Modular number systems and the 2s complement system The 2s complement method The 2s complement is an important method for binary addition and subtraction and is more efficient than the other methods. Hence, most systems implement this technique to perform binary operations. The working of the 2s complement method can be better understood by a graphical representation based on modular arithmetic. Modular Number Systems Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by the German mathematician Gauss. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods If the time is 8:00 now, then 8 hours later it will be 4:00. Simple addition would result in 8 + 8 = 16, but clocks "wrap around" every 12 hours. Because the hour number starts over after it reaches 12, this is arithmetic modulo 12 Circular representation of integers mod N A helpful graphical device for the description of addition of A helpful graphical device for the description of addition of unsigned integers mod N is a circle with the values 0 through N − 1 integers mod N is a marked along its perimeter, unsigned as shown in Figure circle with the values 0 through N − 1 marked along its perimeter, as shown in the figure. Addition Modulo 16 (4 bit numbers) § Consider the case N = 16 § The decimal values 0 through 15 are represented by their 4bit binary values 0000 through 1111 around the outside of the circle § Say we want to check the result of the operation (7+5)%16 § To perform this operation graphically, locate 7 (0111) on the outside of the circle and then move 5 units in the clockwise direction to arrive at the answer 12 (1100). § Similarly, (9 + 14) mod 16 = 7; this is modeled on the circle by locating 9 (1001) and moving 14 units in the clockwise direction past the zero position to arrive at the answer 7 (0111). § This graphical technique works for the computation of (a + b) mod 16 for any unsigned integers a and b; that is, to perform addition, locate a and move b units in the clockwise direction to arrive at (a + b) mod 16. 2s complement method Now consider a different interpretation of the mod 16 circle We will reinterpret the binary vectors outside the circle to represent the signed integers from −8 through +7 in the 2’s-complement representation as shown inside the circle. Let us apply the mod 16 addition technique to the example of adding +7 to −3. The 2’s-complement representation for these numbers is 0111 and 1101, respectively. Here assume the second binary number 1101, to be an unsigned number, which gives us 13. To add these numbers, locate 0111 on the circle, then move 1101 (13) steps in the clockwise direction to arrive at 0100, which yields the correct answer of +4. Note that the 2’s-complement representation of −3 is interpreted as an unsigned value for the number of steps to move. The circular representation of modular arithmetic system can be used as way to interpret the working of binary addition and subtraction using 2s complement method graphically. Modular arithmetic is also part of the reason for discarding the carry-out bit(under no overflow condition),since in modular arithmetic for the number 16, the value next to 1111 is taken as 0000 and not the next binary number 10000.