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 is the oldest and most elementary branch of mathematics.
It deals with basic operations on numbers, namely addition, subtraction, multiplication, and division.
Arithmetic provides the foundation for more advanced topics like algebra and calculus.
The properties of operations such as commutativity, associativity, and distributivity are fundamental in arithmetic.
Understanding number systems, including integers, rational numbers, and real numbers, is essential in arithmetic.
Fractions, decimals, and percentages are key concepts in arithmetic, along with ratio and proportion.

Algebra is a branch of mathematics that generalizes arithmetic, using variables to represent numbers and relationships.
It involves solving equations and inequalities to find unknown values.
Algebraic expressions consist of variables, constants, and mathematical operations.
Polynomials are a fundamental type of algebraic expression, involving sums of terms with variables raised to non-negative integer powers.
Linear equations and systems of linear equations are a core topic in algebra, with applications in various fields.
Quadratic equations and their solutions, including factoring, completing the square, and using the quadratic formula, are important.
Functions, which describe relationships between variables, are a central concept in algebra, with various types like linear, quadratic, and exponential functions.

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
Euclidean geometry, based on Euclid's axioms, deals with shapes like triangles, circles, and polyhedra in a two-dimensional or three-dimensional space.
Coordinate geometry, also known as analytic geometry, uses algebra to describe geometric shapes and solve geometric problems.
Trigonometry studies the relationships between angles and sides of triangles, using trigonometric functions like sine, cosine, and tangent.
Transformations, such as translations, rotations, and reflections, are studied in geometry to understand how shapes can be manipulated.
Concepts like area, volume, perimeter, and surface area are important for measuring geometric shapes.

Calculus is the branch of mathematics that deals with continuous change.
It is divided into two main branches: differential calculus and integral calculus.
Differential calculus focuses on the rate of change of functions and the concept of the derivative.
Integral calculus deals with the accumulation of quantities and the concept of the integral.
Limits are a fundamental concept in calculus, providing the foundation for derivatives and integrals.
Derivatives are used to find slopes of curves, optimization problems, and rates of change in various applications.
Integrals are used to find areas under curves, volumes of solids, and to solve differential equations.
The fundamental theorem of calculus connects differentiation and integration, showing they are inverse operations.

Trigonometry is the study of relationships between angles and sides of triangles.
The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan).
These functions relate angles of a right triangle to the ratios of its sides.
Trigonometric identities are equations that are true for all values of the variables involved.
The unit circle is a valuable tool for understanding trigonometric functions and their values.
Trigonometry is used in various applications, including navigation, surveying, and physics.
Inverse trigonometric functions are used to find angles corresponding to given trigonometric ratios.

Statistics is the science of collecting, analyzing, interpreting, and presenting data.
Descriptive statistics involves summarizing and describing data using measures like mean, median, mode, and standard deviation.
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.
Probability is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
Random variables are variables whose values are numerical outcomes of a random phenomenon.
Probability distributions describe the probabilities of different values of a random variable.