Mar 6, 2014

Characteristics of Mathematics

Mathematics is not just a study of numbers, nor is it simply about calculations.
It is not about applying formulas, either. It can perhaps be better described as “a field of creation through accurate and logical thinking."
Mathematics has a long, rich history and continues to grow rapidly.
Mathematics is a very diverse filed.
Mathematics is an academic discipline of great depth, with a number of unsolved problems.
It has progressed on cumulative contributions from countless mathematicians in the world who have tackled those problems while creating new areas of inquiry.
They illustrate how a number of subfields of mathematics have benefited from and evolved through interactions with other disciplines.
Mathematics, on the other hand, has made considerable contributions to the advancement of other academic fields. In fact, mathematics is often described as the foundation of scientific studies.
One of the major characteristics of mathematics is its general applicability.
One equation, for instance, can represent a particular phenomenon in physics as well as a certain logic in economics. This general nature of mathematical equations enables unified treatment of diverse phenomena in various academic fields.
Furthermore, mathematical theorems are no respecter of age or seniority: any theorem has to be proven through appropriate mathematical procedures whether you are a novice student researcher or an eminent professor of mathematics.
And once proven true, mathematical theorems will never be reversed. This "universality" of mathematics is another important feature that allows the discipline to transcend time and space.

Nature of Mathematics

Mathematics reveals hidden patterns that help us understand the world around us.
Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems.
As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change.
As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth.
The special role of mathematics in education is a consequence of its universal applicability.
The results of mathematics--theorems and theories--are both significant and useful; the best results are also elegant and deep.
Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty.
In addition to theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols.
Experience with mathematical modes of thought builds mathematical power--a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives.
Mathematics empowers us to understand better the information-laden world in which we live.

History of Mathematics

The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.
Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales.
The most ancient mathematical texts available are Plimpton 322(Babylonian mathematics c. 1900 BC), he Rhind Mathematical Papyrus(Egyptian mathematics c. 2000-1800 BC) and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC).
All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction".
Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.
Greek mathematics is thought to have begun with Thales of Miletus (c. 624–c.546 BC) and Pythagoras of Samos (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.
Chinese mathematics made early contributions, including a place value system. Particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.
The Hindu-Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics.
Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations.
Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe.
From ancient times through the Middle Ages, bursts of mathematical creativity were often followed by centuries of stagnation.
Beginning in Renaissance Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day.
Throughout the 19th century mathematics became increasingly abstract. This century saw the development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°.
20th century The 20th century saw mathematics become a major profession. Differential geometry came into its own when Einstein used it in general relativity. Entire new areas of mathematics such as mathematical logic, topology, and John von Neumann's game theory changed the kinds of questions that could be answered by mathematical methods.
21st century In 2000, the Clay Mathematics Institute announced the seven Millennium Prize Problems, and in 2003 the Poincaré conjecture was solved by Grigori Perelman (who declined to accept an award on this point). Most mathematical journals now have online versions as well as print versions, and many online-only journals are launched. There is an increasing drive towards open access publishing, first popularized by the arXiv.

Mar 5, 2014

Indian Mathematicians

Aryabhatta

Aryabhatta was born in 476A.D in Kusumpur, India.
He was the first person to say that Earth is spherical and it revolves around the sun.
He gave the formula (a + b)2 = a2 + b2 + 2ab
He taught the method of solving the following problems:
Aryabhata was the first scientist to provide an approximate value for the mathematical constant π(pi). Generally, we say n 22/ 7, but actually Aryabhata had more to say !
He gives the value correct to five decimal places. A circle of diameter 20,000 units has a circumference approximately equal to (100 + 4) x 8 + 62,000 = 62,832. This can be represented in the form of an equation:
π (circumference / diameter) = (62,832 / 20,000) = 3.1416. What is really remarkable in the work of Aryabhata is the use of the word aasanna, which means approximately. This contribution of Aryabhata was proved only about 8 centuries later !

Brahmagupta

Brahmagupta [597–668 AD] was an Indian mathematician and astronomer.
He wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta (Correctly Established Doctrine of Brahma), written in 628 in Bhinmal. Its 25 chapters contain several unprecedented mathematical results.
Brahmagupta was the first to use zero as a number. He gave rules to compute with zero. Brahmagupta used negative numbers and zero for computing.
A positive number multiplied by a positive number is positive.
A positive number multiplied by a negative number is negative.
A negative number multiplied by a positive number is negative
A negative number multiplied by a negative number is positive.
The modern rule that two negative numbers multiplied together equals a positive number first appears in Brahmasputa Siddhanta.
The book also consisted of many geometrical theories like the ‘Pythagorean Theorem’ for a right angle triangle. Brahmagupta was the one to give the area of a triangle and the important rules of trigonometry such as values of the sin function.
He introduced the formula for cyclic quadrilaterals. Area of a cyclic quadrilateral with side a, b, c, d= √(s -a)(s- b)(s -c)(s- d) where 2s = a + b + c + d Length of its diagonals =
He also gave the value of ‘Pi’ as square root ten to be accurate and 3 as the practical value. Additionally he introduced the concept of negative numbers.
It is composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is not known how Brahmagupta’s mathematics was derived.

BHASKARACHARYA

He was born in a village of Mysore district.
He was the first to give that any number divided by 0 gives infinity (00).
He has written a lot about zero, surds, permutation and combination.
He wrote, “The hundredth part of the circumference of a circle seems to be straight. Our earth is a big sphere and that’s why it appears to be flat.”
He gave the formulae like sin(A ± B) = sinA.cosB ± cosA.sinB

Srinivasa Ramanujan

He was born on 22nd of December 1887 in a small village of Tanjore district, Madras. He failed in English in Intermediate, so his formal studies were stopped but his self-study of mathematics continued.
He sent a set of 120 theorems to Professor Hardy of Cambridge. As a result he invited Ramanujan to England.
Ramanujan showed that any big number can be written as sum of not more than four prime numbers.
He showed 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 since then the number 1729 is called Ramanujan’s number.
In the third century B.C, Archimedes noted that the ratio of circumference of a circle to its diameter is constant. The ratio is now called ‘pi ( Π )’ (the 16th letter in the Greek alphabet series)
The largest numbers the Greeks and the Romans used were 106 whereas Hindus used numbers as big as 1053 with specific names as early as 5000 B.C. during the Vedic period.

SHAKUNTALA DEVI

She was born in 1939
In 1980, she gave the product of two, thirteen digit numbers within 28 seconds, many countries have invited her to demonstrate her extraordinary talent.
In Dallas she competed with a computer to see who give the cube root of 188138517 faster, she won. At university of USA she was asked to give the 23rdroot of9167486769200391580986609275853801624831066801443086224071265164279346570408670965932792057674808067900227830163549248523803357453169351119035965775473400756818688305620821016129132845564895780158806771.She answered in 50seconds.
The answer is 546372891. It took a UNIVAC 1108 computer, full one minute (10 seconds more) to confirm that she was right after it was fed with 13000 instructions.
Now she is known to be Human Computer.