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Groups, rings, fields, modules, and vector spaces are common objects of study in higher algebra.
Modern mathematics is built around a system of axioms, which is a name given to "the rules of the game." Mathematicians then use various methods of formal proof to extend the axioms to come up with surprising and elegant results.
Such methods include proof by induction, and proof by contradiction, for example.
Examples include Mathematical Physics and Mathematical Biology.
Arithmetic is a branch of mathematics and their basic properties under the operations of addition, subtraction, multiplication and division and exponentsas well as percents radicals fractions and decimals In general, more basic properties of the integers belong to arithmetic while deeper or more difficult results belong to number theory, but the boundary is not extremely clear.
In high school, this can the study of examining, manipulating, and solving equations, inequalities, and other mathematical expressions.
Algebra revolves around the concept of the variable, an unknown quantity given a name and usually denoted by a letter or symbol.Applied Maths consists of taking the techniques from Pure Maths and using them to develop models of "the real world." Applied Maths is sometimes considered to be divided into the areas of Dynamical Systems, Approximation Techniques, and Probability & Statistics.There are also various Applied Mathematical disciplines which use a combination of these areas but focus on a particular type of application.Apart from a few procedure texts for finding things like square roots, most Old Babylonian problems are couched in a language of measurement of everyday objects and activities. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. At the same time, a local train traveling 30 miles an hour carrying 40 passengers leaves Phoenix bound for Santa Fe...") trails off with a schoolboy character instead imagining that he is on the train.Students had to find lengths of canals dug, weights of stones, lengths of broken reeds, areas of fields, numbers of bricks used in a construction, and so on. As if to add to the confusion, "algebra" is the name for a certain kind of structure in modern algebra.Modern algebra also arguably contains the field of number theory, which has important applications in computer science.Prior to this, all mathematical problems and solutions were written out in words; the more complicated the problem, the more laborious and convoluted the verbal explanation.Examples of word problems can be found dating back to Babylonian times. Word problems have also been satirised in The Simpsons, when a lengthy word problem ("An express train traveling 60 miles per hour leaves Santa Fe bound for Phoenix, 520 miles away.Such phenomena include the spread of computer viruses on a network, the growth of tumors, the risk associated with certain contracts traded on the stock market, and the formation of turbulence around an aircraft.Mathematics provides a kind of "quality control" for the development of trustworthy theories and equations which are important to people in most modern technical discplines such as engineering and economics.