Questions

To convert decimal numbers to binary, we repeatedly divide the number by 2 and collect the remainders. Then, we read the remainders in reverse order to get the binary representation.

Let's convert 245 and 24 to binary:

1. Decimal 245:

   $$\begin{align*} 245 \div 2 &= 122 \text{ (remainder 1)} \\ 122 \div 2 &= 61 \text{ (remainder 0)} \\ 61 \div 2 &= 30 \text{ (remainder 1)} \\ 30 \div 2 &= 15 \text{ (remainder 0)} \\ 15 \div 2 &= 7 \text{ (remainder 1)} \\ 7 \div 2 &= 3 \text{ (remainder 1)} \\ 3 \div 2 &= 1 \text{ (remainder 1)} \\ 1 \div 2 &= 0 \text{ (remainder 1)} \end{align*}$$

   Reading the remainders in reverse order, we get: $245_{10} = 11110101_2$.

2. Decimal 24:

   $$\begin{align*} 24 \div 2 &= 12 \text{ (remainder 0)} \\ 12 \div 2 &= 6 \text{ (remainder 0)} \\ 6 \div 2 &= 3 \text{ (remainder 0)} \\ 3 \div 2 &= 1 \text{ (remainder 1)} \\ 1 \div 2 &= 0 \text{ (remainder 1)} \end{align*}$$

   Reading the remainders in reverse order, we get: $24_{10} = 11000_2$.

Now, let's compute $24 - 245$ using the subtraction rule in binary:

$$\begin{align*} \text{Binary 24:} & \quad 11000 \\ \text{Binary 245:} & \quad 11110101 \\ \end{align*}$$

Since we're subtracting 245 from 24, we need to take the two's complement of 245 and then add it to 24:

1. Take the one's complement of 245: $00001010$.
2. Add 1 to get the two's complement: $00001011$.

Now, we can add 24 and the two's complement of 245:

$$\begin{align*} & 11000 \\ + & 00001011 \\ \hline & 11001011 \end{align*}$$

So, $24 - 245 = -221$ in binary.