It is impossible to avoid mathematics even for a single day since maths permeates every aspect of our lives. Math is involved in everything that we do in a day. Because of this, mathematics plays a significant role in our day-to-day lives, and it is difficult for us to even conceive of getting through a single day without using some mathematics.
The Greek word “anathema” initially meant “science” or “knowledge,” and it is from this word that we get our word “mathematics.” Mathematical expressions are often written as Math or Maths. There aren’t too many subfields of mathematics, yet throughout the years, mathematics and its subfields have developed.
The use of mathematics has been particularly fruitful in developing technological advances. As a result, many significant subfields of mathematics begin with the fundamentals and progress to the most sophisticated levels. However, many students struggle to grasp the fundamental concepts underlying the many subfields of mathematics.
In addition, doing in-depth research into the many subfields of mathematics is an effective way for students to acquire a solid foundation while improving the clarity of their conceptual organization.
Students are more able to study different ideas of mathematics systematically and better decide which area of mathematics they would want to follow as a profession when they are aware of the contrasts and specifics of the many disciplines of mathematics.
Branches related to Pure Mathematics
The study of mathematical conceptions that are entirely based on mathematics and unconnected to any other concept is called pure mathematics.
The following are the main branches of Pure Mathematics:
- Arithmetic
- Algebra
- Geometry
- Topology
- Number theory
- Combinatorics
Arithmetic
Arithmetic is the most fundamental part of mathematics. It can handle working with numbers ranging from 0 to 9 and odd, prime, or any other kind of number. The three fundamental arithmetic operations of addition, subtraction, and multiplication are included.
Arithmetic is used by everyone, from schoolchildren to working professionals. In addition, we rely on it to carry out essential computations in our day-to-day lives. This is the base upon which pure mathematics is constructed.
Algebra
It is one of the primary subfields that make up pure mathematics. To contribute the appropriate algebraic values, we employ the English alphabet. In algebraic expressions and equations, we utilize them to determine the missing value of an unknown variable.
In addition, algebra includes a plethora of rules of operations, relations and constructions, algebraic structures, and polynomial equations. The fundamental building block of algebra is referred to as introductory algebra.
It serves as a representation of the numbers for us. Algebra is more than simple algebra. Additionally, it may be used in various structures, including groups, rings, and fields.
Geometry
One of the most theoretical as well as applicable subfields of mathematics is geometry. In other words, we may quickly have a fair comprehension of geometry from the actual world if we look at it closely enough. It investigates the relationships between figures’ dimensions and their characteristics.
Points, lines, angles, surfaces, solids, and forms are all essential components of geometry. Every sector relies on geometry to meet the day-to-day demands of its operations. The geometrical concept is essential to the industrial production sector. Even when you’re just walking down the street, you’re still using geometry and GPS to find your route.
Topology
It is merely a much-extended kind of geometry. Studying the qualities of fixed forms, even though their figure dimensions may shift, is one application of this branch’s potential uses.
Topology is a relatively new entry in the long list of subfields that make up mathematics. It focuses on the deformations that occur in various geometrical forms due to stretching, crumpling, twisting, and bedding.
Cutting and ripping are examples of deformations that are not included in topologies. In complex analysis, its use may be seen in differentiable equations, dynamical systems, knot theory, and Riemann surfaces.
Number Theory
Number theory is one of the first subfields of mathematics, and its primary purpose was to determine the nature of the connections between real numbers. The first step in learning number theory is to get familiar with the characteristics of integers, such as addition, subtraction, multiplication, and modulus. From there, students go on to more complicated topics, such as cryptography and game theory.
Combinatorics
Combinatorics is a branch of mathematics that emphasizes the analysis of discrete and finite structures. The study of permutations and combinations and the enumeration of the various sets of elements are included in its scope. It provides a description of mathematical relations and the features they possess.
Combinatorics is a branch of mathematics with numerous applications in other subfields of mathematics, such as probability, graph theory, coding, and cryptography.
Branches related to Applied Mathematics
Mathematical principles are combined with the practical applications of other academic fields in applied mathematics. Said it is the use of mathematics in conjunction with other specialized knowledge.
The following are the branches of applied mathematics:
- Statistics and Probability
- Trigonometry
- Set Theory
- Calculus
Statistics and Probability
Probability and statistics are two subfields of the abstract area of mathematics known as probability, which employs mathematical concepts to organize, examine, and analyze data sets and make predictions about events that are mathematically probable to occur.
It is one of the more recent subfields of mathematics, but it has become more critical due to its applications in the social and natural sciences.
The rules and principles that govern random occurrences and numerical data are the subjects of discussion in this section. The areas of mathematics known as statistics and probability are known for their abundance of unexpected findings.
Trigonometry
The term “trigonometry” originates from the Greek words “trigonal,” which means triangle, and “metron,” which means “measure.” Trigonometry is about angles and sides of triangles to measure distance and length.
The study of the relationship between the angles and sides of a triangle is the subject of trigonometry, one of the most well-known subfields of mathematics.
It is used in technology and science to construct various items. It is all about the various triangles and the attributes of those triangles.
Set Theory
Mathematically speaking, “set theory” refers to the study of well-defined groups, or “sets,” of individual things that are collectively referred to as “members” or “elements” of the set. Because sets are the only topic of discussion in pure set theory, the only sets that may be taken into notice are those that include other sets as members.
The theory of hereditarily-finite sets is analogous to the theory of finite sets. Hereditarily-finite sets are finite sets whose rudiments are also finite sets, the rudiments of which are likewise limited, and so on. Set theory may be thought of as the mathematical theory of the actual infinite instead of the hypothetical infinite since its primary focus is on investigating infinite sets.
Calculus
Calculus is the next area of study in the realm of mathematics. In the 17th century, Isaac Newton and Gottfried Leibniz invented what is now known as calculus. Calculations’ immediate rate of change is the subject of this research.
You may do a simple calculation to determine the whole number by adding up the parts. You have probably been exposed to concepts such as limits, functions, integrals, derivatives, and integration. Calculus encompasses all of these different ideas and notions in its scope.
Differential and integral calculus are the two subfields that fall under the umbrella of calculus. Calculus applies to various disciplines, including science, engineering, and economics.
Branches related to Advanced Mathematics
There is a considerable number of advanced branches that are included as a component of the primary units that were discussed before. These subfields are researched more advanced and incorporate complicated ideas that need competent computing ability. These well-developed subfields are mentioned down below.
- Operations research
- Numerical analysis
- Matrix Algebra
- Complex numbers
- Linear Algebra
- Game theory
- Cartesian Geometry
Operations Research
The mathematical study of issues, including resource allocation, choices, games, uncertainty, scheduling, and optimization, is called operations research (OR). Mathematical modeling and effective problem resolution are possible because of the different sub-disciplines that make OR. These sub-disciplines also provide analytical and computational tools.
Numerical Analysis
Numerical analysis is a branch of mathematics and computer science that focuses on developing, investigating, and using techniques for numerically addressing the issues posed by continuous mathematics.
These issues often stem from the implementation of algebra, geometry, and calculus in the real world, and they include variables whose values undergo a continuous range of change. These issues are prevalent across various academic disciplines, including the natural sciences, the social sciences, medicine, engineering, and business.
Matrix Algebra
In algebra, a matrix is a collection of integers organized in rows and columns to create a rectangular array. The components of the matrix, also known as entries and elements, are the integers. In addition to their use in several mathematics subfields, matrices have widespread use in engineering, economics, physics, and statistics.
Matrices also have vital uses in computer graphics, which represent rotations and other transformations of pictures. Matrices have also been used to describe different types of operations.
Complex Number
Complex numbers may be written as a+ib, whereas a & b are real numbers, and i is an imaginary number denoted by an iota. The value of i is equal to (√ -1). For instance, the expression “7+3i” is a complex number, in which “7” is a natural number and “3i” is an imaginary number.
The most common use for these numbers is to depict periodic movements, such as water waves, alternating currents, light waves, and other similar phenomena, depending on sine or cosine waves, among different locks.
Linear Algebra
Learning linear functions in vector spaces is the focus of the mathematical field known as linear algebra, designated as a subfield of mathematics. A matrix is a product that emerges whenever the information associated with linear functions is presented in an ordered fashion.
Consequently, vector spaces, vectors, linear functions, the system of linear equations, and matrices are the topics that are covered by linear algebra. These ideas are necessary to understand before moving on to related subjects such as geometry and functional analysis.
Game Theory
The term “game theory” refers to a theoretical framework that enables players to make the best possible decisions when applied to a strategic context. The idea that the reward for a player is contingent on the strategies used by other players is central to the discipline of game theory.
It is generally accepted that game theory may be applied to any setting with two or more players in which the outcomes of their actions can be predicted or quantified. This theory assists participants in determining the products that are most likely to occur while also considering the deeds and decisions of others that will impact the development.
Cartesian Geometry
The study of geometry in which coordinate points play a central role is known as coordinate geometry (also known as analytic geometry). It is possible to compute the area of a triangle in the Cartesian plane using coordinate geometry, discover the distance between two points, find the midpoint of a line, divide lines using the m:n ratio, and find the distance between two points.
Conclusion
Since most people have a natural aversion to mathematics, you probably don’t put much stock in it, even though it is one of the most significant aspects of your everyday life because of its practical applications of mathematics.