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I was reading this answer to an amusing comic related question: https://math.stackexchange.com/a/166891/35132 and I understand that in the linked answer, the examples of how four may be expressed used base (expressed in decimal!!) is 10, 4, 3 for 4, 10, 11.

What I can't figure out is what base would need to be used for his last example (100) to equal 4 in decimal?

P.S. Are maths questions like this always this hard to put in words?!

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    Note that the answer holds for "base 10" in all bases N>4. So if by "base 10" you mean base S(S(S(S(S(0))))) (where S is the successor operator ("plus one")), you're okay. You don't have to mean ten.2012-07-05

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Let the required base be $b$. Then, $1 \cdot b^2 + 0 \cdot b^1 + 0 \cdot b^0 = 4 \implies b = 2.$

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$100_{(b)}=b^2+0\cdot b + 0=b^2 \,.$

So 100 in base $b$ is just the number $b^2$.... Can you find $b$ now?

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    ^that is a rather absurd line of reasoning.2016-02-23