Radix and Reduced Radix Complementation

Nearly all computers use some form of number complementation to perform tasks. Using complementary numbers is faster and simpler than using signed numbers (numbers with a "+" or "-" sign) in computer operations. Therefore, it is important to understand the radix and reduced radix complements of numbers when studying computer science.



Radix Complements (b's Complements)

Definition: Two numbers X and Y are considered to be radix complements of each other if:

X + Y = 10bn

Where n is the number of digits in X and Y, and Where b is the base of X and Y
 
 

Consider the case where X is some base b number, six digits long:

X = ******b

Then Y is also a base b number six digits long, but satisfying the condition

X = * * * * * *b

+ Y = * * * * * *b

1 0 0 0 0 0 0b = 10b6

 Here, X is the radix complement of Y, and Y is the radix complement of X.

Some examples of radix complements are the following:

1010112 and 0101012  because  1010112 + 0101012  =  102110

3728 and 4068  because  3728 + 4068  =  1084

904610 and 095410  because  904610 + 095410  =  10104

A62E316 and 59DIC16  because  A62E316 + 59DIC16  =  10165
 
 
 
 

Reduced Radix Complements (b-1's Complements)

Reduced radix complements are very similar to radix complements

Definition: Two numbers X and Y are considered to be reduced radix complements of each other if:

X + Y = 10bn -1

Where n is the number of digits in X and Y, and

Where b is the base of X and Y

 For example, 172510 and 827410 are reduced radix complements of each other because:

172510

+ 827410

999910

Therefore, the reader should notice that if X and Y are numbers n digits long and are in base b, their sum should equal a number, n digits long, where (b-1) is every digit.

This means the sum should equal n number of 9's in base 10, n number of 1's in base 2, n number of 7's base 8 and n number of F's base 16.

Some examples of reduced radix complements are the following:

101102 and 010012 because 101102 + 010012 = 111112 = 102101 - 1

0638 and 7148 because 0638 + 7148 = 7778 = 1083 - 1

587410 and 412510 because 587410 + 412510 = 999910  = 10104 - 1
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Bibliography

Early, G. (1995). Mathematics 2358: Mathematics Supplement. San Marcos,

TX: SWT Print Shop.

Created by: Jay Whitehead, March 1999
Student Learning Assistance Center
Southwest Texas State University