# Different ways to represent Signed Integer

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:

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

**Signed bit Representation:**

| ||||

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 |

| ||||

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**:

- For 0, there are two representations: -0 and +0 which should not be the case as 0 is neither –ve nor +ve.
- 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:**

- No ambiguity in the representation of 0.
- Numbers are in cyclic order i.e. after +7 comes -8.
- Signed Extension works.
- 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.