100 lines
3 KiB
Markdown
100 lines
3 KiB
Markdown
---
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id: gktb
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title: Two's complement
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tags:
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- binary
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created: Tuesday, March 19, 2024
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---
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# Two's complement
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## Summary
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- _Two's complement_ is a method for representing signed numbers (negative
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integers) in binary.
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- It is derived by inverting the values of an unsigned binary integer to create
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its signed equivalent.
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- A benefit is that hardware implementation of the binary arithmetic of signed
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and unsigned numbers can be handled in the same manner as unsigned numbers,
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requiring no additional handling. A drawback is that it halves the
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informational capacity of the given word length for the binary system.
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## Detail
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### Procedural steps
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Two's complement divides the available word length (see
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[[Binary_encoding|binary encoding]]) into two subsets: one for negative integrs
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and one for positive integers.
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Take the binary encoding of decimal five (`0101`). Its complement is `1011`.
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The procedure for deriving the complement is as follows.
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To derive the complement of an unsigned number:
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1. Take the unsigned number and invert its digits: `0` becomes `1`, `1` becomes
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`0`
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2. Add one
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To derive the unsigned equivalent of a signed number you invert the process but
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still make the smallest digit `1`:
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### Formal expression
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$$
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2^n - x
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$$
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- where $x$ is the negative integer in binary that we wish to derive
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- where $n$ is the word length of the binary system in bits.
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Applied to the earlier example we have $2^4 -5$ which is:
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$$
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16 - 5 = 11
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$$
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When we convert the decimal `11` to binary we get `1011` which is identical to
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the signed version of the unsigned integer.
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We can confirm the correctness of the derviation by summing the signed and
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unsigned binary values. If this results in zeros (ignoring the overflow bit),
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the derivation is correct as the two values effectively cancel each other out:
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$$
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1011 + 0101 = 0000
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$$
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### Advantages
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- The circuit implementation of arithmetic involving positive and negative
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integers is the same as the implementation of positive integers. There is no
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need for additional harware or special handling of the values.
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- This can be contrasted with the alternative approaches to signing numbers such
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as **signed magnitude representation** which uses certain bits as designators
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of negative/positive status.
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### Limitations
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- Two's complement reduces the overall informational capacity of the given
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binary word length, effectively halving the total number of unique values.
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- In a 4-bit system instead of 16 total unique encodings of integers you have 8
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encodings for positive integers and 8 encodings for the their signed
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equivalent. For integers larger than denary 8 you would need to increase the
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bit length of the system
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- Consequently two's complement can necessitate larger overall word lengths.
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## Related notes
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[[Signed_and_unsigned_numbers|signed_and_unsigned_numbers]],
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[[Binary_addition|binary addition]], [[Signed_magnitude_representation]]
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