Chapter 3, Problem 39E

Explanation of Solution

a.

To prove: “Adding one to nine’s complement of a number is equal to ten’s complement of the same number”.

Proof:

Formula for ten’s complement to compute the number:

$\text{Negative}\left(\text{I}\right)={\text{ 10}}^{\text{k}}-\text{ I }$        (1)

In the above formula,

• “k” represents the number of digits.
• “I” represents the number to be calculated.

Formula for nine’s complement to compute the number:

$\text{Negative[I]=9999}\mathrm{....}-\text{I}$

In the above formula,

• “I” represents the number to be calculated.
• “9999….” represents all 9s in which the number of 9s should be equal to the number of digits in a given number “I”.

Adding one to Nine’s complement formula to calculate the numbers:

$\text{Negative[I]=(9999}\mathrm{....}-\text{I)+1}$        (2)

Comparing Equation (1) and Equation (2),

$\begin{array}{l}{\text{10}}^{\text{k}}-\text{I=(9999}\mathrm{....}-\text{I)+1}\\ {\text{10}}^{\text{k}}-\text{I=9999}\mathrm{....}-\text{I+1}\\ {\text{10}}^{\text{k}}-\cancel{\text{I}}\text{+}\cancel{\text{I}}\text{=9999}\mathrm{}\end{array}$

Explanation of Solution

b.

Calculating ten’s complements using nine’s complement plus one formula for “-35768”:

Nine’s complement plus one formula to compute the numbers:

$\text{Negative[I]=(9999}\mathrm{....}-\text{I)+1}$

$\begin{array}{c}\text{Negative[35768]=(99999}-\text{35768)+1}\\ \text{=64231+1}\\ \text{=64232}\end{array}$

Therefore, ten’s complement using nine’s complement plus one formula for “-35768” is “64232”.

Calculating ten’s complements using nine’s complement plus one formula for “-444455”:

Nine’s complement plus one formula to compute the numbers:

$\text{Negative[I]=(9999}\mathrm{...}$

Explanation of Solution

c.

Nine’s complement plus one method is easier.

Justification:

• Borrowing of values are needed for subtraction in direct ten’s complement method but in nine’ complement plus one method, borrowing of values are not needed for subtraction...