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I want to convert a decimal (base 10) number to its binary (base 10) equivalent. The binary string has to be of infinite length. Is any of the following functions correct for non-negative integers $x$: $$ x = \sum_{i=0}^\infty 2^i $$ or $$ x = \sum 2^i ; i \in \{ 0,1,2,...\} $$ for unique $i$, in both cases.

Edit: I know that a more appropriate function would be $$ x = \sum_{i=0}^\infty y_i2^i ; y_i \in \{ 0,1\} $$ but I wanted to know if any of the above two formulations would be equivalent to this.
Thanks

user51013
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3 Answers3

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If your $x$ is between $0$ and $1$, you can write $x=\sum_{i=1}^\infty a_i2^{-i}$ where $a_i \in \{0,1\}$ are binary digits of the expansion. If it is not, you can add the integral part of $x$ converted to binary to this expression. You can't have an infinite binary string to the left of the fraction point as the value would be infinite.

Ross Millikan
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  • Thanks, but will there be a problem if there are infinite zeros to the left of the binary equivalent number, as in $ ...0001010$ for decimal 10 – user51013 Aug 07 '15 at 19:56
  • I edited the question while you were answering. could you kindly address the present question. Thanks again – user51013 Aug 07 '15 at 19:57
  • Neither of the others is equivalent because they do not express the fact that the binary bits can be different. Both say $x=1+2+4+8+\dots$, which is not what you want to do. Your last is not correct because the exponents of $2$ must be negative as I pointed out. – Ross Millikan Aug 07 '15 at 20:01
  • You mean to say that a binary value with infinite leading zeros is not feasible. But if i do not mean the actual binary value and just use it as an infinite binary string, then will the third function be ok? – user51013 Aug 07 '15 at 20:12
  • If you are just working with strings, why use a summation? Your third is fine as long as the $y_i$ are eventually all zeros. In that case it is really a finite sum with a finite result. I feel I still do not understand what you are looking for. – Ross Millikan Aug 07 '15 at 20:33
  • I was just looking for a notation to depict the string. Thanks a lot for your help. – user51013 Aug 07 '15 at 20:35
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Trying to catch the direction of the question, probably with something like taking advantage of the fact that 10=5*2 y making use of modular arithmetic to calculate the decimal overflow, i could'n t imagine other way without entry in a kind of complex "iterating" stuff... and then transform to binary.

pd: Like a detour... https://en.wikipedia.org/wiki/Quote_notation

sigma2sigma
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I have a function that does just that:

This function counts the number of digits of the binary representation of 'n':$$P(n)=⌈\log_{2}(n)⌉-⌈\log_{2}(n)-⌊\log_{2}(n)⌋⌉+1$$

And this function computes the decimal representation of the n in binary:

$$B(n)=\sum_{i=1}^{P(n)} (⌊\frac{n}{2^{i-1}}⌋\mod2)10^{i-1}$$

Iaroslav
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