🥨 Conversion to Binary Complement Forms
Preface When we represent signed integers in binary, we want to optimise it for addition
As seen in derived expression , we have 2 ways to achieve it, through finding radix complement or the diminished radix complement
Would we be able to do subtraction through addition through ignoring carry-overs at place-values bigger than the numbers in operation
While the positive binary numbers can be represented in its original form, the negative binary numbers are to be represented in their complement forms
As we will see, being binary allows conversion to 1s complement and 2s complement to be no more than simple bit inversion
🧠 Signed Binary Representation via Radix Complements
For signed numbers in binary, since we want to represent them in 1’s complement or 2’s complement form, we always represent the absolute value of that number in either form.
The sign is fixed to be 0 or 1 to represent positive or negative.
🔁 1’s Complement
We compute:
2^(n+1) - 1 MINUS the binary numberThis is a string of all 1s—same length as the binary number—minus the binary number itself.
11111...111Because the differences between 1 and the numbers in base 2 are:
1 - 1 = 0
1 - 0 = 1The result is always the bit-flipped version of the original number, which is equivalent to the NOT logical operation.
Because bit-flipping at MSB also turns positive sign to negative sign We can disregard treating the magnitude part separately from the sign.
Converting to Negative with Signed Bit Notice that polarity change——flipping from positive to negative——is only a feature because the number is expressed in signed form
- It is an optimisation we make in binary signed-forms where the bits exactly represent 2 states——correlating with positive and negative for the signed bit
- Thus we can directly take the positive binary number and convert it to its negative complement form
This isn’t an inherent property of complements, its just a specific to hardware optimisation specific to binary signed-forms
bitflips , its reversal is also the same procedure🐬 1’s Complement Conversion Procedure
1’s Complement Conversion—and Reversal
Flip every bit of the number (including the sign bit)
🔁 2’s Complement
Since 2’s complement is 1 more than 1’s complement:
We can simply Perform the 1’s complement——flip every bit (including the sign bit) Add 1 to the result
🌊 Ripple Carry Optimisation
Addition of 1 at the LSB causes a ripple carry that flips the string of 1s from the LSB rightwards until it hits the first 0
From ...0111...111
To ...1000...000Since we performed a full number bitflip in the first step
All the bits that were consecutive 0s from the LSB up to the first 1, won’t have net change because it would have:
- Bit flip the first time to a string of 1s from LSB up to the first 0
- Bit flip back to the string of 0s from LSB up to the first 1—when adding one.
The bits that are after the first 1, will only have flipped once and thus will have net change——bit-flip
[!Fun Fact] Permanent Bit
Because succession——adding 1——is always a bitflip, the LSB will always flip twice, and therefore is the only bit that is always preserved in 2's complement conversions
bitflips , its reversal is also the same procedure🐬 2’s Complement Conversion Procedure
2’s Complement Conversion—and Reversal
- Scan from LSB toward MSB
- Preserve all bits up to and including the first
1encountered
🧩 Philosophical Closure
Such simplicity arises from applying the method of complements using radixes, and then applying it to the design of signed binary values.
This is thus no trick, but a intertwined series of optimisations from the properties of complements and binary