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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.
In this web page, I'll discuss the mathematical meaning of "real number."
Before that, I want to discuss this more elementary question: where did the name
"real" come from? (It turns out to have little to do with the deeper properties
of real numbers.) To answer that question, I first need to talk about complex
numbers.
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