Add two positive 8-bit binary integers
📚 Data Representation: Adding Two Positive 8‑bit Binary Integers
What is an 8‑bit Binary Integer?
An 8‑bit binary integer uses 8 binary digits (bits) to represent a number. Each bit can be 0 or 1, and the value of the number is calculated as: $$\displaystyle N = b_7\cdot2^7 + b_6\cdot2^6 + \dots + b_1\cdot2^1 + b_0\cdot2^0$$ where $b_7$ is the most significant bit (MSB) and $b_0$ the least significant bit (LSB).
🔢 Example: Adding 01100101 and 01011011
- Write the numbers one under the other, aligning the bits:
- Start from the rightmost bit (LSB) and add column by column, keeping track of the carry.
- Use the binary addition table:
- Continue until all bits are processed. The final binary result is 11000000.
- Convert to decimal to check: $11000000_2 = 128 + 64 = 192$.
01100101 + 01011011 --------
| Bit 1 | Bit 2 | Sum | Carry |
|---|---|---|---|
| 0 | 1 | 1 | 0 |
| 1 | 1 | 0 | 1 |
🧮 Quick Check: Why 192?
01100101₂ = 101 + 64 + 32 + 8 + 1 = 101 + 64 + 32 + 8 + 1 = 106 01011011₂ = 64 + 32 + 8 + 2 + 1 = 107 106 + 107 = 213, but we only have 8 bits, so we ignore the overflow beyond 8 bits. The correct sum within 8 bits is 11000000₂ = 192. (In this example we actually need to consider overflow; the real sum 213 would require 9 bits.)
💡 Exam Tip: Handling Overflow
• Always check if the result fits in 8 bits. • If there is a carry out of the MSB, it is discarded in unsigned 8‑bit addition. • In exam questions, they may ask you to state whether overflow occurs. • Remember: 255 (11111111₂) is the maximum unsigned 8‑bit value. Anything above it wraps around.
🤔 Analogy: Binary Addition as Counting Cups
Imagine each bit as a cup that can hold either 0 or 1 cup of water. When you add two cups, if both have water (1+1), you pour the water into a bigger cup (carry) and leave 0 in the original cup. This is exactly how binary addition works: 1+1 = 0 with a carry of 1. Visualising it this way helps you remember the carry rule without getting lost in numbers.
📝 Quick Practice Problem
- Add 10101010 and 01010101.
- Show your work column by column.
- State the final 8‑bit result and its decimal equivalent.
- Did overflow occur? Explain.
Revision
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