Binary Arithmetic and Signed Numbers
A bit pattern's numeric meaning depends on its word width and whether it is interpreted as unsigned or signed. Binary addition propagates carries between columns.
Summary
A bit pattern's numeric meaning depends on its word width and whether it is interpreted as unsigned or signed. Binary addition propagates carries between columns. Fixed-width arithmetic retains only the lowest result bits.
| Representation | Negative-value encoding | Zero patterns |
|---|---|---|
| Sign-magnitude | Sign bit plus magnitude bits | Separate positive and negative zero |
| One's complement | Invert the positive representation | Separate positive and negative zero |
| Two's complement | Invert all bits and add | One zero pattern |
An -bit two's complement integer ranges from to . Its highest bit has negative positional weight. Negation inverts every bit and adds within the fixed width. Subtraction uses the same adder circuitry as addition: .
🧠 Key Concepts
- Fixed Word Width
- Binary Addition
- Sign Magnitude
- One's Complement
- Two's Complement
- Signed Overflow
- Unsigned Carry
- Sign Extension
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Binary Arithmetic and Signed Numbers
📘 Overview Binary arithmetic operates on bit patterns whose numeric meaning depends on word width and signed representation. Two's complement is the standard signed integer representation because addition and subtraction can use the same binary adder circuitry.
🧠 Key Idea The same bit pattern can represent different unsigned and signed values. In two's complement, arithmetic retains a fixed number of bits, so a valid bit-level result may still overflow the signed numeric range.
⚔️ Core Details: - Binary addition carries into the next column: and
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