Radix Complement

Radix Complement Representation:

In the decimal number system, the radix complement is the 10’s complement. In radix complement representation system, the complement of an n-digit number is obtained by subtracting the number from 10_n.

Let us consider some examples of 3-digit numbers and their radix complement in decimal system.

Decimal Number

948

607

155

735

Radix Complement

393

845

265

br>From the above discussion we find that a subtraction operation is to be preformed to get the 10’s complement of a number, say, N. This subtraction operation can be avoided by rewriting 10_n as (10_n - 1) + 1 and 10_n - N as {(10_n - 1) - N} + 1. The number 10_n - 1 is of the form 999...99 consisting of n digits. If the complement of a digit be defined as (9 - the concerned digit), then (10_n - 1) - N is obtained by complementing the digits of N.

Therefore, the 10’s complement of the number N is obtained by subtracting each digit of the number from 9 and then adding 1 to the LSD of the number so formed.

For instance, the 10’s complement of 172 is (827 + 1) or 828 and that of 405 is (594 + 1) or 595.

For the binary number system the radix complement is the two’s complement. The 2’s complement of a binary number is obtained by subtracting each bit of the number from the radix diminished by 1 i.e. from (2 - 1) or 1 and adding an 1 to the LSB. The application of this rule is very simple. We have to just change 1 to 0 and 0 to 1 in every bit and then add 1 to the LSB of the number so formed. For example, the 2’s complement of the binary number 11011 is (00100 + 1) or 00101 and that of 10110 is (01001 + 1) or 01010.

If the number be in signed magnitude representation, it is positive if the MSB is 0 and negative if the MSB is 1. The decimal equivalent of a 2’s complement binary number, in the case of signed-magnitude representation, is computed in the same way as for an unsigned number except that the weight of the MSB is -2^n-1 instead of +2^n-1 for an n-bit binary number.

Let us observe some examples of 8-bit binary numbers and their 2’s complement are shown below: