Radix & Diminished Radix Complement | Digital Circuits - Electronics and Communication Engineering (ECE) PDF Download

Introduction

In the above formulas,

• The n is the number of digits in the number.
• The N is the given number.
• The r is the radix or base of the number.

Advantages of r's complement

These are the following advantages of using r's complement:

• In r's complement, we can further use existing addition circuitry means there is no special circuitry.
• There is no need to determine whether the minuend and subtrahend are larger or not because the result has the right sign automatically.
• The negative zeros are eliminated by r's complement.

Let's take some examples to understand how we can calculate the r's and (r-1)'s complement of binary, decimal, octal, and hexadecimal numbers.

Example 1: (1011000)₂
This number has a base of 2, which means it is a binary number. So, for the binary numbers, the value of r is 2, and r-1 is 2-1=1. So, we can calculate the 1's and 2's complement of the number.
1's complement of the number 1011000 is calculated as:
= {(2⁷ )10^-1}-(1011000)₂
= {(128)₁₀-1}-(1011000)₂
= {(127)₁₀}-(1011000)₂
= 1111111₂ - 1011000₂
= 0100111
2's complement of the number 1011000 is calculated as:
= (27 )₁₀ - (1011000)₂
= (128)₁₀ - (1011000)₂
= 10000000₂ - 1011000₂
= 0101000₂

Example 2: (155)₁₀
This number has a base of 10, which means it is a decimal number. So, for the decimal numbers, the value of r is 10, and r-1 is 10-1=9. So, we can calculate the 10's and 9's complement of the number.
9's complement of the number 155 is calculated as:
= {(10³)10^-1}-(155)₁₀
= (1000-1)-155
= 999-155
= (844)₁₀
10's complement of the number 1011000 is calculated as:
= (10³)₁₀ - (155)₁₀
= 1000 - 155
= (845)₁₀

Example 3: (172)₈
This number has a base of 8, which means it is an octal number. So, for the octal numbers, the value of r is 8, and r-1 is 8-1=7. So, we can calculate the 8's and 7's complement of the number.
7's complement of the number 172 is calculated as:
= {(8³)₁₀ - 1} - (172)₈
= ((512)₁₀ - 1) - (132)₈
= (511)₁₀ - (122)₁₀
= (389)₁₀
= (605)₈
8's complement of the number 172 is calculated as:
= (8³)₁₀-(172)₈
= (512)₁₀ - 172₈
= 512₁₀ - 122₁₀
= 390₁₀
= 606₈

Example 4: (F9)₁₆
This number has a base of 16, which means it is a hexadecimal number. So, for the hexadecimal numbers, the value of r is 16, and r-1 is 16-1=15. So, we can calculate the 16's and 15's complement of the number.
15's complement of the number F9 is calculated as:
{(16²)₁₀ - 1}-(F9)₁₆
(256-1)₁₀-F9₁₆
255₁₀ - 249₁₀
(6)₁₀
(6)₁₆ 
16's complement of the number F9 is calculated as:
{(16² )₁₀} - (F9)₁₆
256₁₀ - 249₁₀
(7)₁₀
(7)₁₆