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Question

Answers

[a] ${{\left( 111 \right)}_{2}}$ and ${{\left( 100 \right)}_{2}}$

[b] ${{\left( 100 \right)}_{2}}$ and ${{\left( 111 \right)}_{2}}$

[c] ${{\left( 101 \right)}_{2}}$ and ${{\left( 101 \right)}_{2}}$

[d] ${{\left( 100 \right)}_{2}}$ and ${{\left( 100 \right)}_{2}}$

Answer
Verified

Hint: Convert the binary numbers in decimal form. Divide to find quotient and remainder. Then convert the remainder and quotient back in binary form to get the answer.

Complete step-by-step answer:

Binary system: In the binary system, every number is represented as a string of 0â€™s and 1â€™s. The place value of each place increases two times moving right to the left, e.g. ${{\left( 111 \right)}_{2}}$. The first 1 to the left has place value 4, the second 1 from left has place value 2, and the last one has place value 1.

Decimal system: In the Decimal system, every number is represented using the digits 0-9. The place value of each place increases ten times moving right to the left, e.g. ${{\left( 121 \right)}_{10}}$. The first 1 to the left has place value 100, digit 2 has place value $2\times 10$ , and the last one has place value 1.

Procedure for converting Binary to Decimal:

Find the place value of each digit in the number. The place value \[Digit\times {{2}^{\text{Number of digits to the right of the digit till ''}\centerdot \text{'' }}}\]

e.g., in the number (111011) in binary representation, the place of highlighted 1 = $1\times {{2}^{3}}={{2}^{3}}=8$.

Add the place values of each digit to get the number in decimal representation.

Converting Decimal to Binary.

.Divide P by 2 and note the quotient and remainder.

If quotient = 0, Stop the process

Otherwise, set P = quotient and repeat the above process.

Write all remainders in reverse order.

This is the binary representation of the number.

The above process will become clear in the following example.

Let P = 72

Dividing P by 2 gives quotient = 36 and remainder = 0

So we set P = 36

Dividing P by 2 gives quotient = 18 and remainder = 0

So wet set P = 18

Dividing P by 2 gives quotient = 9 and remainder = 0

So, we set P = 9

Dividing P by 2 gives quotient = 4 and remainder = 1

So, we set P =4

Dividing P by 2 gives quotient = 2 and remainder = 0.

We set P =2

Dividing P by 2 gives quotient = 1 and remainder = 0

We set P = 1

Dividing P by 2 gives quotient = 0 and remainder = 1.

Since quotient = 0 we stop

Writing remainders in reverse order gives 72 = (1001000) in binary representation

Converting 101110 in decimal representation

$\begin{align}

& =1\times {{2}^{5}}+0\times {{2}^{4}}+1\times {{2}^{3}}+1\times {{2}^{2}}+1\times 2+0\times {{2}^{0}} \\

& =32+8+4+2 \\

& =46 \\

\end{align}$

Converting 110 in decimal representation

\[\begin{align}

& =1\times {{2}^{2}}+1\times 2+0\times {{2}^{0}} \\

& =4+2 \\

& =6 \\

\end{align}\]

Dividing 46 by 6 gives quotient = 7 and remainder = 4

Converting 7 in binary representation gives $7\text{ }={{\left( 111 \right)}_{2}}$

Converting 4 in binary representation gives \[4\text{ }={{\left( 100 \right)}_{2}}\]

Hence quotient = ${{\left( 111 \right)}_{2}}$ and remainder = ${{\left( 100 \right)}_{2}}$

Note: [1] We converted the above binary division in decimal form because it is easier to perform division in decimal representation since we are acquainted with it in our day to day life.

[2] The above method for converting decimal to binary is known as an algorithm.

Complete step-by-step answer:

Binary system: In the binary system, every number is represented as a string of 0â€™s and 1â€™s. The place value of each place increases two times moving right to the left, e.g. ${{\left( 111 \right)}_{2}}$. The first 1 to the left has place value 4, the second 1 from left has place value 2, and the last one has place value 1.

Decimal system: In the Decimal system, every number is represented using the digits 0-9. The place value of each place increases ten times moving right to the left, e.g. ${{\left( 121 \right)}_{10}}$. The first 1 to the left has place value 100, digit 2 has place value $2\times 10$ , and the last one has place value 1.

Procedure for converting Binary to Decimal:

Find the place value of each digit in the number. The place value \[Digit\times {{2}^{\text{Number of digits to the right of the digit till ''}\centerdot \text{'' }}}\]

e.g., in the number (111011) in binary representation, the place of highlighted 1 = $1\times {{2}^{3}}={{2}^{3}}=8$.

Add the place values of each digit to get the number in decimal representation.

Converting Decimal to Binary.

.Divide P by 2 and note the quotient and remainder.

If quotient = 0, Stop the process

Otherwise, set P = quotient and repeat the above process.

Write all remainders in reverse order.

This is the binary representation of the number.

The above process will become clear in the following example.

Let P = 72

Dividing P by 2 gives quotient = 36 and remainder = 0

So we set P = 36

Dividing P by 2 gives quotient = 18 and remainder = 0

So wet set P = 18

Dividing P by 2 gives quotient = 9 and remainder = 0

So, we set P = 9

Dividing P by 2 gives quotient = 4 and remainder = 1

So, we set P =4

Dividing P by 2 gives quotient = 2 and remainder = 0.

We set P =2

Dividing P by 2 gives quotient = 1 and remainder = 0

We set P = 1

Dividing P by 2 gives quotient = 0 and remainder = 1.

Since quotient = 0 we stop

Writing remainders in reverse order gives 72 = (1001000) in binary representation

Converting 101110 in decimal representation

$\begin{align}

& =1\times {{2}^{5}}+0\times {{2}^{4}}+1\times {{2}^{3}}+1\times {{2}^{2}}+1\times 2+0\times {{2}^{0}} \\

& =32+8+4+2 \\

& =46 \\

\end{align}$

Converting 110 in decimal representation

\[\begin{align}

& =1\times {{2}^{2}}+1\times 2+0\times {{2}^{0}} \\

& =4+2 \\

& =6 \\

\end{align}\]

Dividing 46 by 6 gives quotient = 7 and remainder = 4

Converting 7 in binary representation gives $7\text{ }={{\left( 111 \right)}_{2}}$

Converting 4 in binary representation gives \[4\text{ }={{\left( 100 \right)}_{2}}\]

Hence quotient = ${{\left( 111 \right)}_{2}}$ and remainder = ${{\left( 100 \right)}_{2}}$

Note: [1] We converted the above binary division in decimal form because it is easier to perform division in decimal representation since we are acquainted with it in our day to day life.

[2] The above method for converting decimal to binary is known as an algorithm.

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