Convert the following $\newline$ a. $\newline$ $165.078125_{(10)}$ to binary $\newline$ b. $\newline$ $1101101101.10011_{(2)}$ to denary.

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Grade 7

Convert between mixed numbers and improper fractions

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Q. Convert the following

\newline

\newline

165.078125_{(10)}

to binary

\newline

\newline

1101101101.10011_{(2)}

to denary.

Convert Integer to Binary: To convert the decimal number $165.078125$ to binary, we start by converting the integer part $(165)$ to binary.
Convert Fraction to Binary: Divide $165$ by $2$ , which gives us $82$ with a remainder of $1$ . The remainder is the least significant bit (LSB) of the binary representation.
Combine Integer and Fraction: Divide $82$ by $2$ , which gives us $41$ with a remainder of $0$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ . Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ . Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ . Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ . Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ . Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part. The binary representation of the integer part $20$ $5$ is $20$ $6$ . Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step. Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ . Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ .Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ .Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ .After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ .Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ .After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ .Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here.
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ . Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ . Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ . Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ . Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ . Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part. The binary representation of the integer part $20$ $5$ is $20$ $6$ . Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step. Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ . Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ . Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ . Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ . After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ . Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here. The binary representation of the fractional part $20$ $7$ is $2$ $6$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ .Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ .After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ .Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here.The binary representation of the fractional part $20$ $7$ is $2$ $6$ .Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ .Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ .After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ .Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here.The binary representation of the fractional part $20$ $7$ is $2$ $6$ .Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ .Now, let's convert the binary number $2$ $9$ to decimal, starting with the integer part.
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ . Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ . Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ . Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ . Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ . Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part. The binary representation of the integer part $20$ $5$ is $20$ $6$ . Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step. Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ . Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ . Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ . Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ . After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ . Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here. The binary representation of the fractional part $20$ $7$ is $2$ $6$ . Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ . Now, let's convert the binary number $2$ $9$ to decimal, starting with the integer part. Starting from the right, each digit represents an increasing power of $2$ . The rightmost digit is $10$ $1$ , the next is $10$ $2$ , and so on.
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ . Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ . Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ . Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ . Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ . Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part. The binary representation of the integer part $20$ $5$ is $20$ $6$ . Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step. Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ . Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ . Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ . Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ . After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ . Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here. The binary representation of the fractional part $20$ $7$ is $2$ $6$ . Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ . Now, let's convert the binary number $2$ $9$ to decimal, starting with the integer part. Starting from the right, each digit represents an increasing power of $2$ . The rightmost digit is $10$ $1$ , the next is $10$ $2$ , and so on. Calculate the decimal value of the integer part: $10$ $3$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ . Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ . Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ . Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ . Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ . Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part. The binary representation of the integer part $20$ $5$ is $20$ $6$ . Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step. Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ . Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ . Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ . Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ . After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ . Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here. The binary representation of the fractional part $20$ $7$ is $2$ $6$ . Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ . Now, let's convert the binary number $2$ $9$ to decimal, starting with the integer part. Starting from the right, each digit represents an increasing power of $2$ . The rightmost digit is $10$ $1$ , the next is $10$ $2$ , and so on. Calculate the decimal value of the integer part: $10$ $3$ . Perform the calculation: $10$ $4$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ .Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ .After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ .Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here.The binary representation of the fractional part $20$ $7$ is $2$ $6$ .Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ .Now, let's convert the binary number $2$ $9$ to decimal, starting with the integer part.Starting from the right, each digit represents an increasing power of $2$ . The rightmost digit is $10$ $1$ , the next is $10$ $2$ , and so on.Calculate the decimal value of the integer part: $10$ $3$ .Perform the calculation: $10$ $4$ .Add the values: $10$ $5$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ .Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ .After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ .Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here.The binary representation of the fractional part $20$ $7$ is $2$ $6$ .Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ .Now, let's convert the binary number $2$ $9$ to decimal, starting with the integer part.Starting from the right, each digit represents an increasing power of $2$ . The rightmost digit is $10$ $1$ , the next is $10$ $2$ , and so on.Calculate the decimal value of the integer part: $10$ $3$ .Perform the calculation: $10$ $4$ .Add the values: $10$ $5$ .Now, let's convert the fractional part $10$ $6$ to decimal.
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ . Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ . Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ . Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ . Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ . Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part. The binary representation of the integer part $20$ $5$ is $20$ $6$ . Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step. Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ . Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ . Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ . Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ . After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ . Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here. The binary representation of the fractional part $20$ $7$ is $2$ $6$ . Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ . Now, let's convert the binary number $2$ $9$ to decimal, starting with the integer part. Starting from the right, each digit represents an increasing power of $2$ . The rightmost digit is $10$ $1$ , the next is $10$ $2$ , and so on. Calculate the decimal value of the integer part: $10$ $3$ . Perform the calculation: $10$ $4$ . Add the values: $10$ $5$ . Now, let's convert the fractional part $10$ $6$ to decimal. Starting from the left, each digit after the decimal point represents a decreasing power of $2$ . The first digit after the decimal is $10$ $8$ , the next is $10$ $9$ , and so on.
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ . Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ . Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ . Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ . Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ . Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part. The binary representation of the integer part $20$ $5$ is $20$ $6$ . Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step. Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ . Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ . Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ . Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ . After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ . Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here. The binary representation of the fractional part $20$ $7$ is $2$ $6$ . Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ . Now, let's convert the binary number $2$ $9$ to decimal, starting with the integer part. Starting from the right, each digit represents an increasing power of $2$ . The rightmost digit is $10$ $1$ , the next is $10$ $2$ , and so on. Calculate the decimal value of the integer part: $10$ $3$ . Perform the calculation: $10$ $4$ . Add the values: $10$ $5$ . Now, let's convert the fractional part $10$ $6$ to decimal. Starting from the left, each digit after the decimal point represents a decreasing power of $2$ . The first digit after the decimal is $10$ $8$ , the next is $10$ $9$ , and so on. Calculate the decimal value of the fractional part: $0$ $0$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ .Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ .After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ .Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here.The binary representation of the fractional part $20$ $7$ is $2$ $6$ .Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ .Now, let's convert the binary number $2$ $9$ to decimal, starting with the integer part.Starting from the right, each digit represents an increasing power of $2$ . The rightmost digit is $10$ $1$ , the next is $10$ $2$ , and so on.Calculate the decimal value of the integer part: $10$ $3$ .Perform the calculation: $10$ $4$ .Add the values: $10$ $5$ .Now, let's convert the fractional part $10$ $6$ to decimal.Starting from the left, each digit after the decimal point represents a decreasing power of $2$ . The first digit after the decimal is $10$ $8$ , the next is $10$ $9$ , and so on.Calculate the decimal value of the fractional part: $0$ $0$ .Perform the calculation: $0$ $1$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ .Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ .After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ .Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here.The binary representation of the fractional part $20$ $7$ is $2$ $6$ .Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ .Now, let's convert the binary number $2$ $9$ to decimal, starting with the integer part.Starting from the right, each digit represents an increasing power of $2$ . The rightmost digit is $10$ $1$ , the next is $10$ $2$ , and so on.Calculate the decimal value of the integer part: $10$ $3$ .Perform the calculation: $10$ $4$ .Add the values: $10$ $5$ .Now, let's convert the fractional part $10$ $6$ to decimal.Starting from the left, each digit after the decimal point represents a decreasing power of $2$ . The first digit after the decimal is $10$ $8$ , the next is $10$ $9$ , and so on.Calculate the decimal value of the fractional part: $0$ $0$ .Perform the calculation: $0$ $1$ .Add the values: $0$ $2$ .
Convert Binary to Decimal: Divide $41$ by $2$ , which gives us $20$ with a remainder of $1$ .Divide $20$ by $2$ , which gives us $10$ with a remainder of $0$ .Divide $10$ by $2$ , which gives us $2$ $0$ with a remainder of $0$ .Divide $2$ $0$ by $2$ , which gives us $2$ with a remainder of $1$ .Divide $2$ by $2$ , which gives us $1$ with a remainder of $0$ .Divide $1$ by $2$ , which gives us $0$ with a remainder of $1$ . We have now reached $0$ , so we stop dividing and take all the remainders in reverse order to get the binary representation of the integer part.The binary representation of the integer part $20$ $5$ is $20$ $6$ .Now, let's convert the fractional part $20$ $7$ to binary by multiplying by $2$ and taking note of the integer part at each step.Multiply $20$ $7$ by $2$ , which gives us $1$ $1$ . The integer part is $0$ .Multiply $1$ $1$ by $2$ , which gives us $1$ $5$ . The integer part is $0$ .Multiply $1$ $5$ by $2$ , which gives us $1$ $9$ . The integer part is $0$ .Multiply $1$ $9$ by $2$ , which gives us $20$ $3$ . The integer part is $1$ .After taking out the integer part ( $1$ ), we are left with $20$ $6$ . Multiply $20$ $6$ by $2$ , which gives us $20$ $9$ . The integer part is $0$ .Multiply $20$ $9$ by $2$ , which gives us $2$ $3$ . The integer part is $1$ . Since we have reached an exact value, we stop here.The binary representation of the fractional part $20$ $7$ is $2$ $6$ .Combining the integer and fractional parts, the binary representation of $2$ $7$ is $2$ $8$ .Now, let's convert the binary number $2$ $9$ to decimal, starting with the integer part.Starting from the right, each digit represents an increasing power of $2$ . The rightmost digit is $10$ $1$ , the next is $10$ $2$ , and so on.Calculate the decimal value of the integer part: $10$ $3$ .Perform the calculation: $10$ $4$ .Add the values: $10$ $5$ .Now, let's convert the fractional part $10$ $6$ to decimal.Starting from the left, each digit after the decimal point represents a decreasing power of $2$ . The first digit after the decimal is $10$ $8$ , the next is $10$ $9$ , and so on.Calculate the decimal value of the fractional part: $0$ $0$ .Perform the calculation: $0$ $1$ .Add the values: $0$ $2$ .Combining the integer and fractional parts, the decimal representation of $2$ $9$ is $0$ $4$ .

Convert the following\newlinea. \newline165.078125(10)165.078125_{(10)}165.078125(10)​ to binary\newlineb. \newline1101101101.10011(2)1101101101.10011_{(2)}1101101101.10011(2)​ to denary.

Convert the following $\newline$ a. $\newline$ $165.078125_{(10)}$ to binary $\newline$ b. $\newline$ $1101101101.10011_{(2)}$ to denary.