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Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. # ISO certification in India
Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated to have certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.# ISO certification in India
Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application (and are therefore called pure mathematics), but often later find practical applications. The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks.# ISO certification in India
Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid’s Elements. Since its beginning, mathematics was essentially divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new areas. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method, which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than 60 first-level areas of mathematics.# ISO certification in India
Etymology
The word mathematics comes from Ancient Greek máthÄ“ma (μάθημα), meaning ‘that which is learnt’, ‘what one gets to know’, hence also ‘study’ and ‘science’. The word came to have the narrower and more technical meaning of ‘mathematical study’ even in Classical times. Its adjective is mathÄ“matikós (μαθηματικός), meaning ‘related to learning’ or ‘studious’, which likewise further came to mean ‘mathematical’. In particular, mathÄ“matikḗ tékhnÄ“ (μαθηματικὴ Ï„Îχνη; Latin: ars mathematica) meant ‘the mathematical art’.# ISO certification in India
Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathÄ“matikoi (μαθηματικοί)—which at the time meant ‘learners’ rather than ‘mathematicians’ in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established.# ISO certification in India
In Latin, and in English until around 1700, the term mathematics more commonly meant ‘astrology’ (or sometimes ‘astronomy’) rather than ‘mathematics’; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine’s warning that Christians should beware of mathematici, meaning ‘astrologers’, is sometimes mistranslated as a condemnation of mathematicians.# ISO certification in India
The apparent plural form in English goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathÄ“matiká (Ï„á½° μαθηματικά) and means roughly “all things mathematical”, although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math.# ISO certification in India