How to Represent Continuous Values in Programming

A signed integer is an integer with a positive '+' or negative sign '-' associated with it. Since the computer only understands binary, it is necessary to represent these signed integers in binary form.

In binary, signed Integer can be represented in three ways:

1. Signed bit.
2. 1's Complement.
3. 2's Complement.

Let us see why 2's complement is considered to be the best method.

Signed bit Representation

In the signed integer representation method the following rules are followed:

1. The MSB (Most Significant Bit) represents the sign of the Integer.
2. Magnitude is represented by other bits other than MSB i.e. (n-1) bits where n is the no. of bits.
3. If the number is positive, MSB is 0 else 1.
4. The range of signed integer representation of an n-bit number is given as –(2^{n-1}-1) to (2)^{n-1}-1.

Example:

Let n = 4

Range:
 –(2^{4-1}-1)    to 2^{4-1}-1
= -(2^{3}-1)    to 2^{3}-1
= -(7)   to+7

For 4 bit representation, minimum value=-7 and maximum value=+7

Signed bit Representation:

Positive Numbers
Sign	Magnitude			Decimal Representation
0	0	0	0	+0
0	0	0	1	+1
0	0	1	0	+2
0	0	1	1	+3
0	1	0	0	+4
0	1	0	1	+5
0	1	1	0	+6
0	1	1	1	+7
Negative Numbers
Sign	Magnitude			Decimal Representation
1	0	0	0	-0
1	0	0	1	-1
1	0	1	0	-2
1	0	1	1	-3
1	1	0	0	-4
1	1	0	1	-5
1	1	1	0	-6
1	1	1	1	-7

Drawbacks:

1. For 0, there are two representations: -0 and +0 which should not be the case as 0 is neither –ve nor +ve.
2. Out of 2^n bits for representation, we are able to utilize only 2^{n-1}  bits.
3. Numbers are not in cyclic order i.e. After the largest number (in this, for example, +7) the next number is not the least number (in this, for example, +0).
4. For negative numbers signed extension does not work.

Example:
Signed extension for +5

Signed extension for -5

5. As we can see above, for +ve representation, if 4 bits are extended to 5 bits there is a need to just append 0 in MSB.
6. But if the same is done in –ve representation we won't get the same number. i.e. 10101 ≠ 11101.

1's Complement representation of a signed integer

In 1's complement representation the following rules are used:

1. For +ve numbers the representation rules are the same as signed integer representation.
2. For –ve numbers, we can follow any one of the two approaches:

• Write the +ve number in binary and take 1's complement of it.

1's complement of 0 = 1 and 1's complement of 1 = 0

Example:
(-5) in 1's complement:
+5 = 0101
-5 = 1010

• Write Unsigned representation of 2^n-1-X for –X.

Example:
–X = -5      for n=4
2^4-1-5=10 ->1010(Unsigned)

3. The range of 1's complement integer representation of n-bit number is given as –(2^{n-1}-1) to 2^{n-1}-1.

1's Complement Representation:

Positive Numbers
Sign	Magnitude	Number
0	0   0   0	+0
0	0   0   1	+1
0	0   1   0	+2
0	0   1   1	+3
0	1   0   0	+4
0	1   0   1	+5
0	1   1   0	+6
0	1   1   1	+7
Negative Numbers
Sign	Magnitude	Number
1	0   0   0	-7
1	0   0   1	-6
1	0   1   0	-5
1	0   1   1	-4
1	1   0   0	-3
1	1   0   1	-2
1	1   1   0	-1
1	1   1   1	-0

Drawbacks:

1. For 0, there are two representations: -0 and +0 which should not be the case as 0 is neither –ve nor +ve.
2. Out of 2^n bits for representation, we are able to utilize only 2^{n-1} bits.

Merits over Signed bit representation:

1. Numbers are in cyclic order i.e. after the largest number (in this, for example, +7) the next number is the least number (in this, for example, -7).
2. For negative number signed extension works.

Example: Signed extension for +5

Signed extension for -5

3. As it can be seen above, for +ve as well as -ve representation, if 4 bits are extended to 5 bits there is a need to just append 0/1 respectively in MSB.

2's Complement representation

In 2's Complement representation the following rules are used:

1. For +ve numbers, the representation rules are the same as signed integer representation.
2. For –ve numbers, there are two different ways we can represent the number.

• Write an unsigned representation of 2^n-X for –X in n-bit representation.

Example:
(-5) in 4-bit representation
 2^4-5=11   -→1011(unsigned)

• Write a representation of +X and take 2's Complement.

To take 2's complement simply take 1's complement and add 1 to it.

Example:
(-5) in 2's complement
(+5) = 0101
1's complement of (+5) = 1010
 Add 1 in 1010:  1010+1 = 1011
Therefore (-5) = 1011

3. Range of representation of n-bit is –(2^{n-1} )  to (2)^{(n-1)-1}.

2's Complement representation (4 bits)

Merits:

1. No ambiguity in the representation of 0.
2. Numbers are in cyclic order i.e. after +7 comes -8.
3. Signed Extension works.
4. The range of numbers that can be represented using 2's complement is very high.

Due to all of the above merits of 2's complement representation of a signed integer, binary numbers are represented using 2's complement method instead of signed bit and 1's complement.

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Source: https://www.geeksforgeeks.org/different-ways-to-represent-signed-integer/

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