Mark Meckes recently commented on my post on writing Astounding Math Stories that students “usually think of decimal expansions as formal expressions”. His point is well taken, but I would go further and say that they think of real numbers as decimal expansions, hence as formal expressions.

For example, 1/3 is approximately 0.333… However, 1/3 is exactly 1/3. The expression “1/3” is more exact than the decimal expansion. Similarly \sqrt 2 is defined exactly as the positive real number whose square is 2. And of course there are still other representations of some or all real numbers, for example binary notation, representations as limits, as solutions of equations, and so on.

The main thing to understand is that every interesting mathematical object has several representations, each representation coming from a different system of metaphors. And if you are going to understand math you have to be aware of various representations of the same object and hold (some of) the details of several of them in your head at once. Even “2 + 3 = 5″ is talking about two different representations of a number simultaneously. William Thurston once said that it was a revelation to him as a child that when you divide 127 by 23 you get 127/23. That notation”127/23” tells you two things: exactly what the number is and one way it is related to other numbers. That kind of phenomenon is what makes math work.

I went on and on about this stuff in abstractmath.org here and here.

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