What is Mathematics?
The word Mathematics comes from the Greek word máthēma means knowledge, study, learning.
It includes the study of such topics as:
- Quantity (Number theory)
- Structure (Algebra)
- Space (Geometry)
- Change (Mathematical Analysis)
Simply, The Science or study of numbers, quantities or shapes.
It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter.
Maths is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.
A German Mathematician Johann Carl Friedrich Gauss said: Mathematics is the father of all sciences.
Who is the father of mathematics?
Archimedes is known as the Father Of Mathematics.
Applied vs Pure Mathematics
If you want to keep studying maths, you might have to choose between applied and pure mathematics.
But what’s the difference?
The Main Differences
The easiest way to think of it is that pure maths is maths done for its own sake, while applied maths is maths with a practical use. But in fact, it’s not that simple, because even the most abstract maths can have unexpected applications. For example, the branch of mathematics known as “number theory” was once considered one of the most “useless”, but now plays a vital part in computer encryption systems. If you’ve ever bought something online, you can thank number theorists for letting you do it safely.
You could also think about how maths relates to other subjects and to the real world. Applied maths tries to model, predict and explain things in the real world: for example, one area of applied mathematics is fluid mechanics, which analyses how fluids are affected by forces. Other examples of applied maths might be statistics or probability theory.
Pure maths, on the other hand, is separate from the physical world. It solves problems, finds facts and answers questions that don’t depend on the world around us, but on the rules of mathematics itself.
Unfortunately, there is no perfect way to decide what pure maths is and what applied maths is. Even mathematicians can’t agree on it!
Applied Mathematics
It is a branch of mathematics that concerns itself with the application of mathematical knowledge to other domains. Such applications include numerical analysis, mathematics of engineering, linear programming, optimization and operations research, continuous modelling, mathematical biology and bioinformatics, information theory, game theory, probability and statistics, financial mathematics, actuarial science, cryptography and hence combinatorics and even finite geometry to some extent, graph theory as applied to network analysis, and a great deal of what is called computer science.
Pure Mathematics
It is the study of the basic concepts and structures that underlie mathematics. Its purpose is to search for a deeper understanding and an expanded knowledge of mathematics itself.
Traditionally, pure mathematics has been classified into three general fields: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry.
Branches of Mathematics
Different Branches of Mathematics
Arithmetic
“Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.”– Bertrand Russell
It is one of the most important branch because its fundamentals are used in daily life.
Arithmetic deals with numbers and their applications in many ways. Addition, subtraction, multiplication, and division form its basic groundwork as they are used to solve a large number of questions and progress into more complex concepts like exponents, limits, and many other types of calculations.
Algebra
“The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing.”– Rudolf Clausius
A broad field and a fascinating branch of mathematics i.e., ALGEBRA , it involves complicated solutions and formulas to derive answers to the problems.
Algebra deals with solving generic algebraic expressions and manipulating them to arrive at results. Unknown quantities denoted by alphabets that form a part of an equation are solved for and the value of the variable is determined.
Geometry
“The description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn.”-Isaac Newton
Do you curious about the shapes and sizes of various objects??
Then Geometry is the branch where you must explore.
- Geometry dealing with the shape, sizes, and volumes of figures.
It is a practical branch of mathematics that focuses on the study of polygons, shapes, and geometric objects in both two-dimensions and three-dimensions. Congruence of objects is studied at the same time focussing on their special properties and calculation of their area, volume, and perimeter.
Trigonometry
"A surprise trigonometry quiz that everyone in class fails? Must be in the Lord's plan to give us challenges."-Nicholas Sparks
It is derived from Greek words “trigonon” meaning triangle and “metron” meaning “measure”.
Trigonometry focuses on studying angles and sides of triangles to measure the distance and length. Trigonometry is a study of the correlation between the angles and sides of the triangle. It is all about different triangles and their properties!
Calculus
“Calculus is the most powerful weapon of thought yet devised by the wit of man.”– Wallace B. Smith
It is one of the advanced branches of mathematics and studies the rate of change. Earlier maths could only work on static objects but with calculus, mathematical principles began to be applied to objects in motion.
A branch with mind-numbing questions, calculus is an interesting concept introduced to students at a later stage of their study in mathematics.
Probability and Statistics
"Facts are stubborn, but Statics are more pliable" -Mark Twain
"Probability theory is nothing but common sense reduced to calculations"-Pierre Simon Laplace
The abstract branch of mathematics, probability and statistics use mathematical concepts to predict events that are likely to happen and organize, analyze, and interpret a collection of data.
The scope of this branch involves studying the laws and principles governing numerical data and random events. Presenting an interesting study, statistics, and probability is a branch full of surprises.
Number Theory
“Mathematics is the queen of the sciences, and number theory is the queen of mathematics.”– Carl Friedrich Gauss
Number theory is one of the oldest branches of Mathematics which established a relationship between numbers belonging to the set of real numbers.
The basic level of Number Theory includes introduction to properties of integers like addition, subtraction, multiplication, modulus and builds up to complex systems like cryptography, game theory and more.
Topology
“The basic ideas and simplest facts of set-theoretic topology are needed in the most diverse areas of mathematics; the concepts of topological and metric spaces, of compactness, the properties of continuous functions and the like are often indispensable.”– Pavel Sergeevich Aleksandrov
Topology is a much recent addition into the branches of Mathematics list.Its application can be observed in differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex analysis.
It is concerned with the deformations in different geometrical shapes under stretching, crumpling, twisting and bedding. Deformations like cutting and tearing are not included in topologies.
Advanced Branches of Mathematics
These branches are studied at an advanced level and involve complex concepts that need strong computational skills. Such advanced branches are listed below:
Linear Algebra
Numerical Analysis
Operation Research
Game Theory
Real Analysis
Complex Analysis
Cartesian Geometry
Combinatorics
Differential Equations
Integral Equations
Linear Algebra
"Mathematics is not about the numbers, equations, computations, or algorithms; it is about Understand" -William Paul Thurston
It is a branch of mathematics that is concerned with mathematical structures closed under the operations of addition and scalar multiplication and that includes the theory of systems of linear equations, matrices, determinants, vector spaces, and linear transformations.
Numerical Analysis
"I don't like to end my talk with a 700 million dollar loss, even if it shows the importance of Numerical Analysis" -Richard A. Falk
It is the branch of mathematics that deals with the development and use of numerical methods for solving problems.
Operation Research
"So never lose an opportunity of urging a practical beginning, however small, for it's wonderful how often in such matters the mustard-seed germinates and roots" -Florence Nightingale
A method of mathematically based analysis for providing a quantitive basis for management decisions.
"Operation Research is concerned with optimal decision making in and modelling of deterministic and probabilistic systems that originate from real life" -Hiller and Lieberman
Game Theory
"In terms of the game theory, we might say the Universe is so constituted as to maximize play. The best games are not those in which all goes smoothly and steadily toward a certain conclusion, but those in which the outcome is always in doubt." -George B. Leonard
It is the branch of mathematics concerned with the analysis of strategies for dealing with competitive situations where the outcome of a participant's choice of action depends critically on the actions of other participants. Game theory has been applied to contexts in war, business, and biology.
Real Analysis
"The ultimate authority must always rest with the individual's own reason and critical analysis." -Dalai Lama
It is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions.
It includes convergence, limits, continuity, smoothness, differentiability and integrability.
Complex Analysis
"The more complex the world situation becomes, the more scientific and rational analysis, you have to have, the less you can do with simple good will and sentiment." -Reinhold Niebuhr
Complex analysis is the study of complex numbers together with their derivatives, manipulation, and other properties. Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of physical problems. Contour integration, for example, provides a method of computing difficult integrals by investigating the singularities of the function in regions of the complex plane near and between the limits of integration.
Cartesian Geometry
"When you corrdinate your mind and body, you have unlimited access to the wisdom of the universe." -Koichi Tohei
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions).
Combinatorics
Combinatorics, also called combinatorial mathematics, the field of mathematics concerned with problems of selection, arrangement, and operation within a finite or discrete system. Included is the closely related area of combinatorial geometry.
Differential Equations
"Science is a differential equation. Religion is a boundary condition." -Alan Turing
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives.
In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Integral Equations
"Integral Reality is the world's transparency, a perceiving of the world as truth; a mutual perceiving and imparting of the truth of the world and of man and of all that transluces both" -Jean Gebser
In mathematics, integral equations are equations in which an unknown function appears under an integral sign.
There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Green's function, Fredholm theory, and Maxwell's equations.