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Signed representation by 2’s Complement

2´s Complement Representation:

2´s complement representation is the most used representation on modern computers. This notation allows a computer to perform addition and subtraction using the same operations. Sign bit 0 and 1 represent positive and negative numbers respectively. The magnitude part represents the 2’s complement of the actual number.

Example 1:

Representation of +10 and -10:

Example 2:

-5 represented by 2’s complement, first write +5 → 0101 then take its 2’s complement.1011→ represents the -5 in terms of 2’s complement.

Properties of 2´s complement Representation:

1. 2´s complement extends the same hardware for addition and subtraction.

2. 2´s complement is a weighted system. It is the only code that assigns ‘-ve’ weight to all sign bit.

3. If a signed number in 2´s complement form is 

Then its decimal equivalent will be 

Example:

If ${\mathbf{\left(}\mathbf{1001}\mathbf{\right)}}_{\mathbf{2}}$ is a number represented in 2´s complement form, then its decimal equivalent will be 

4. With n bits the range of numbers that can be represented in 2´s complement form will be 

Example:

If consider 8 bits, we can represent -128 to +127.

If consider 4 bits, we can represent -8 to +7. 

5. Sign Extension in 2´s complement form:

The rule of sign extension is the same as that of in 1´s complement form. Here also sign bit is copied to all extended positions.

Example:

Representation of -7 in 8 bits (without sign extension)

Representation of -7 in 8 bits (with sign extension)

Advantages of 2´s complement representation over the signed magnitude and its complement

1. 2´s complement has only one value for zero, i.e. 2´s complement has the same representation of +0 and -0, unlike other representations.

2. 2´s complement arithmetic is faster than1´s complement arithmetic as there is no need for correction of end-around carry. (EAC)

Question:

(-13) 2’s Complement form?

Answer:

-5 represented by 2’s complement, first write +13 → 01101.

After that takes its 2’s complement .10011→ represents the -13 in terms of 2’s complement.

Question:

A number in 2’s complement form as 1010. Find the equivalent decimal.

Answer:

1010

Let another 2’s complement form 11010

11010

Important note:

If we notice 1010 and 11010 two 2’s complement forms give the same result -6.

So, it is clear that at +ve or –ve number in 2’s complement or 1’s complement form, if we replicate the MSB, then the value remains the same.

11111111010 it also gives you the same value -6.

Let +ve number (1’s or 2’s complement)

+6

↓

00110 – if you copy MSB 00000110 it same +6.

Let 1’s Complement (-ve number)

-6

First, write +6→ 0110

After 1’s Complement 

-6 → 1001

So, (1001) = -6

Let copy MSB 11111001 is same -6.