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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.
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.
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.
Many contest problems test one's fluency with algebraic manipulation.
Algebra can be used to solve different types of equations, but algebra is also many other things Modern algebra (or "higher", or "abstract" algebra) deals (in part) with generalisations of the normal operations seen arithmetic and high school algebra.The second example, however, does not necessarily have to be "real-life" to a high school student, who may find that it is easier to handle the following problem: Word problems are a common way to train and test understanding of underlying concepts within a descriptive problem, instead of solely testing the student's capability to perform algebraic manipulation or other "mechanical" skills.The modern notation that enables mathematical ideas to be expressed symbolically was developed in Europe from the sixteenth century onwards.Groups, rings, fields, modules, and vector spaces are common objects of study in higher algebra.Algebra Involving Equation Algebra can be used to solve equations as simple as 3x=9 but in some cases so complex that mathematicians have not figured how to solve the particular equation yet.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. The relevance of these situations to the students is varying.The situation in the first example is well-known to most people and may be useful in helping primary school students to understand the concept of subtraction.The logico-mathematical properties can be classified in numerous ways, but one such scheme is to classify the quantities in the problem (assuming the word problem is primarily numerical) into known quantities (the values given in the text of the problem), wanted quantities (the values that need to be found) and auxiliary quantities (values that may need to be found as intermediate stages of the problem).These examples are not only intended to force the students into developing mathematical models on their own, but may also be used to promote mathematical interest and understanding by relating the subject to real-life situations.