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The mathematics we use today is based on the decimal system. I was wondering if some or all of these proofs would hold true in other numeral systems. For example binary or hexadecimal. Is there any field which works with questions like this?

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    $\begingroup$ "The mathematics we use today is general mathematics." I don't know what that means. Theorems that depend on the base may be true in one base and false in another. Theorems that don't depend on the base (e.g., Fermat's Last Theorem) are perfectly fine regardless of the base. If what I've written reads like trivialities and platitudes, it's because it's very difficult to make sense of your question. Is there some specific proof you are worried about? $\endgroup$ – Gerry Myerson Aug 9 '16 at 13:14
  • $\begingroup$ @GerryMyerson Is it true that such a discussion leans toward science fiction? Hexadecimal beings have been dreamed about for quite a while. As for mathematics, the vast majority of theorems are base-independent, but the way we would think about these theorems would change due to our different perception of numbers. $\endgroup$ – Parcly Taxel Aug 9 '16 at 13:18
  • $\begingroup$ I have never dreamed about a hexadecimal being, and I don't know what the term means. I think the discussion leads toward nowhere, because you haven't defined any of your terms precisely enough to support a meaningful discussion. $\endgroup$ – Gerry Myerson Aug 9 '16 at 13:25
  • $\begingroup$ "The mathematics we use today is based on the decimal system." I'm sorry, but this just shows an ignorance of Mathematics. Again, is there any particular proof that you are worried about? $\endgroup$ – Gerry Myerson Aug 9 '16 at 13:28
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With rare exception, mathematics works with numbers, not numerals, so the choice of numeral system you use to represent numbers is completely irrelevant to the mathematics.

The typical exceptions are things that explicitly reference numerals; e.g. elementary school algorithms for arithmetic of numbers expressed in a numeral form.

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