Chapter 2, Problem 1E

Explanation of Solution

a)

Converting 458 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 458. The number that can be subtracted from the given number 458 with the power of 3 is 243. The number 243 is less than 458. So subtract 243 from 458.

$\begin{array}{l}458\\ {\text{-243=3}}^5\text{x1}\\ \text{------------}\\ 215\end{array}$

Step 2:

The number that can be subtracted from the given number 215 with the power of 3 is 81 and it should be multiplied by 2 in order to get the nearest number of 215. The number 162 is less than 215. So subtract 162 from 215.

$\begin{array}{l}215\\ {\text{-162=3}}^4\text{x2}\\ \text{------------}\\ 53\end{array}$

Step 3:

The number that can be subtracted from the given number 53 with the power of 3 is 27. The number 27 is less than 53. So subtract 27 from 53.

$\begin{array}{l}53\\ {\text{-27=3}}^3\text{x1}\\ \text{------------}\\ 26\end{array}$

Step 4:

The number that can be subtracted from the given number 26 with the power of 3 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.

$\begin{array}{l}26\\ {\text{-18=3}}^2\text{x2}\\ \text{------------}\\ 8\end{array}$

Step 5:

The number that can be subtracted from the given number 8 with the power of 3 is 3 and it should be multiplied by 2 in order to get the nearest number of 8. The number 6 is less than 8. So subtract 6 from 8.

$\begin{array}{l}8\\ {\text{-6=3}}^1\text{x2}\\ \text{------------}\\ 2\end{array}$

Step 6:

The number that can be subtracted from the given number 2 with the power of 3 is 1 and it should be multiplied by 2 in order to get the nearest number of 2. So subtract 2 from 2. The result will be 0. The process is stopped as the remainder is 0.

$\begin{array}{l}2\\ {\text{-2=3}}^0\text{x2}\\ \text{------------}\\ 0\end{array}$

Therefore, the equivalent value of ${458}_{10}$ is “$121222_2$”.

Explanation of Solution

b)

Converting 677 base 10 to base 5 using subtraction method:

Step 1:

Check the possibility of multiplying any integer with the powers of 5 which may result in lower number than 677. The number that can be subtracted from the given number 677 with the power of 5 is 625. The number 625 is less than 677. So subtract 625 from 677.

$\begin{array}{l}677\\ {\text{-625=5}}^4\text{x1}\\ \text{------------}\\ \text{52}\end{array}$

Step 2:

Take 5 to the power of 3 that is 125. The number 125 is greater than 52. So make the value as 0.

$\begin{array}{l}52\\ {\text{-0=5}}^3\text{x0}\\ \text{------------}\\ 52\end{array}$

Step 3:

The number that can be subtracted from the given number 52 with the power of 5 is 25 and it should be multiplied by 2 in order to get the nearest number of 52. The number 50 is less than 52. So subtract 50 from 52.

$\begin{array}{l}52\\ {\text{-50=5}}^2\text{x2}\\ \text{------------}\\ 2\end{array}$

Step 4:

Take 5 to the power of 1 that is 5. The number 5 is greater than 2. So make the value as 0.

$\begin{array}{l}2\\ {\text{-0=5}}^1\text{x0}\\ \text{------------}\\ 2\end{array}$

Step 5:

The number that can be subtracted from the given number 2 with the power of 5 is 1 and it should be multiplied by 2 in order to get the nearest number of 2. So subtract 2 from 2. The result will be 0. The process is stopped as the remainder is 0.

Therefore, the equivalent value of ${677}_{10}$ is “$10202_2$”.

Explanation of Solution

c)

Converting 1518 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 1518. The number that can be subtracted from the given number 1518 with the power of 7 is 343 and it should be multiplied by 4 in order to get the nearest number of 1518. The number 1372 is less than 1518. So subtract 1372 from 1518.

$\begin{array}{l}1518\\ {\text{-1372=7}}^3\text{x4}\\ \text{------------}\\ 146\end{array}$

Step 2:

The number that can be subtracted from the given number 146 with the power of 7 is 49 and it should be multiplied by 2 in order to get the nearest number of 146. The number 98 is less than 146. So subtract 98 from 146.

$\begin{array}{l}146\\ {\text{-98=7}}^2\text{x2}\\ \text{------------}\\ 48\end{array}$

Step 3:

The number that can be subtracted from the given number 48 with the power of 7 is 7 and it should be multiplied by 6 in order to get the nearest number of 48. The number 42 is less than 48. So subtract 42 from 48.

$\begin{array}{l}48\\ {\text{-42=7}}^1\text{x6}\\ \text{------------}\\ 6\end{array}$

Step 4:

The number that can be subtracted from the given number 6 with the power of 7 is 1 and it should be multiplied by 6 in order to get the nearest number of 6. So subtract 6 from 6. The result will be 0. The process is stopped as the remainder is 0.

$\begin{array}{l}6\\ {\text{-6=7}}^0\text{x6}\\ \text{------------}\\ 0\end{array}$

Therefore, the equivalent value of ${1518}_{10}$ is “$4266_2$”.

Explanation of Solution

d)

Converting 4401 base 10 to base 9 using subtraction method:

Step 1:

Check the possibility of multiplying any integer with the powers of 9 which may result in lower number than 4401. The number that can be subtracted from the given number 4401 with the power of 9 is 729 and it should be multiplied by 6 in order to get the nearest number of 4410. The number 4374 is less than 4401. So subtract 4374 from 4401.

$\begin{array}{l}4410\\ {\text{-4374=9}}^3\text{x6}\\ \text{------------}\\ 27\end{array}$

Step 2:

Take 9 to the power of 2 that is 81. The number 81 is greater than 27. So make the value as 0.

$\begin{array}{l}27\\ {\text{-0=9}}^2\text{x0}\\ \text{------------}\\ 27\end{array}$

Step 3:

The number that can be subtracted from the given number 27 with the power of 9 is 9 and it should be multiplied by 3 in order to get the nearest number of 27. So subtract 27 from 27.

$\begin{array}{l}27\\ {\text{-27=9}}^1\text{x3}\\ \text{------------}\\ 0\end{array}$

Step 4:

Take 9 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.

$\begin{array}{l}0\\ {\text{-0=9}}^0\text{x0}\\ \text{------------}\\ 0\end{array}$

Therefore, the equivalent value of ${4401}_{10}$ is “$6030_2$”.