What are the "real numbers," really?
The short, simple answer used in calculus courses is that a real number is a point on the number line. That's not the whole truth, but it is adequate for the needs of freshman calculus. The freshman calculus course (at most universities nowadays) follows the 17th century style of Newton and Leibniz, emphasizing computations and omitting many proofs. The omitted proofs depend on a careful explanation of what the "real numbers" really are. That explanation and those proofs were not discovered until the 19th century, after Newton and Leibniz were long dead.
A proper explanation of the real numbers nowadays is covered, if at all, in a course in "real analysis" in the junior or senior year of students who are majoring in mathematics. Surprisingly few students take such a course; perhaps that's because it is too algebraic for the analysts' taste and too analytic to please the algebraists.
|Maria Luisa Duarte|
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Native speaker of: English, Portuguese
PRO pts in category: 4