Introduction to Core Mathematics

• Mathematics is the abstract science of number, quantity, and space
• Mathematics can be studied in its own right (pure mathematics) or as it is applied to other disciplines like physics and engineering (applied mathematics)
• Mathematics is an essential tool used globally in many fields, including natural science, engineering, medicine, finance, and social sciences

Core Areas of Mathematics

• Arithmetic studies numbers and operations like addition, subtraction, multiplication, and division
• Algebra studies mathematical symbols and the rules for manipulating them
• Geometry studies shapes, sizes, positions of figures, and the properties of space
• Trigonometry studies relationships between angles and sides of triangles
• Calculus studies continuous change, dealing with rates of change and accumulation

Number Systems

• Natural numbers (1, 2, 3, ...) are used for counting
• Integers (... -2, -1, 0, 1, 2, ...) include natural numbers, zero, and negative numbers
• Rational numbers can be expressed as a ratio of two integers (e.g., 1/2, -3/4)
• Real numbers include rational and irrational numbers (those that cannot be expressed as a ratio of two integers, like √2, π)
• Complex numbers have a real and an imaginary part (of the form a + bi, where i is the imaginary unit, √-1)

Arithmetic Operations

• Addition (+) combines two numbers to find their sum
• Subtraction (-) finds the difference between two numbers
• Multiplication (× or *) combines two numbers to find their product
• Division (÷ or /) finds how many times one number is contained in another
• Exponentiation raises a number to a power (e.g., 2^3 = 8)
• Root extraction finds a number that, when raised to a power, equals a given number (e.g., √9 = 3)

Algebraic Concepts

• Variables are symbols representing unknown or changing quantities
• Expressions are combinations of variables, numbers, and operations
• Equations are statements that two expressions are equal
• Solving equations involves finding the values of variables that make the equation true
• Functions are mathematical relationships that map inputs to outputs
• Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents

Geometric Concepts

• Points are locations in space
• Lines are straight paths extending infinitely in both directions
• Planes are flat surfaces extending infinitely in all directions
• Angles are formed by two lines or rays sharing a common endpoint
• Triangles are three-sided polygons
• Circles are sets of points equidistant from a center point
• Polygons are closed figures formed by line segments

Trigonometric Functions

• Sine (sin), cosine (cos), and tangent (tan) are functions relating angles of a right triangle to ratios of its sides
• Sine is the ratio of the opposite side to the hypotenuse
• Cosine is the ratio of the adjacent side to the hypotenuse
• Tangent is the ratio of the opposite side to the adjacent side
• These functions are extensively used in navigation, physics, and engineering

Calculus Concepts

• Limits describe the behavior of a function as its input approaches a certain value
• Derivatives measure the instantaneous rate of change of a function
• Integrals calculate the area under a curve, and can be used to find accumulation
• Differential equations are equations involving derivatives, used to model various phenomena in science and engineering
• Sequences are ordered lists of numbers
• Series are sums of terms in a sequence

Mathematical Logic

• Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics
• Propositional logic deals with propositions (statements that are either true or false) and logical connectives (e.g., AND, OR, NOT, IF...THEN)
• Predicate logic extends propositional logic by introducing predicates (statements about variables) and quantifiers (e.g., FOR ALL, THERE EXISTS)
• Proofs are arguments demonstrating the truth of a mathematical statement
• Common proof techniques include direct proof, proof by contradiction, and proof by induction

Statistics and Probability

• Statistics is the science of collecting, analyzing, interpreting, and presenting data
• Probability is the measure of the likelihood that an event will occur
• Descriptive statistics summarize and describe the main features of a dataset such as mean, median, mode, standard deviation
• Inferential statistics uses sample data to make inferences about larger populations
• Random variables are variables whose values are numerical outcomes of a random phenomenon
• Probability distributions describe the likelihood of different values of a random variable

Discrete Mathematics

• Discrete mathematics studies mathematical structures that are fundamentally discrete rather than continuous
• Set theory studies sets, which are collections of objects
• Combinatorics deals with counting and arranging objects
• Graph theory studies graphs, which are mathematical structures used to model relationships between objects
• Number theory studies properties of integers

Applications of Mathematics

• Physics uses mathematics to model and explain natural phenomena
• Engineering uses mathematics to design and analyze systems and structures
• Computer science relies on mathematical concepts such as algorithms, data structures, and logic
• Economics and finance use mathematical models to analyze markets and make predictions
• Cryptography uses mathematical techniques to secure communications