Elementary Technical Mathematics

Published by Brooks Cole
ISBN 10: 1285199197
ISBN 13: 978-1-28519-919-1

Chapter 16 - Section 16.3 - Subtraction of Binary Numbers - Exercise - Page 554: 28

Answer

$0 0 0 0 1$.

Work Step by Step

$$\text{Solution}$$ Here $A = 10000, B = 1111 .$ Find $A - B = ?$ using 1's complement $\rightarrow$ First, find 1's complement of $B = 1111 $ $\rightarrow $ Write out the numbers as if you were subtracting decimals. $$-\begin{array}{r|r} 10000 \\ 1111 \\ \hline \end{array}$$ $\rightarrow $ Let's use the 1's Complement Method $\rightarrow $ 1's complement of a number is obtained by subtracting all bits from $11$ $\rightarrow $ $\text{1's complement of 1111 is}$ $$-\begin{array}{r|r} 1 1 1 1 1 \\ 0 1 1 1 1\\ \hline 1 0 0 0 0 \end{array}$$ $\rightarrow $ Now Add this 1's complement of B to A Note $\rightarrow$ Addition Rules $0 + 0 = 0$ $0 + 1 = 1$ $1 + 0 = 1$ $1 + 1 = 10$ ("$0$" in the column, carry $1$ to the next bit) So, $$+\begin{array}{r|r} 1 0 0 0 0 \\ 1 0 0 0 0 \\ \hline 1 0 0 0 0 0 \end{array}$$ $\rightarrow$ left most bit of the above result is called carry and add it to the rest part of the result $00000$. $$+\begin{array}{r|r} 00000 \\ 1 \\ \hline 0 0 0 0 1 \end{array}$$ so the Answer is $0 0 0 0 1$.
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