## Thinking Mathematically (6th Edition)

Base five Numeral is ${{12344}_{\text{five}}}$
First write the given Roman numeral as Hindu Arabian Numeral CMLXXIV$\,=\left( 1000-100 \right)+\left( 50+10 \right)+10+\left( 5-1 \right)=974$ Now let’s find out the base five numeral using base ten Hindu Arabic numeral, The Place values in base five numeral are ${{5}^{1}},\,{{5}^{2}},{{5}^{3}},{{5}^{4}},\ldots$ or $5,25,125,625,\ldots$ Using highest divisor from these place values which is less than dividend for the first time and then continue in same manner, we get \begin{align} 974=625\overset{1}{\overline{\left){974}\right.}} & \\ \underline{625} & \\ 349 & \\ \end{align} Then, \begin{align} 125\overset{2}{\overline{\left){349}\right.}} & \\ \underline{250} & \\ 99 & \\ \end{align} Then, \begin{align} 25\overset{3}{\overline{\left){99}\right.}} & \\ \underline{75} & \\ 24 & \\ \end{align} Then, \begin{align} 245\overset{4}{\overline{\left){24}\right.}} & \\ \underline{20} & \\ 4 & \\ \end{align} So, ${{974}_{\text{ten}}}=\,{{12344}_{\text{five}}}$ Using the four quotients and the last remainder, we can directly write the Base Five Numeral as above.