepts for calculus

Trigonometry

• Trigonometry is the branch of mathematics that studies relationships between angles and sides of triangles
• Sine, cosine, and tangent are fundamental trigonometric functions, relating angles to ratios of sides in a right triangle
• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved
• The Law of Sines and Law of Cosines are used to solve triangles (find unknown angles and sides)

Statistics and Probability

• Statistics involves the collection, analysis, interpretation, presentation, and organization of data
• Descriptive statistics summarize and describe the characteristics of a data set using measures like mean, median, mode, and standard deviation
• Inferential statistics uses sample data to make inferences and predictions about a larger population
• Probability is the measure of the likelihood that an event will occur
• Probability theory provides the framework for analyzing random events
• Common probability distributions include the normal distribution, binomial distribution, and Poisson distribution