<text><span class="style10">athematics and its Applications (1 of 4)</span><span class="style7">Many people think of mathematics in terms of rules to be learned in order to manipulate symbols or study numbers or shapes in the abstract for their own sake. Mathematical theory does develop in the abstract; it need have no dependence on anything outside itself. The truth of the theory is measured by logic rather than experiment. However, one of its most valuable uses is in describing or modeling processes in the real world, and thus there is constant interaction between pure mathematics and applied mathematics.Mathematics may be considered as the very general study of the structure of systems. Since the study is unrelated to the physical world, rigorous formal proofs are sought, rather than experimental verifications. Theory is presented in terms of a small number of given truths (known as </span><span class="style26">axioms</span><span class="style7">) from which the entire theory can be inferred.Thus, the aims are for generality in approach and rigor in proof, aims that explain the traditional concern of mathematicians for the unification of seemingly different branches of mathematics. As an example, Descartes showed that geometrical figures could be described in terms of algebra, enabling geometric proofs to be established in terms of arithmetic, so that both generality and rigor were advanced.</span><span class="style10">Applied mathematics and modeling</span><span class="style7">There is no sharp boundary between the study of mathematical systems in the abstract (the field of </span><span class="style26">pure mathematics</span><span class="style7">) and the study of such systems to make inferences about certain physical systems that are described by the mathematical theory (the field of </span><span class="style26">applied mathematics</span><span class="style7">). In principle, any branch of mathematics may turn out to describe some physical, economic, biological, medical, or other system. </span><span class="style26">Modeling</span><span class="style7"> a physical system consists of seeking a formal mathematical theory that conforms with the properties of the physical system. Often, as for example in computer simulations of space travel, the mathematical theories are very large and complex, but sometimes the model can be quite simple. Sometimes, known mathematics can describe and predict the behavior of the system; at other times, the modeling can give rise to completely new branches of mathematics.Applied mathematics encompasses many specialized fields in which the relation ships between the experimental findings and the mathematical theories are well established. Although the subject can include the application of statistical theory to such areas as sociology, the term is usually restricted to the application of the methods of advanced calculus, linear algebra and other branches of advanced mathematics to physical and technological processes.</span><span class="style10">Triangulation, geometry and trigonometry</span><span class="style7">A simple example of a mathematical model is the representation of a portion of the Earth's surface by a set of interlocking triangles, from the measurement of which maps may be constructed. The triangulation model uses the rules of geometry and trigonometry to derive angles and distances that cannot be measured directly.Geometry establishes that two triangles each have angles of the same sizes if, and only if, corresponding pairs of sides are in the same proportions.</span></text>
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<text>ΓÇó ASTRONOMYΓÇó PHYSICSΓÇó CHEMISTRYΓÇó THE HISTORY OF SCIENCE</text>
