Thinking Mathematically (6th Edition)

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

Chapter 4 - Number Representation and Calculation - 4.3 Computation in Positional Systems - Exercise Set 4.3 - Page 234: 35

Answer

The quotient of the given division in base four is, \[{{20}_{\text{four}}}\].

Work Step by Step

The division in base four can be performed similar to that in base ten. First, divide the first two digits of the dividend,\[{{10}_{\text{four}}}\]by \[{{2}_{\text{four}}}\]. In the given table, the largest product, in the vertical column of 2, that is less than or equal to \[{{10}_{\text{four}}}\] is \[{{10}_{\text{four}}}\]. Since, \[{{2}_{\text{four}}}\times {{2}_{\text{four}}}={{10}_{\text{four}}}\], the first number in the quotient is \[{{2}_{\text{four}}}\]: \[{{2}_{\text{four}}}\overset{2}{\overline{\left){{{100}_{\text{four}}}}\right.}}\] Now, perform the multiplication \[{{2}_{\text{four}}}\times {{2}_{\text{four}}}={{10}_{\text{four}}}\] and write \[{{10}_{\text{four}}}\]under the first two digits of the dividend: \[{{2}_{\text{four}}}\overset{2}{\overline{\left){\begin{align} & {{100}_{\text{four}}} \\ & \underline{10} \\ \end{align}}\right.}}\] Now, perform the subtraction: \[{{10}_{\text{four}}}-{{10}_{\text{four}}}=0\] \[{{2}_{\text{four}}}\overset{2}{\overline{\left){\begin{align} & {{100}_{\text{four}}} \\ & \underline{10} \\ \end{align}}\right.}}\] 0 Now, drop down the next digit in the dividend, \[0\]: \[{{2}_{\text{four}}}\overset{2}{\overline{\left){\begin{align} & {{100}_{\text{four}}} \\ & \underline{10} \\ \end{align}}\right.}}\] 00 Since,0 is less than 2, write a 0 after 2 in the quotient: \[{{2}_{\text{four}}}\overset{{{20}_{\text{four}}}}{\overline{\left){\begin{align} & {{100}_{\text{four}}} \\ & \underline{10} \\ \end{align}}\right.}}\] 00 Thus, the obtained quotient is \[{{20}_{\text{four}}}\]. Now, to check whether the above obtained solution is correct, perform the division by converting thedivisor, the dividend and the quotient into base ten: \[{{2}_{\text{four}}}=2\], \[{{100}_{\text{four}}}=16\] and \[{{20}_{\text{four}}}=8\] Since,\[2\overset{8}{\overline{\left){16}\right.}}\] is indeed true, the answer obtained is correct. Hence, the quotient of the given division in base four is, \[{{20}_{\text{four}}}\].
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