Convert the following a. 165.078125_(10) to binary b. 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.Divide 20 by 2, which gives us 10 with a remainder of 0.Divide 10 by 2, which gives us 5 with a remainder of 0.Divide 5 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 165 is 10100101.Now, let's convert the fractional part 0.078125 to binary by multiplying by 2 and taking note of the integer part at each step.Multiply 0.078125 by 2, which gives us 0.15625. The integer part is 0.Multiply 0.15625 by 2, which gives us 0.3125. The integer part is 0.Multiply 0.3125 by 2, which gives us 0.625. The integer part is 0.Multiply 0.625 by 2, which gives us 1.25. The integer part is 1.After taking out the integer part (1), we are left with 0.25. Multiply 0.25 by 2, which gives us 0.5. The integer part is 0.Multiply 0.5 by 2, which gives us 1.0. The integer part is 1. Since we have reached an exact value, we stop here.The binary representation of the fractional part 0.078125 is .0001.Combining the integer and fractional parts, the binary representation of 165.078125 is 10100101.0001.Now, let's convert the binary number 1101101101.10011 to decimal, starting with the integer part.Starting from the right, each digit represents an increasing power of 2. The rightmost digit is 2^0, the next is 2^1, and so on.Calculate the decimal value of the integer part: 1*2^9 + 1*2^8 + 0*2^7 + 1*2^6 + 1*2^5 + 0*2^4 + 1*2^3 + 1*2^2 + 0*2^1 + 1*2^0.Perform the calculation: 1*512 + 1*256 + 0*128 + 1*64 + 1*32 + 0*16 + 1*8 + 1*4 + 0*2 + 1*1.Add the values: 512 + 256 + 64 + 32 + 8 + 4 + 1 = 877.Now, let's convert the fractional part 0.10011 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 2^-1, the next is 2^-2, and so on.Calculate the decimal value of the fractional part: 1*2^-1 + 0*2^-2 + 0*2^-3 + 1*2^-4 + 1*2^-5.Perform the calculation: 1*0.5 + 0*0.25 + 0*0.125 + 1*0.0625 + 1*0.03125.Add the values: 0.5 + 0 + 0 + 0.0625 + 0.03125 = 0.59375.Combining the integer and fractional parts, the decimal representation of 1101101101.10011 is 877.59375.

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

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Math Problems

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

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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.