How I convert Decimal Using negetive Base 2? I dont have any idea how i convert deciaml using negetive base 2?
i googling today long time but couldn't find any solution.
can any tell me process?
i need to learn it .
 A: You can do it by recursion. Let me write numbers in base $-2$ in brackets, and decimal numbers without, so that $[0]=0$, $[1]=1$, $[10]=-2$, and $[11]=-1$. I listed these for a reason: They are all the numbers in base $-2$ with at most two digits. Now note that if you add a zero to the right end of a number in this base, the number is multiplied by $-2$. Therefore, if you add two zeros at the end, the number is multiplied by four. This is useful.
To write a number $n$ in base $-2$, reduce it by writing $n=4q+r$ with the “remainder” $r\in\{-2,-1,0,1\}$. Write $r$ as two digits, figure out the representation of $k$ recursively, and append the digits of $r$ at the end.
Here is an example:
$$
  -17=-4\cdot4-1=(-1\cdot4)\cdot4+[11]=([11]\cdot4)\cdot4+[11]\\
  =[1100]\cdot4+[11]=[110000]+[11]=[110011].
$$
To check that: The final bracket is $-32+16-2+1=-17$.
A: One can use the standard algorithm for converting from one base to another. Divide by the target base $b$ to get an integer quotient and a remainder in the chosen range of digits, generally but not always $\{0,\dots,|b|-1\}$; replace the original number by the quotient and repeat; continue until you get a quotient of $0$, and read the remainders in reverse order. For example, to convert $-19$ to base $-2$:
$$\begin{align*}
-19&=-2\cdot 10+\color{red}{1}\\
10&=-2\cdot(-5)+\color{red}{0}\\
-5&=-2\cdot3+\color{red}{1}\\
3&=-2\cdot(-1)+\color{red}{1}\\
-1&=-2\cdot1+\color{red}{1}\\
1&=-2\cdot0+\color{red}{1}\\
0&=-2\cdot0
\end{align*}$$
Reading the remainders (in red) from the bottom up, we find that $-19=111101_{-2}$. As a check,
$$\begin{align*}
1\cdot(-2)^5&+1\cdot(-2)^4+1\cdot(-2)^3+1\cdot(-2)^2+0\cdot(-2)^1+1\cdot(-2)^0\\
&=-32+16-8+4+1\\
&=-19\;.
\end{align*}$$
Added: Here are a couple more examples.
$$\begin{align*}
4&=-2\cdot(\color{blue}{-2})+\color{red}{0}\\
\color{blue}{-2}&=-2\cdot\color{green}{1}+\color{red}{0}\\
\color{green}{1}&=-2\cdot0+\color{red}{1}\;,
\end{align*}$$
so $4=100_{-2}$.
$$\begin{align*}
7&=-2\cdot(\color{blue}{-3})+\color{red}{1}\\
\color{blue}{-3}&=-2\cdot\color{green}{2}+\color{red}{1}\\
\color{green}{2}&=-2\cdot(\color{blue}{-1})+\color{red}{0}\\
\color{blue}{-1}&=-2\cdot\color{green}{1}+\color{red}{1}\\
\color{green}{1}&=-2\cdot0+\color{red}{1}\;,
\end{align*}$$
so $7=11011_{-2}$.
