# Method of Complements in Base 17 Given a base 10 number.

Given the following table which shows the symbols I am using when representing numbers in base 17.

\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline 0_{10} & 1_{10} & 2_{10} & 3_{10} & 4_{10} & 5_{10} & 6_{10} & 7_{10} & 8_{10} & 9_{10} & 10_{10} & 11_{10} & 12_{10} & 13_{10} & 14_{10} & 15_{10} & 16_{10} \\ \hline 0_{17} & 1_{17} & 2_{17} & 3_{17} & 4_{17} & 5_{17} & 6_{17} & 7_{17} & 8_{17} & 9_{17} & a_{17} & b_{17} & c_{17} & d_{17} & e_{17} & f_{17} & g_{17} \\ \hline \end{array}

If I want to convert $-9_{10}$ to a number in base $17$ using the method of complemnts, what is the correct algorithm to do this? Currently what am doing is the following.

1. Convert $9_{10}$ to base 17 by continuously dividing $9_{10}$ by $17_{10}$ until the quotient is $0$.

\begin{array}{|c|c|} \hline Division & Remainder \\ \hline 9_{10} / 17_{10} & 9_{10} \\ \hline 0_{10} & \\ \hline \end{array}

1. So $9_{10} \rightarrow 9_{17}$. Now I convert $9_{17}$ to its base 17's complement using the following algorithm. I subtract it from the largest number $16_{10}$ or $g_{17}$, then I add $1$.

$$g - 9 = 7$$ $$7 + 1 = 8$$

Therefore, using what I think to be the method of complements, is $8_{17}$ the complements form of $9_{17}$? If not what is the correct algorithm to get the $17's$ complement of $-9_{10}$ in base $17$?