What are the greater uses of classifying numbers in advanced mathematics

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In high school throughout college, we have been thought what is rational, integers etc. But how does is fit to the greater scheme of things. Trigonometry, Calculus, Mensuration have very obvious applications in real life. What does this have?

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Trigonometry and calculus require a consistent definition of real number if they are to be well-founded. Such a definition is typically arrived at by an iterative process which defines each of the natural numbers, the integers, the rational numbers, and the real numbers in terms of the preceding number types.