A binary number is a number whose representation is a combination of only two digits, 0 and 1. For example, 01011001 and 010 are two binary numbers.
Methods to represent signed binary numbers
There are three well-known methods to represent signed binary numbers: sign-magnitude, ones’ complement, and two’s complement.
Sign-magnitude
This method allocates the most significant bit (MSB) for representing the sign and the other bits for the number magnitude. A number is negative if its MSB is 1 and positive otherwise. For example, 0101 represents 5, whereas 1101 represents -5.
Given n bits, the range of the numbers this method can handle is (-(2^(n-1) - 1), 2^(n-1) - 1).
It should be noted that this method yields two representations of zero. For example, given 4 bits, the two representations of zero are 1000 and 0000.
Ones’ complement
A ones’ complement of a binary number is the result of applying bitwise NOT operation on that binary number. For example, the ones’ complement of 10010 is 01101. Using this method, a negative binary number is the ones’ complement of its positive counterpart. For example, in a 4-bit system, 0101 represents 5 and 1010 represents -5.
Similar to sign-magnitude, given n bits, the range of the numbers represented by this method is (-(2^(n-1) - 1), 2^(n-1) - 1).
There are also two presentations of zero using this method. For example, given 4 bits, the two representations of zero are 0000 and 1111.
Two’s complement
A two’s complement of a binary number of n bits is the result of subtracting the number from 2^n. Since the sum of a binary number of n bits and its ones’ complement is a number of n bits 1, the two’s complement of the binary number is also equal to the sum of its ones’ complement and 1. For example, the two’s complement of 10101 is equal to 1 + 01010 = 01011. Using this method, a negative binary number is the two’s complement of its positive counterpart.
There are two ways to compute the two’s complement of a binary number. The first one is mentioned above, which computes the two’s complement by adding 1 to the ones’ complement of the binary number. The second one is a bit more convenient for computing by hand. Specifically, it has two step: it finds the first 1 bit starting from the least significant bit, and then flips all the bits on the right of that 1 bits. For example, to find the two’s complement of 10110, this second approach keeps 10, flips 101 into 010, and finally gives the result of 01010.
Arithmetic
Addition
Rules:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0, carry 1
Example:
1 1 1 1 1 (carried)
0 1 1 0 1
+ 1 0 1 1 1
-----------
= 1 0 0 1 0 0
Subtraction
Rules:
0 - 0 = 0
0 - 1 = 1, borrow 1
1 - 0 = 1
1 - 1 = 0
Example:
1 1 1 1 (borrowed)
1 1 0 1 1 1 0
- 1 0 1 1 1
----------------
= 1 0 1 0 1 1 1
Substraction can also be done by adding the substrahend to the two’s complement of the minuend, i.e., x - y = x + ((~y) + 1).
Multiplication
Multiplication is performed similarly to that of decimal numbers, but with the following rules:
x * 0 = 0
x * 1 = x
Example:
1 0 1 1
* 1 0
---------
0 0 0 0
+ 1 0 1 1
---------
= 1 0 1 1 0
Division
Division is performed similarly to that of decimal numbers.
Example: 11011/101 gives the quotient 101 and the remainder 10.
1 0 1) 1 1 0 1 1
- 1 0 1 -> 1
-----
1 1 -> 0
1 1 1
- 1 0 1 -> 1
-----
1 0
Further reading
[1] https://en.wikipedia.org/wiki/Binary_number
[2] http://www.electronics-tutorials.ws/binary/signed-binary-numbers.html
[3] https://en.wikipedia.org/wiki/Signed_number_representations