- #1

robertjford80

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In mathematics, a real number is a value that represents a quantity along a continuous line.

But then it turns out that that definition is not rigorous:

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics.

Since we don't have a rigorous definition of a real number it is hard to justify that pi is a real number. Let's discuss what it means to 'represent a quantity'. Say I were to ask you what is the quantity of x, ultimately, you would have to give me an answer which is a complete sentence with a period at the end, in other words, you're explanation of the quantity would have to end. You would then indicate where on the number line the quantity exists. You can't indicate where pi exists on the number line. If you say that it is between 3.14 and 3.15 then you haven't told me where it exists exactly you have only told where it exists roughly. This could go on ad infinitum. Hence, you can't "represent the quantity" of pi.

Admittedly, though in order to give a real rigorous proof that pi is not real, we would need a rigorous definition of real number, something much more precise then represent the quantity and we don't have that yet.