Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 4 - Number Representation and Calculation - 4.4 Looking Back at Early Numeration Systems - Exercise Set 4.4 - Page 242: 63


Base five Numeral is \[{{1232}_{\text{five}}}\]

Work Step by Step

Let’s convert by Using Roman Numeral System, First write the given Roman numeral as Hindu Arabian Numeral CXCII\[\,=100+\left( 100-10 \right)+2=192\] 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 \[\begin{align} 192=125\overset{1}{\overline{\left){192}\right.}} & \\ \text{ }\underline{125} & \\ \text{ 67} & \\ \end{align}\] Then, \[\begin{align} & 25\overset{2}{\overline{\left){67}\right.}} \\ & \text{ }\underline{50} \\ & \text{ 17} \\ \end{align}\] Then, \[\begin{align} & 5\overset{3}{\overline{\left){17}\right.}} \\ & \text{ }\underline{15} \\ & \text{ }2 \\ \end{align}\] So, ${{192}_{\text{ten}}}={{1232}_{\text{five}}}$ Using the three quotients and the last remainder, we can directly write the Base five Numeral as above.
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