Introduction to Mathematics: Arithmetic and Algebra

• Mathematics encompasses a vast and interconnected body of knowledge, including arithmetic, algebra, calculus, geometry, and statistics.
• It provides essential tools for problem-solving, logical reasoning, and quantitative analysis across diverse fields.

Arithmetic

• Arithmetic deals with basic operations: addition, subtraction, multiplication, and division.
• Number systems include integers, rational numbers (fractions), real numbers, and complex numbers.
• Order of operations (PEMDAS/BODMAS) dictates the sequence of calculations: parentheses, exponents, multiplication and division, addition and subtraction.
• Prime numbers are only divisible by 1 and themselves; composite numbers have more than two factors.
• The greatest common divisor (GCD) is the largest number that divides two or more integers without a remainder.
• The least common multiple (LCM) is the smallest number that is a multiple of two or more integers.

Algebra

• Algebra uses symbols (variables) to represent unknown quantities and relationships.
• Linear equations involve variables raised to the first power; solving them involves isolating the variable.
• Quadratic equations involve variables raised to the second power; they 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; solutions satisfy all equations simultaneously.
• Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication.
• Factoring polynomials involves expressing them as a product of simpler polynomials.
• Functions are relations that assign a unique output to each input; they can be represented graphically or algebraically.
• Domain of a function is the set of allowable inputs; the range is the set of corresponding outputs.
• Exponential functions involve a constant base raised to a variable exponent and model growth or decay.
• Logarithmic functions are the inverses of exponential functions and are used to solve exponential equations.

Geometry

• Geometry deals with the properties and relationships of points, lines, angles, surfaces, and solids.
• Basic shapes include triangles, squares, circles, cubes, spheres, pyramids, etc.
• Euclidean geometry is based on postulates by Euclid, including parallel postulate.
• Angles are measured in degrees or radians; complementary angles add up to 90 degrees, supplementary angles add up to 180 degrees.
• Triangles can be classified by their angles (acute, obtuse, right) or sides (equilateral, isosceles, scalene).
• The Pythagorean theorem relates the sides of a right triangle: a² + b² = c², where c is the hypotenuse.
• Area is the measure of the surface of a two-dimensional shape; perimeter is the distance around the boundary.
• Volume is the measure of the space occupied by a three-dimensional object; surface area is the total area of its surfaces.
• Coordinate geometry (analytic geometry) uses coordinates to represent geometric shapes and relationships.
• Transformations include translations, rotations, reflections, and dilations.

Calculus

• Calculus deals with rates of change and accumulation.
• Differential calculus focuses on derivatives, which measure the instantaneous rate of change of a function.
• Integral calculus focuses on integrals, which measure the accumulation of a quantity over an interval.
• Limits describe the behavior of a function as its input approaches a certain value.
• Derivatives are used to find maxima and minima of functions, as well as points of inflection.
• Integrals are used to calculate areas under curves, volumes of solids, and work done by a force.
• The fundamental theorem of calculus connects differentiation and integration.
• Techniques of integration include substitution, integration by parts, and partial fractions.
• Applications of calculus include optimization, related rates, and differential equations.

Statistics

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Descriptive statistics summarize and describe the main features of a dataset.
• Measures of central tendency include mean (average), median (middle value), and mode (most frequent value).
• Measures of dispersion include range (difference between max and min), variance, and standard deviation.
• Probability is the measure of the likelihood of an event occurring.
• Probability distributions describe the probabilities of different outcomes in a random experiment (e.g., normal distribution, binomial distribution).
• Inferential statistics uses sample data to draw conclusions about a population.
• Hypothesis testing involves formulating a null hypothesis and testing it against an alternative hypothesis.
• Regression analysis involves finding the relationship between a dependent variable and one or more independent variables.
• Correlation measures the strength and direction of a linear relationship between two variables.

Discrete Mathematics

• Discrete mathematics studies mathematical structures that are fundamentally discrete rather than continuous.
• Logic deals with reasoning and argumentation, including propositional logic and predicate logic.
• Set theory studies collections of objects (sets) and their properties.
• Relations define connections between elements of sets.
• Functions map elements from one set to another.
• Combinatorics deals with counting and arrangements of objects.
• Graph theory studies graphs, which are mathematical structures used to model pairwise relations between objects.
• Algorithms are step-by-step procedures for solving problems.

Trigonometry

• Trigonometry studies the relationships between angles and sides of triangles.
• Trigonometric functions include sine, cosine, tangent, cotangent, secant, and cosecant.
• These functions relate angles to ratios of sides in a right triangle.
• The unit circle provides a visual representation of trigonometric functions for all angles.
• Trigonometric identities are equations that are true for all values of the variables.
• The law of sines and the law of cosines relate the sides and angles of any triangle.
• Trigonometric equations involve finding the angles that satisfy a given equation.
• Applications of trigonometry include navigation, surveying, and physics.

Number Theory

• Number theory studies the properties of integers.
• Divisibility rules provide shortcuts for determining whether a number is divisible by another number.
• Prime factorization involves expressing a number as a product of prime numbers.
• Congruences relate integers that have the same remainder when divided by a given number.
• Diophantine equations are equations where the solutions are required to be integers.
• Cryptography utilizes number theory principles to create secure communication methods.