Complete the table by converting the 8-bit binary number '11110000' into denary, and the denary number '105' into 8-bit binary, and convert '00011110' into denary.

Steps to Solve

1. Convert Binary to Denary (11110000)

To convert the binary number $11110000$ to denary, we multiply each bit by $2^n$, where $n$ is the position of the bit from the right (starting from $0$).

Calculating each bit:

• $1 \times 2^7 = 128$
• $1 \times 2^6 = 64$
• $1 \times 2^5 = 32$
• $1 \times 2^4 = 16$
• $0 \times 2^3 = 0$
• $0 \times 2^2 = 0$
• $0 \times 2^1 = 0$
• $0 \times 2^0 = 0$

Now, sum them up: $$ 128 + 64 + 32 + 16 + 0 + 0 + 0 + 0 = 240 $$

2. Convert Denary to Binary (105)

To convert the denary number $105$ into binary, we repeatedly divide by $2$ and note the remainders.

• $105 \div 2 = 52$ with a remainder of $1$
• $52 \div 2 = 26$ with a remainder of $0$
• $26 \div 2 = 13$ with a remainder of $0$
• $13 \div 2 = 6$ with a remainder of $1$
• $6 \div 2 = 3$ with a remainder of $0$
• $3 \div 2 = 1$ with a remainder of $1$
• $1 \div 2 = 0$ with a remainder of $1$

Now, reading the remainders from bottom to top gives the binary representation: $$ 1101001 $$

To ensure it is an 8-bit representation, add leading zeros: $$ 01101001 $$

3. Convert Binary to Denary (00011110)

To convert the binary number $00011110$ to denary, we again multiply each bit by $2^n$.

Calculating each bit:

• $0 \times 2^7 = 0$
• $0 \times 2^6 = 0$
• $0 \times 2^5 = 0$
• $1 \times 2^4 = 16$
• $1 \times 2^3 = 8$
• $1 \times 2^2 = 4$
• $1 \times 2^1 = 2$
• $0 \times 2^0 = 0$

Now, sum them up: $$ 0 + 0 + 0 + 16 + 8 + 4 + 2 + 0 = 30 $$