Efforts by the Mathematician and Difference between Pure and Applied Mathematics

Satpal Panika, Department of Mathematics, Kalinga University, Naya Raipur

What is the importance of mathematics? If such a question is asked by a vegetable seller, perhaps he will say, ” due to mathematics, I can tell how many profits I have earned today” it is correct that the second name of the book of accounting in ordinary language is mathematics. What is the time now? How many works are still overflowing? It is also mathematics to tell the distance of office from my home, what date will be next Monday from today? Etc. are some of the questions that are numerical, so it is mathematics to give a statement a numerical form. As a traditional definition, it has been called a scientific study of quantities in which they were represented by numbers or symbols. Karl Frederick Gauss called mathematics the “Queen of Science”, from old times to the present; it was believed that mathematics could play an important role if other subjects were to be developed and interesting. It is a tool that can enhance the ability and reasoning in a person to develop leadership qualities such as decision making. It is the language of scientists as well as understanding engineering branches, it is necessary to have knowledge of mathematics. It is the past, present and future of all sciences.

Our country has given many mathematicians whose mathematical concepts present many events in the universe, whether geographical or astronomical, to the people of the world .There are some mathematicians and their contributions given as below:

Aryabhatta:

His birth place was held in 476 BC in Tarnaga, a city in Bihar, India. It is believed that for his studies he visited Kusumapara (currently Patna). He is believed to have been the principal of Nalanda University.

• The famous work done by him” Aryabhatiya “and the”Arya‐siddhanta”.Aryabhatiya cover arithmetic algebra, plane and spherical trigonometry and the Arya‐siddhanta has the work on astronomical computation.
• Concept of  (Pie): He worked on the approximation for pi (π) came to the conclusion that π is not a rational number it is an irrational number. He defined the pie as the ratio of circumference to diameter is 3.1416.
• He was the person who firstly said “Earth is spherical and it revolves around the Sun”.

BRAHMAGUPT:

He was born in 598 A.D.in Bhinmal city, Rajasthan in India. Being a mathematician and astronomer, he wrote sevaral important works related to mathematics and astronomy.

• “Brahmasphuta‐Siddhanta” is the best work done by him, written in 628 AD in Bhinmal.
•  Firstly he use zero as a number in the field of mathematics and computation method with zero.
• He gave following formulae, used in G.P. series a+ar+ar² +ar³ +……….+ar^n-1 =a(rⁿ  ‐1)/( r‐1) .
• He gave the following formulae known as Brahmagupta’s formula: Area of a cyclic

    quadrilateral with sides a, b, c, d =square root of (s‐a)(s‐b)(s‐c)(s‐d), where 2s= a+b+c+d.

BHASKARACHARYA: 

He was born in Bijapur in Karnataka. There is a significant contribution in the field of mathematics & astronomic in the 12th century.

• His work “Siddhanta Shiromani” is divided into four parts called Lilawati, Bijaganit, Grahaganita and Goladhyaya. These four sections deal with arithmetic, algebra, mathematics of planets and spheres respectively.
•  Firstly he gave the concept that any number divided by zero gives infinity.
•  He wrote a lot about zero, surds, permutation and combination.
• He wrote” The hundredth part of the circumference of a circle is straight. Our earth is a big sphere   and that’s why it appears to be flat.”
• He gave the formulae :  
• He gave derivations of Trigonometric functions and formulae.
• He developed spherical trigonometry using number of other trigonometric results.
• He explained solution of quadratic, cubic and quartic indeterminate equations.
• He developed a proof of Pythagoras Theorem.
• He developed general method for finding the solution of the problem x² – ny² = 1 (Pell’s equation).

Ramanujan:

He was born on 22^nd of December 1887 in a small village of Tanjore district, Madras. He gave great contributions as below:

• He showed that any big number can be written as sum of not more than four prime numbers.
• He gave the concept that how to divide the number into two or more squares or cubes.
• When Mr. Litlewood came to see Ramanujan in taxi number 1729, Ramanujan said that 1729 is the smallest number which can be written in the form of sum of cubes of two numbers in two ways, i.e. 1729 = 9³ + 10³ = 1³ + 12³ .The number 1729 is called as Ramanujan’s number.
• By 17, he had conducted his own mathematical research on Bernoulli numbers and the Euler‐Mascheroni constan.
• In 1918, Ramanujan and Hardy studied the partition function P(n) extensively and gave a non‐convergent asymptotic series that permits exact computation of the number of partition of an integer.
• He discovered mock theta function in the last year of his life .For many years these functions were a mystry, but they are now known to be the holomorphic parts of harmonic weak mass forms.

There are some famous mathematicians from all over the world whose contributions can not be forgotten.

ALBERT EINSTEIN:

He was born in Ulm, Germany on 14^th March 1879. His education was completed in Germany, Italy and Switzerland after that he worked in Bern, Zurich and Prague as a Professor and then he became Director of Kaiser Wilhelm Institute for Physics in Berlin in 1914. For advanced study in Princeton he became a professor in the School of Mathematics at the Institute. He died in Princeton on April, 1955.

• His development of general relativity, still the best theory of gravitation that we have.
• He proposed the light quantum hypothesis, which developed into the theory of the photon, the first elementary particle to be given a quantum treatment.
• His quantum theory of solids, which provided the basis for explaining the anomalous low-temperature behavior of crystalline solids.
• His explanation of Planck’s law based on the introduction of the A & B coefficients, which placed the concept of transition probabilities at the center of atomic physics.
•  The Einstein-Podolsky-Rosen “paradox,” which highlighted the nature of the quantum entanglement of two or more systems.
•  His work on Bose-Einstein statistics, leading to his prediction of the existence of Bose-Einstein condensates, only recently confirmed.
• The Einstein-Infeld-Hoffmann derivation of the equations of motion of massive bodies from the field equations of general relativity – the list could go on indefinitely.

Euclid:

• He was a Greek Mathematician called as the “Father of Geometry.”
• Euclid’s Elements is recognized as the most prominent books on mathematics in history. In this book, he has included information related to number theory, algebra and geometry.
• There are several kinds of math namely Euclidean geometry, Euclidean number and the Euclidean algorithm.

Gottfried Leibniz:

He was a great polymath who knew almost everything that could be known at the time about any subject or intellectual enterprise. He made important contributions to philosophy, mathematics, physics, law, politics, philology and theology.

• His greatest achievement was the discovery of a modern mathematical method called calculus. Scientists use to deal with quantities that are constantly varying. Newton had devised a similar method for his work on gravity. Therefore, there was a harsh debate about who had been first.
• Newton began working on his version in 1665, but Leibniz published his results in 1684, almost three years before Newton. However, the consensus is that they discovered the method simultaneously.
• Leibniz also discovered the binary number system and invented the first calculating machine that could add, subtract, multiply and divide.
• He formulated the famous theory of monads which explained the relation between soul and the body.
• Leibniz is often known as the founder of symbolic logic as he developed the universal characteristic, a symbolic language in which any item of information can be represented in a natural and systematic way.

Difference between Pure and Applied Mathematics:

Mathematics can be divided into two parts. First type of mathematics pure mathematics and second type mathematics applied mathematics. Basic concepts in pure mathematics are studied and their structures, provides special contributions in the field of intensive study and extended knowledge research. Traditionally, pure mathematics is divided into three common areas: the first analysis, which highlights the continuing aspects of mathematics; the second is algebra, which is based on discrete aspects; and the third form is geometry. Subjects belong to the pure mathematics are as below:

Pure Mathematics:

• Calculus and Analysis
• Geometry and Trigonometry
• Combinatory
• Algebra
• Logic
• Number Theory   etc.

Applied Mathematics:

• Dynamic systems and differential equations
• Mathematical Physics
• Numerical Methods
• Information theory and signal processing
• Probability and statistics
• Game Theory
• Operation Research

Applied mathematics involves the application of concepts of pure mathematics which are studied in the areas of engineering, practical economics, marketing and management, etc. Definitions of pure and applied mathematics give the principal difference between them. Pure Mathematics is dedicated mainly to procure one’s own knowledge of this discipline, from a certain degree of abstraction while applied mathematics employ this knowledge to solve problems of real life, as well as to convert into verifiable facts the different scientific postulates about the realities and laws of their objects of study.

References:

1. The Distinction Between Pure And Applied Mathematics,  R. O. Robinson (1963)
2. Leibniz on Mathematics, Methodology and thegood : A reconsideration of the Place of Mathematics in Leibniz’s Philosophy ,Early Science and Medicine ,2006
3. http://famous-mathematicians.org/euclid/
4. Exact Definitions of mathematics by Dharmendra Kumar yadav, 2017.