Let's solve the two problems from your image, both requiring binary arithmetic using an 8-bit system.

Problem 1: Compute each as a computer would, using binary code on an 8-bit system.

(a) $26 - 15$

1. Convert both numbers to binary:

   • $26$ in binary (8 bits): $00011010$
   • $15$ in binary (8 bits): $00001111$

2. Perform binary subtraction:

   • $00011010 - 00001111$

Let's compute that using binary subtraction step-by-step:

• Borrowing as needed, the binary result is $00001011$, which is $11$ in decimal.

(b) $-31 - 6$

This requires two's complement because we're dealing with negative numbers in binary.

1. Convert $31$ and $6$ to binary:

   • $31$ in binary (positive): $00011111$
   • $6$ in binary (positive): $00000110$

2. Convert $31$ to its negative using two's complement:

   • Flip the bits of $31$: $11100000$
   • Add 1: $11100001$ (this is $-31$ in two's complement form)

3. Now subtract $6$, which is the same as adding $-6$:

   • Two's complement of $6$: Flip $00000110$ to get $11111001$, then add 1: $11111010$ (this is $-6$).

4. Add $-31$ and $-6$ in binary: $11100001 + 11111010 = 11011011$ The result $11011011$ is in two's complement form, and when converted back to decimal, this equals $-37$.

Problem 2: Subtract 29 from 23 as a computer would, using binary code.

1. Convert $23$ and $29$ to binary:

   • $23$ in binary: $00010111$
   • $29$ in binary: $00011101$

2. Convert $29$ to its negative using two's complement:

   • Flip $00011101$: $11100010$
   • Add 1: $11100011$ (this is $-29$ in two's complement form)

3. Now, add $23$ and $-29$: $00010111 + 11100011 = 11111010$ The result $11111010$ in two's complement is $-6$ in decimal.

Would you like more details on how binary arithmetic works, or further clarifications?

Here are 5 related questions you might find helpful:

1. How does two's complement represent negative numbers?
2. What are the rules for binary subtraction?
3. How do you handle overflow in binary arithmetic?
4. Can binary arithmetic be extended beyond 8-bit systems?
5. How do computers detect and correct arithmetic errors?

Tip: When subtracting binary numbers, converting to two's complement allows you to turn subtraction into addition, simplifying the operation!