What is 2's Complement Number?

Question

What is 2's Complement Number?
Why do we take 1's Complement and add 1 to it? Why don't we subtract 1 after taking 1's Complement?
Why do computers use 2's Complement?

Answer 1

What is 2's Complement Number?

Complementary number system is used to represent negative numbers. So, 2's Complement number system is used to represent negative numbers.

UPDATE

Q:  What “2’s Complement System” says?

A: The negative equivalent of binary number is its 2’s complement. (1’s Complement + 1)

Note: 1 extra bit is required to represent the sign of a number. MSB (Most Significant Bit) is used as sign bit. If MSB is 0, then the number is positive. If MSB is 1, then the number is negative.

1’s Complement  Value   2’s Complement
    011         +3          011
    010         +2          010
    001         +1          001
    000         +0          000
    111         -0          000
    110         -1          111
    101         -2          110
    100         -3          101
                -4          100

How '100' (3 bits) is -4?

MSB is used as sign, if 1, its negative, if 0 it is positive.

-1 * 2^2 + 0*2^1 + 0*2^0 = -4 + 0 + 0 = -4

Similarly 101 (3 bits) is -3

-1 * 2^2 + 0*2^1 + 1*2^0 = -4 + 0 + 1 = -3

Observations:

•   In 1’s complement, using 3 bits, we represented 2^3 = 8 numbers i.e from -3 to +3.
•   In 1’s complement, -0 and +0 are having 2 representation. (+0 is ‘000’ and -0 is ‘111’).
    But mathematically +0 and -0 are same.
•   In 2’s complement, using 3 bits, we represented only 2^3 = 8 numbers i.e from -4 to +3.
•   In 2’s complement, -0 and +0 are having same representation.
•   Since +0 and -0 in 2’s complement is having same representation, 
    we are left out with one more combination which is ‘100’ = -4.

Why do we take 1's Complement and add 1 to it? Why don't we subtract 1 after taking 1's Complement?

Refer "Why Inversion and Adding One Works" topic in the below link. If I start explaining, this post will grow big. http://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html

Why computer uses 2' Complement?

• Cos' of less hardware. If the computer is using 2' Complement means, it does subtraction using addition circuit. So, less hardware!!!
• As seen in the above example, +0 and -0 have same representation. (1's complement and sign magnitude representation have 2 different representation for +0 and -0).
• (Not an important) You will be able to represent one extra number using 2's complement. (in the above example its -4 which is '100' in binary using 3 bits).