Introduction to Geometry

• Mathematics is the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of relations of space and number.
• It is a science that deals with the logic of quantity and shape and arrangement.
• Mathematics is used around the world as an essential tool in many fields, including natural science, engineering, medicine, finance, and social sciences.
• Applied mathematics concerns itself with the use of mathematical tools to solve problems in natural science, engineering, medicine, finance, and social sciences.

Geometry

• Geometry is concerned with questions of shape, size, relative position of figures, and the properties of space.
• It is a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids.
• Plane geometry deals with two-dimensional shapes (e.g., squares, circles, triangles) that can be drawn on a flat surface.
• Solid geometry deals with three-dimensional shapes (e.g., cubes, spheres, cylinders).
• Analytic geometry uses algebra to study geometric properties, using coordinate systems to describe geometric figures.
• Differential geometry uses calculus to study the properties of curves and surfaces.
• Euclidean geometry is the study of geometrical shapes based on axioms and theorems.
• Non-Euclidean geometry includes hyperbolic and elliptic geometry, which differ from Euclidean geometry in their axioms about parallel lines.
• Topology is a branch of geometry that deals with properties preserved under continuous deformations, like stretching and bending.

Arithmetic

• Arithmetic is the branch of mathematics dealing with numerical calculations.
• It involves the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication, division, exponentiation, and extraction of roots.
• The fundamental theorem of arithmetic states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors.
• Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.
• The four basic operations in arithmetic are addition, subtraction, multiplication, and division.
• Addition is the process of combining two or more numbers to find their total.
• Subtraction is the process of finding the difference between two numbers.
• Multiplication is the process of repeated addition.
• Division is the process of splitting a number into equal parts or groups.
• Fractions represent a part of a whole and consist of a numerator and a denominator.
• Decimals are a way of writing numbers that are not whole numbers.
• Percentages are a way of expressing a number as a fraction of 100.

Algebra

• Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols.
• It is a unifying thread of almost all of mathematics.
• Elementary algebra deals with solving equations and inequalities involving real numbers and variables.
• Linear algebra deals with vector spaces, linear transformations, and systems of linear equations.
• Abstract algebra studies algebraic structures such as groups, rings, and fields.
• Boolean algebra is a branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0, respectively.
• Algebraic expressions are combinations of variables, constants, and operations.
• Equations are statements that two algebraic expressions are equal.
• Variables are symbols that represent unknown quantities.
• Constants are fixed values that do not change.
• Polynomials are algebraic expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Factoring is the process of breaking down an algebraic expression into its constituent factors.

Calculus

• Calculus is the mathematical study of continuous change.
• It has two major branches, differential calculus and integral calculus.
• Differential calculus concerns instantaneous rates of change and the slopes of curves.
• Integral calculus concerns the accumulation of quantities and the areas under and between curves.
• Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a certain value.
• Derivatives measure the instantaneous rate of change of a function with respect to its variable.
• Integrals are used to find the area under a curve or the accumulation of a quantity.
• The fundamental theorem of calculus establishes the relationship between differentiation and integration.
• Sequences are ordered lists of numbers (or other elements), and series are the sum of the elements of a sequence.
• Multivariable calculus extends the concepts of calculus to functions of multiple variables.
• Differential equations are equations that relate a function with its derivatives.

Statistics

• Statistics is the science of collecting, analyzing, presenting, and interpreting data.
• It involves methods for gathering and summarizing data, and for making inferences and predictions based on data.
• Descriptive statistics involves summarizing and presenting data using measures such as 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.
• Random variables are variables whose values are numerical outcomes of a random phenomenon.
• Statistical distributions describe the probability of different outcomes in a random experiment.
• Hypothesis testing is a method for testing a claim or hypothesis about a population based on sample data.
• Regression analysis is a statistical technique used to model the relationship between two or more variables.
• Data visualization is the graphical representation of data to help understand patterns and trends.
• Statistical significance is a measure of the probability that a result occurred by chance.