I'm wondering if there are number systems with bases other than integers? For example, with a fractional, imaginary, irrational, transcendental basis, or with the basis "infinity"? If there are, then how is the translation made in and between them?
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What should the digits be in such a system ? – Peter Jul 17 '23 at 13:19
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Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Jul 17 '23 at 13:25
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2Sure. See here. The way to produce the positional representation of a number is the same as always, division with remainder. You divide the number $x$ by the base $b$ and get $g,r$ such that $x=bg+r$ and $0\leq r<b$. Then $r$ is the first digit from the right. Dividing $g$ again you get the next digit, etc. – NDB Jul 17 '23 at 13:32
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2Maybe this one is a more specific link. – NDB Jul 17 '23 at 13:34
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I like this example ... in $\mathbb C$, take base $-1+i$ with digit set $D := \{0,1\}$. Numbers of the form $$ \sum_{k=0}^K a_k (-1+i)^k,\qquad a_k \in \{0,1\} $$ are the Gaussian integers . And numbers of the form $$ \sum_{k=-\infty}^K a_k (-1+i)^k,\qquad a_k \in \{0,1\} $$ give us all of $\mathbb{C}$. And the "fractions" for this number system $$ \sum_{k=-\infty}^{-1} a_k (-1+i)^k,\qquad a_k \in \{0,1\} $$ form the fractal known as the twindragon.
GEdgar
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