Mathematics is the art of effective reasoning. Progress in mathematics is encoded and communicated via its own language, a universal language that is understood and accepted throughout the world. Indeed, in appropriate contexts, communication via the language of mathematics is typically easier than communication via a natural language. This is because mathematical language is clear and concise.
Mathematical language is more than just the formulae that are instantly recognised as part of mathematics. The language is sometimes called mathematical “vernacular”1 because it is a form of natural language with its own style, idioms and conventions. It is impossible to teach mathematical vernacular, but we can summarise some of its vocabulary. This chapter is about elements of the vocabulary that occur in almost all mathematical texts, namely sets, predicates, functions and relations. Later chapters expand on these topics in further detail.
Mathematical notation is an integral part of the language of mathematics, so notational issues are a recurring theme in the chapter. Well-designed mathematical notation is of tremendous value in problem solving. By far the best example is the positional notation we use to denote numbers (for example, 12, 101, 1.31; note the use of “0” and the decimal point). Thanks to this notation, we are all able to perform numerical calculations that are unimaginable ...
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