# A function for decimal to binary conversion

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

• How would different values of x arise from your formulation? Aug 7 '15 at 19:25
• Thanks @DJohnM, edited the question. Will it be correct now Aug 7 '15 at 19:30

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.
• 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 Aug 7 '15 at 19:56
• 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. Aug 7 '15 at 20:01
• 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. Aug 7 '15 at 20:33