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mathematics Made by =Jayant kumar sharma CLASS=IX-B ROLL NO=23


mathematics Mathematics is the abstract study of topics encompassing quantity, structure, space, change, and others; it has no generally accepted definition. Mathematics seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. The research required to solve mathematical problems can take years or even centuries of sustained inquiry

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Through the use of abstraction and logical reasoning, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements . Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

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Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering,  medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disciplines, such as statistics and game theory.

Carl Friedrich Gauss, known as the "prince of mathematicians".:

Carl Friedrich Gauss , known as the "prince of mathematicians ".


Etymology The word "mathematics" comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means what one learns, what one gets to know, hence also study and science, and in modern Greek just lesson. The word máthēma is derived from μανθάνω (manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of which mean to learn. In Greece, the word for "mathematics" came to have the narrower and more technical meaning "mathematical study", even in Classical times.

Definitions of mathematics:

Definitions of mathematics Defined mathematics as "the science of quantity," and this definition prevailed until the 18th century. Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. Today, no consensus on the definition of mathematics prevails, even among professionals.

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There is not even consensus on whether mathematics is an art or a science. A great many professional mathematicians take no interest in a definition of mathematics, or consider it indefinable. Some just say, "Mathematics is what mathematicians do." A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic" (1903). L. E. J. Brouwer, identify mathematics with certain mental phenomena.

 History of mathematics:

History of mathematics The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals, was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members. In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time – days, seasons, years. Elementary arithmetic (addition, subtraction,  multiplication and division) naturally followed.

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Since numeracy pre-dated writing, further steps were needed for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data. Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.

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The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC

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Greek mathematician Pythagoras (c. 570–c. 495 BC), commonly credited with discovering the Pythagorean theorem .

 Mathematical beauty:

Mathematical beauty Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.

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Sir Isaac Newton ( 1643-1727 ), an inventor of infinitesimal calculus.

Fields of mathematics:

Fields of mathematics Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change. In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty

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Pure mathematics


Quantity The study of quantity starts with numbers, first the familiar natural numbers and integers and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem . The twin prime conjecture and Goldbach's conjecture are two unsolved problems in number theory.


Structure Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations .


Space The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem .


Change Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis.

Applied mathematics:

Applied mathematics Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems

Statistics and other decision sciences:

Statistics and other decision sciences Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians "create data that makes sense" with random sampling and with randomized experiments; the design of a statistical sample or experiment specifies the analysis of the data.

Computational mathematics:

Computational mathematics Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory.

Mathematics as profession:

Mathematics as profession Arguably the most prestigious award in mathematics is the Fields Medal , established in 1936 and now awarded every 4 years. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize . The Wolf Prize in Mathematics , instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize , was introduced in 2003. The Chern Medal was introduced in 2010 to recognize lifetime achievement .

Leonhard Euler, who created and popularized much of the mathematical notation used today :

Leonhard Euler , who created and popularized much of the mathematical notation used today

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Indian mathematics

Indian mathematics:

Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BC until the end of the 18th century. In the classical period of Indian mathematics (400 AD to 1200 AD), important contributions were made by scholars like Aryabhata , Brahmagupta , and Bhaskara II . The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers , arithmetic , and algebra . In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East , China , and Europe ] and led to further developments that now form the foundations of many areas of mathematics.

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Ancient and medieval Indian mathematical works, all composed in Sanskrit , usually consisted of a section of sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered so important as the ideas involved.

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All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent is the birch bark Bakhshali Manuscript , discovered in 1881 in the village of Bakhshali , near Peshawar (modern day Pakistan) and is likely from the 7th century CE.

Vedic mathematics:

Vedic mathematics Vedic mathematics is a system of mathematics consisting of a list of 16 basic sūtras , or aphorisms . They were presented by a Hindu scholar and mathematician, Bharati Krishna Tirthaji Maharaja , during the early part of the 20th century. Tirthaji claimed that he found the sūtras after years of studying the Vedas , a set of sacred ancient Hindu texts. However, Vedas do not contain any of the "Vedic mathematics" sutras. The calculation strategies provided by Vedic mathematics are said to be creative and useful, and can be applied in a number of ways to calculation methods in arithmetic and algebra, most notably within the education system. Some of its methods share similarities with the Trachtenberg system .

Great mathematician-astronomers :Aryabhata :

Great mathematician-astronomers :Aryabhata Aryabhata ( IAST : Āryabhaṭa, Sanskrit: आर्यभट्ट) or Aryabhata I (476–550 CE )was the first in the line of great mathematician - astronomers from the classical age of Indian mathematics and Indian astronomy . His most famous works are the Āryabhaṭīya (499 CE, when he was 23 years old) and the Arya- siddhanta . The works of Aryabhata dealt with mainly mathematics and astronomy. He also worked on the approximation for pi.


Bhāskara Bhāskara also known as Bhāskara II and Bhāskarāchārya ("Bhāskara the teacher"), (1114–1185), was an Indian mathematician and astronomer . He was born near Vijjadavida (Bijāpur in modern Karnataka). Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of ancient India. He lived in the Sahyadri region. Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India. His main work Siddhānta Shiromani, (Sanskrit for "Crown of treatises," is divided into four parts called Lilāvati, Bijaganita, Grahaganita and Golādhyāya These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karan Kautoohal

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Bhāskara's work on calculus predates Newton and Leibniz by half a millennium. He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhāskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.


Brahmagupta Brahmagupta (598–668 AD) was an Indian mathematician and astronomer who 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. The modern rule that two negative numbers multiplied together equals a positive number first appears in Brahmaputa siddhanta. 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. The modern rule that two negative numbers multiplied together equals a positive number first appears in Brahmasputa siddhanta.

Brahmagupta: Indian mathematician:

Brahmagupta: Indian mathematician Algebra :Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta Arithmetic :Four fundamental operations (addition, subtraction, multiplication and division) were known to many cultures before Brahmagupta. This current system is based on the Hindu Arabic number system and first appeared in Brahmasputa siddhanta. Series :Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers. He gives the sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6 and the sum of the cubes of the first n natural numbers as (n(n+1)/2)². Zero :Brahmagupta's Brahmasphuṭasiddhanta is the very first book that mentions zero as a number, hence Brahmagupta is considered as the man who found zero. He gave rules of using zero with negative and positive numbers. Zero plus a positive number is the positive number and negative number plus zero is a negative number etc. Pythagorean triples :In chapter twelve of his Brahmasphutasiddhanta , Brahmagupta finds Pythagorean triples. Brahmagupta's formula :Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,

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Srinivasa Ramanujan (22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics , made extraordinary contributions to mathematical analysis , number theory , infinite series , and continued fractions . Living in India with no access to the larger mathematical community, which was centered in Europe at the time, Ramanujan developed his own mathematical research in isolation. As a result, he sometimes rediscovered known theorems in addition to producing new work. Ramanujan was said to be a natural genius by the English mathematician G.H. Hardy , in the same league as mathematicians like Euler and Gauss .

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