Mathematics: Branches and Concepts Overview

Study Notes

Mathematics is a field of study that deals with numbers, quantity, structure, space, and change. It has no generally accepted definition, but some common definitions follow:

Mathematics is the science that draws certain conclusions from uncertain premises.

Bertrand Russell

Mathematics is divided into several branches, each studying different aspects of mathematical structures or systems. Some of the main branches include arithmetic, geometry, algebra, real numbers, and polynomials.

Arithmetic

Arithmetic involves performing operations such as addition, subtraction, multiplication, division, and exponentiation on numbers. This branch includes elementary arithmetic and more advanced concepts like modular arithmetic.

Elementary Arithmetic

Elementary arithmetic is what most people think of when they hear the word "arithmetic." It covers basic number facts and follows the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Modular Arithmetic

Modular arithmetic is another part of arithmetic that extends the ordinary arithmetic done using integers. Instead of dealing with the negative integers, we take only the positive ones (positive residues), which makes calculations easier since it eliminates all the rules associated with negative integers.

Geometry

Geometry is a branch of mathematics concerned with properties and relationships of points, lines, angles, surfaces, and solids. Its name comes from the Greek words 'geometria', meaning "to measure the earth," indicating its ancient origins and broad scope.

Algebra

Algebra is one of the major areas of mathematics that studies structures and their properties. A structure consists of objects and relations between those objects. For example, integers and addition form an abelian group, while matrices and matrix multiplication form a monoid. The formalism of algebra is extended by abstract algebra, where the structures are allowed to have additional structure. In this sense, every algebra forms a subset of an abstraction of itself called the free algebra over a set.

Real Number

Real numbers are real numbers which may be expressed in the decimal system, which includes rational numbers and irrational numbers. They can be positive, zero and negative. They are called'real' because they represent continuous quantities, unlike whole numbers which do not have a decimal expansion.

Polynomial

Polynomials are expressions consisting of variables raised to non-negative integral powers combined with coefficients and sums, products, and differences of these terms. They are named for having a degree, which is the highest power of the variable appearing in the expression. When polynomials are written in standard form, they start with the term of highest degree.

In conclusion, mathematics is a vast field encompassing various branches including arithmetic, geometry, algebra, real numbers, and polynomials. Each branch offers unique perspectives and challenges, contributing significantly to our understanding of patterns and structures in the world around us.

Description

Explore the diverse branches of mathematics such as arithmetic, geometry, algebra, real numbers, and polynomials. Learn about elementary arithmetic, modular arithmetic, properties of points and lines in geometry, structures and relations in algebra, real numbers in decimal system, and polynomials with variables and coefficients.