Signed Number Representation || Computer Organisation and Architecture || Saral Bhasha me

Signed Number Representation || Computer Organisation and Architecture || Saral Bhasha me

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In computing, signed number representations are required to encode negative numbers in binary number systems.

In mathematics, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in computer hardware, numbers are represented only as sequences of bits, without extra symbols. The four best-known methods of extending the binary numeral system to represent signed numbers are: sign-and-magnitude, ones' complement, two's complement, and offset binary. Some of the alternative methods use implicit instead of explicit signs, such as negative binary, using the base −2. Corresponding methods can be devised for other bases, whether positive, negative, fractional, or other elaborations on such themes.

There is no definitive criterion by which any of the representations is universally superior. For integers, the representation used in most current computing devices is two's complement, although the Unisys ClearPath Dorado series mainframes use ones' complement.

Signed magnitude representation
This representation is also called "sign–magnitude" or "sign and magnitude" representation. In this approach, a number's sign is represented with a sign bit: setting that bit (often the most significant bit) to 0 for a positive number or positive zero, and setting it to 1 for a negative number or negative zero. The remaining bits in the number indicate the magnitude (or absolute value). For example, in an eight-bit byte, only seven bits represent the magnitude, which can range from 0000000 (0) to 1111111 (127). Thus numbers ranging from −12710 to +12710 can be represented once the sign bit (the eighth bit) is added. For example, −4310 encoded in an eight-bit byte is 10101011 while 4310 is 00101011. Using signed magnitude representation has multiple consequences which makes them more intricate to implement:[5]

There are two ways to represent zero, 00000000 (0) and 10000000 (−0).
Addition and subtraction require different behavior depending on the sign bit, whereas one's complement can ignore the sign bit and just do an end-around carry, and two's complement can ignore the sign bit and depend on the overflow behavior.
Comparison also require inspecting the sign bit, whereas in two's complement, one can simply subtract the two numbers, and check if the outcome is positive or negative.
The minimum negative number is −127 instead of −128 in the case of two's complement.
This approach is directly comparable to the common way of showing a sign (placing a "+" or "−" next to the number's magnitude). Some early binary computers (e.g., IBM 7090) use this representation, perhaps because of its natural relation to common usage. Signed magnitude is the most common way of representing the significand in floating-point values.

Ones' complement
Eight-bit ones' complement
Binary value Ones' complement interpretation Unsigned interpretation
00000000 +0 0
00000001 1 1
⋮ ⋮ ⋮
01111101 125 125
01111110 126 126
01111111 127 127
10000000 −127 128
10000001 −126 129
10000010 −125 130
⋮ ⋮ ⋮
11111101 −2 253
11111110 −1 254
11111111 −0 255
Main article: Ones' complement
Alternatively, a system known as ones' complement[6] can be used to represent negative numbers. The ones' complement form of a negative binary number is the bitwise NOT applied to it, i.e. the "complement" of its positive counterpart. Like sign-and-magnitude representation, ones' complement has two representations of 0: 00000000 (+0) and 11111111 (−0).[7]

As an example, the ones' complement form of 00101011 (4310) becomes 11010100 (−4310). The range of signed numbers using ones' complement is represented by −(2N−1 − 1) to (2N−1 − 1) and ±0. A conventional eight-bit byte is −12710 to +12710 with zero being either 00000000 (+0) or 11111111 (−0).

To add two numbers represented in this system, one does a conventional binary addition, but it is then necessary to do an end-around carry: that is, add any resulting carry back into the resulting sum.[8] To see why this is necessary, consider the following example showing the case of the addition of −1 (11111110) to +2 (00000010):

binary decimal
11111110 −1
+ 00000010 +2
─────────── ──
1 00000000 0 ← Not the correct answer
1 +1 ← Add carry
─────────── ──
00000001 1 ← Correct answer
In the previous example, the first binary addition gives 00000000, which is incorrect. The correct result (00000001) only appears when the carry is added back in.

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