Fundamentals of Mathematics

• Mathematics encompasses a broad field of study, including arithmetic, algebra, calculus, geometry, and trigonometry, among others.

Arithmetic

• Arithmetic deals with basic operations on numbers, including addition, subtraction, multiplication, and division.
• Integers are whole numbers, which can be positive, negative, or zero.
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q is not zero.
• Irrational numbers cannot be expressed as a fraction and have non-repeating, non-terminating decimal representations (e.g., π, √2).
• Real numbers include both rational and irrational numbers.
• Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).

Algebra

• Algebra involves the use of variables and symbols to represent numbers and quantities in mathematical expressions and equations.
• 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, generally expressed as ax² + bx + c = 0.
• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Factoring involves breaking down a polynomial into simpler terms or expressions that, when multiplied together, give the original polynomial.

Calculus

• Calculus is the study of continuous change, divided into differential calculus and integral calculus.
• Differential calculus deals with rates of change and slopes of curves.
• Derivatives measure the instantaneous rate of change of a function.
• Integral calculus deals with the accumulation of quantities and the areas under and between curves.
• Integration involves finding the integral of a function, which represents the area under the curve of the function.
• Limits describe the value that a function approaches as the input approaches some value.

Geometry

• Geometry deals with the properties and relations of points, lines, surfaces, and solids.
• Euclidean geometry is based on a set of axioms and theorems developed by Euclid.
• Analytical geometry uses algebraic techniques to study geometric shapes.
• Trigonometry studies the relationships between the angles and sides of triangles.
• Triangles are polygons with three sides and three angles.
• Circles are sets of points equidistant from a center point.
• Polygons are closed figures formed by line segments.
• Three-dimensional shapes include spheres, cubes, prisms, and pyramids.

Trigonometry

• Trigonometry is the study of relationships between angles and sides of triangles.
• Sine (sin), cosine (cos), and tangent (tan) are fundamental trigonometric functions that relate angles to ratios of sides in right triangles.
• The unit circle is a circle with a radius of 1, 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.

Statistics

• Statistics is the science of collecting, analyzing, presenting, and interpreting data.
• Descriptive statistics involves methods for summarizing and presenting data, such as mean, median, mode, standard deviation, and variance.
• Inferential statistics involves making inferences and generalizations about a population based on a sample of data.
• Probability is the measure of the likelihood that an event will occur.
• Distributions describe the probabilities of different outcomes in a population.

Logic

• Logic is the study of the principles of valid reasoning and inference.
• Propositional logic deals with propositions and logical connectives such as AND, OR, NOT, and IF-THEN.
• Predicate logic extends propositional logic to include predicates, quantifiers, and variables.
• Set theory is the branch of mathematical logic that studies sets, which are collections of objects.

Number Theory

• Number theory is a branch of mathematics that deals with the properties and relationships of numbers, especially integers.
• Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.
• Divisibility rules are shortcuts for determining whether one number is divisible by another.
• Modular arithmetic involves operations on integers with remainders after division by a fixed number (the modulus).
• Cryptography uses number theory to design secure encryption methods.

Mathematical Analysis

• Mathematical analysis is a branch of mathematics that deals with the concepts of limits, continuity, differentiation, and integration.
• Sequences are ordered lists of numbers, while series are sums of sequences.
• Convergence and divergence describe the behavior of sequences and series as they approach infinity.
• Real analysis focuses on the properties of real numbers and real-valued functions.
• Complex analysis studies the behavior of complex numbers and complex-valued functions.

Discrete Mathematics

• Discrete mathematics deals with mathematical structures that are fundamentally discrete rather than continuous.
• Combinatorics involves counting and arranging objects, including permutations and combinations.
• Graph theory studies the properties of graphs, which are mathematical structures used to model relationships between objects.
• Algorithms are step-by-step procedures for solving problems.
• Automata theory deals with abstract machines and their computational capabilities.

Explore the fundamental concepts of mathematics, including arithmetic and algebra. Arithmetic covers basic operations, number types and algebra introduces variables, equations, and functions. Grasp key principles in these mathematical areas.