Chapter 2, Problem 2E

Explanation of Solution

a)

Converting 588 base 10 to base 3 using subtraction method:

Step 1:

Check the possibility of multiplying any integer with the powers of 3 which may result in lower number than 588. The number that can be subtracted from the given number 588 with the power of 3 is 243 and it should be multiplied by 2 in order to get the nearest number of 588. The number 486 is less than 588. So subtract 486 from 588.

$\begin{array}{l}588\\ {\text{-486=3}}^5\text{x2}\\ \text{------------}\\ 102\end{array}$

Step 2:

The number that can be subtracted from the given number 102 with the power of 3 is 81. The number 81 is less than 102. So subtract 81 from 102.

$\begin{array}{l}102\\ {\text{-81=3}}^4\text{x1}\\ \text{------------}\\ 21\end{array}$

Step 3:

Take 3 to the power of 3 that is 27. The number 27 is greater than 21. So make the value as 0.

$\begin{array}{l}21\\ {\text{-0=3}}^3\text{x0}\\ \text{------------}\\ 21\end{array}$

Step 4:

The number that can be subtracted from the given number 21 with the power of 3 is 9 and it should be multiplied by 2 in order to get the nearest number of 21. The number 18 is less than 21. So subtract 18 from 21.

$\begin{array}{l}21\\ {\text{-18=3}}^2\text{x2}\\ \text{------------}\\ 3\end{array}$

Step 5:

The number that can be subtracted from the given number 3 with the power of 3 is 3. So subtract 3 from 3.

$\begin{array}{l}3\\ {\text{-3=3}}^1\text{x1}\\ \text{------------}\\ 0\end{array}$

Step 6:

Take 3 to the power of 0 that is 1. So make the value as 0. The result will be 0. The process is stopped as the remainder is 0.

$\begin{array}{l}0\\ {\text{-0=3}}^0\text{x0}\\ \text{------------}\\ 0\end{array}$

Therefore, the equivalent value of ${588}_{10}$ is “$210210_2$”.

Explanation of Solution

b)

Converting 2254 base 10 to base 5 using subtraction methods:

Step 1:

Check the possibility of multiplying any integer with the powers of 5 which may result in lower number than 2254. The number that can be subtracted from the given number 2254 with the power of 5 is 625 and it should be multiplied by 3 in order to get the nearest number of 2254. The number 1875 is less than 2254. So subtract 1875 from 2254.

$\begin{array}{l}2254\\ {\text{-1875=5}}^4\text{x3}\\ \text{------------}\\ 379\end{array}$

Step 2:

The number that can be subtracted from the given number 379 with the power of 5 is 125 and it should be multiplied by 3 in order to get the nearest number of 379. The number 375 is less than 379. So subtract 375 from 379.

$\begin{array}{l}379\\ {\text{-375=5}}^3\text{x3}\\ \text{------------}\\ 4\end{array}$

Step 3:

Take 5 to the power of 2 that is 25. The number 25 is greater than 4. So make the value as 0.

$\begin{array}{l}4\\ {\text{-0=5}}^2\text{x0}\\ \text{------------}\\ 4\end{array}$

Step 4:

Take 5 to the power of 1 that is 5. The number 5 is greater than 4. So make the value as 0.

$\begin{array}{l}4\\ {\text{-0=5}}^1\text{x0}\\ \text{------------}\\ 4\end{array}$

Step 5:

The number that can be subtracted from the given number 4 with the power of 5 is 1 and it should be multiplied by 4 in order to get the nearest number of 4. So subtract 4 from . The result will be 0. The process is stopped as the remainder is 0.

$\begin{array}{l}4\\ {\text{-4=5}}^1\text{x4}\\ \text{------------}\\ 0\end{array}$

Therefore, the equivalent value of ${2254}_{10}$ is “$33004_2$”.

Explanation of Solution

c)

Converting 652 base 10 to base 7 using subtraction method:

Step 1:

Check the possibility of multiplying any integer with the powers of 7 which may result in lower number than 652. The number that can be subtracted from the given number 652 with the power of 7 is 343. The number 343 is less than 652. So subtract 343 from 652.

$\begin{array}{l}652\\ {\text{-343=7}}^3\text{x1}\\ \text{------------}\\ 309\end{array}$

Step 2:

The number that can be subtracted from the given number 309 with the power of 7 is 49 and it should be multiplied by 6 in order to get the nearest number of 309. The number 294 is less than 309. So subtract 294 from 309.

$\begin{array}{l}309\\ {\text{-294=7}}^2\text{x6}\\ \text{------------}\\ 15\end{array}$

Step 3:

The number that can be subtracted from the given number 15 with the power of 7 is 7 and it should be multiplied by 2 in order to get the nearest number of 15. The number 14 is less than 15. So subtract 14 from 15.

$\begin{array}{l}15\\ {\text{-14=7}}^1\text{x2}\\ \text{------------}\\ 1\end{array}$

Step 4:

The number that can be subtracted from the given number 1 with the power of 7 is 1. So subtract 1 from 1. The result will be 0. The process is stopped as the remainder is 0.

$\begin{array}{l}1\\ {\text{-1=7}}^0\text{x1}\\ \text{------------}\\ 0\end{array}$

Therefore, the equivalent value of ${652}_{10}$ is “$1621_2$”.

Explanation of Solution

d)

Converting 3104 base 10 to base 9 using subtraction methods:

Step 1:

Check the possibility of multiplying any integer with the powers of 9 which may result in lower number than 3104. The number that can be subtracted from the given number 3104 with the power of 9 is 729 and it should be multiplied by 4 in order to get the nearest number of 3104. The number 2916 is less than 3104. So subtract 2916 from 3014.

$\begin{array}{l}3104\\ {\text{-2916=9}}^3\text{x4}\\ \text{------------}\\ 188\end{array}$

Step 2:

The number that can be subtracted from the given number 188 with the power of 9 is 81 and it should be multiplied by 2 in order to get the nearest number of 188. The number 162 is less than 188. So subtract 162 from 188.

$\begin{array}{l}188\\ {\text{-162=9}}^2\text{x2}\\ \text{------------}\\ 26\end{array}$

Step 3:

The number that can be subtracted from the given number 26 with the power of 9 is 9 and it should be multiplied by 2 in order to get the nearest number of 26. The number 18 is less than 26. So subtract 18 from 26.

$\begin{array}{l}26\\ {\text{-8=9}}^1\text{x2}\\ \text{------------}\\ 8\end{array}$

Step 4:

The number that can be subtracted from the given number 8 with the power of 9 is 1 and it should be multiplied by 8 in order to get the nearest number of 8. So subtract 8 from 8. The result will be 0. The process is stopped as the remainder is 0.

Therefore, the equivalent value of ${3104}_{10}$ is “$4228_2$”.