Signed Number's Binary Addition

Question

I'm going to discuss about Signed Number's Binary addition, I searched about it and even read books. Now I make little changes in it's logic and start my own logic to solve it.

Let me show 4 bit example by Book Method.

-5+3

Book told me that I should take two's complement of 5 to add in 3's binary. 1011 is two's complement of 5. now add with 3's binary. 0011.

1011 <----- -5 ( 5's two's complement )
0011 <-----  3
---
1110 <----- -2 ( leftsigned bit is 1 therefore it is in two's complement form )
0010 <-----  2 ( Again two's complement of answer to show original answer )

But the answer sign bit is positive that's mean answer is 2?

Now I tried again and again and find a stupid method to solve the above or any signed number's binary addition. Let me explain with my stupid logic.

Note: I will never change sign bit during once complement.

First I will take the complement of 5 with no effect on sign bit during once complement.

1101 <----- -5 ( Sign bit showing it's -5 )
1010 <----- -5's once complement ( never change sign bit )
   1 
----
1011 <----- -5 two's complement.

Now start addition with 3's binary.

1011 <----- -5 (two's complement)
0011 <----- 3 ( with positive sign bit )
----
1110 <-- Signbit 1 mean it's in two's complement form. Again two's complement to get answer
1001 <----- Once complement (never change sign bit during Once Complement)
   1
----
1010 <----- -2 ( Sign bit is negative(1). it's showing that it is -2 )

Now I just want to clear that could my method satisfy the Signed number's binary addition? or it is a stupid method.

Answer 1

Here's a good page that explains adding signed and unsigned binary numbers, and using the 4-bit 2's complement.

Hope that helps.

EDIT: Just noticed this was asked 4 months ago; I hope he managed to find an answer. :-)

Answer 2

Let's take a second look at the "book" method of adding $-5$ and $3$ in four-bit two's-complement arithmetic.

In four-bit two's-complement binary, $-5$ is represented by $1011$ and $3$ is represented by $0011.$ The sum is $1011 + 0011 = 1110,$ which is the four-bit two's-complement representation of $-2.$ And that's the answer: $-2.$

The only reason to do any further operations on the result is if you want to convert the result back to ordinary decimal notation and don't instantly recognize that $1110$ represents $-2.$ One way to do that is to take the two's complement of $1110,$ which is $0010.$ Taking the two's complement of a number changes the sign of the number in the two's-complement representation. If reversing the sign of $1110$ gives us $0010,$ which represents $2,$ then what does $1110$ represent? It can only represent $-2.$ Again, that 's the answer: $-2.$

---

The rest of the question presents an interesting procedure for adding binary representations of integers. Instead of using two's-complement exclusively, however, this method begins with the two operands ($-5$ and $3$) in a four-bit signed-magnitude representation, and ends with the result ($-2$) in four-bit signed-magnitude representation.

The method has three major steps:

1. Convert the signed-magnitude representation to two's complement.
2. Add the two's-complement representations.
3. Convert the result from two's complement to signed magnitude.

The two's-complement addition is performed in the conventional way, so we merely have to check that the conversions from signed-magnitude to two's-complement and back again are correct.

Positive numbers are "converted" by leaving their bits completely unchanged. Only negative numbers have their bits changed by the conversion. In order to be able to reason about the conversion mathematically, I'll evaluate each four-bit binary number as an unsigned integer, although it can still also be interpreted as signed magnitude or two's complement.

If a four-bit signed-magnitude binary number represents $x,$ where $x < 0,$ then the bits of that representation have unsigned value $8 - \lvert x\rvert.$ (The sign bit has place value $8,$ and the rightmost three bits have value $\lvert x\rvert.$) The first step is to take the one's complement of the three magnitude bits, which is equivalent to subtracting them from the three-bit number $111,$ that is, from $7.$ The sign bit remains unchanged, so after this step we have a four-bit number with unsigned value $8 + (7 - \lvert x\rvert) = 15 - \lvert x\rvert.$ The second step is to add $1.$ The resulting number has unsigned value $(15 - \lvert x\rvert) + 1 = 16 - \lvert x\rvert = 16 + x,$ since $x < 0.$ And $16 + x$ is the unsigned value of the four-bit two's-complement representation of a number $x$ when $x < 0.$ Hence the conversion is always correct.

More generally, for an $n$-bit signed-magnitude representation of a negative number $x,$ the unsigned value of the signed-magnitude representation is $2^{n-1} + \lvert x\rvert,$ after taking the one's complement of the magnitude we have $2^{n-1} + (2^{n-1} - 1 - \lvert x\rvert) = 2^n - 1 - \lvert x\rvert,$ and after adding $1$ we have $(2^n - 1 - \lvert x\rvert) + 1 = 2^n - \lvert x\rvert = 2^n + x,$ the two's complement representation of $x.$ Since the signed-magnitude representation can only represent a number $x$ if $\lvert x\rvert \leq 2^{n-1} - 1,$ there is no possibility of overflow, and this conversion will always work.

Note that if the signed-magnitude number is $-0$ (sign bit $1$ with magnitude $0$), the procedure above produces a number with all bits set to $1$ after the one's complement, and adding $1$ to this results in all bits set to $0.$ That is, this signed-magnitude representation is correctly converted to $0$ in two's complement.

To convert from two's complement to signed magnitude, again positive numbers are not changed, and only negative numbers require manipulation of their bits. Clearly the procedure cannot work when $x = -2^{n-1},$ which is the most negative possible number that can be represented in $n$-bit two's complement, because the least number that can be represented in $n$-bit signed magnitude is $-(2^{n-1} - 1).$ So let's assume that $-(2^{n-1} - 1) \leq x \leq -1,$ that is, $x$ can be any of the possible negative integers in $n$-bit two's-complement binary representation except $-2^{n-1}$ Then the $n$-bit two's-complement representation of $x$ is a binary number with unsigned value $2^n + x = 2^n - \lvert x \rvert \geq 2^{n-1} + (2^{n-1} - \lvert x \rvert).$ That is, the rightmost $n-1$ bits have unsigned value $2^{n-1} - \lvert x \rvert.$ We take the one's complement of these bits, $(2^{n-1} - 1) - (2^{n-1} - \lvert x \rvert) = \lvert x \rvert - 1,$ and put these to the right of the sign bit, so we have a binary number with unsigned value $2^{n-1} + \lvert x \rvert - 1.$ The next step is to add $1$ to the this, with the result $2^{n-1} + \lvert x \rvert,$ which is the $n$-bit signed-magnitude representation of $x$ when $x < 0.$

In summary, the proposed method does indeed give correct results in all cases except when overflow occurs, and those are precisely the cases where it is not possible to represent the correct result in signed magnitude (that is, those are the cases in which no algorithm can give the correct answer).