Base complement and (base-1) complement of any Number

Negative numbers in all number systems can be represented in complementary form. For any number system there exist two types of complements and they are:

Base complement
(Base-1) complement

1. (Base-1) Complement Calculation :

To calculate (base-1) complement of a given number we have to subtract each and every digit from (base-1) value. That is, if base or radix of a number is “x” then to find the (base-1) complement of that number we have to subtract each and every digit from (x-1).

Example: Calculate the 8’s complement of the base 9 number -6350.

If 9 is the base, then (base-1) value is 8. Therefore 8's complement is:

                              8888-6350 = 2538

2. Base Complement Calculation :

Method I
To calculate the base complement of a given number proceeds as follows:

Step 1: Scan the number from the right side and copy all leading zeros until the first non-zero digit is encountered.
Step 2: Subtract the first non-zero digit from “base value” and the rest of the digits from “base-1 value”.

Example: Calculate the base complement of the base 9 number -6350.

Scanning from right the first non-zero digit is 5. 
Subtract 5 from 9 and the remaining digits from 8. 
9-5 = 4
8-3 = 5
8-6 = 2

Therefore 9's complement of base 9 number -6350 is 2540.

Method II

Step 1: Calculate (base-1) complement of the number.
Step 2: Add 1 to the (base-1) complement value.

Example: Calculate the base-1 complement of the base 9 number -6350.

The 8's complement of -6350 is 2538.

Therefore 9's complement of base 9 number -6350 is:
2538+1 = 2540

Practice Questions:
(1) Calculate base complement of base 4 number -12300.
(2) Calculate base complement of decimal number -198021.
(3) Calculate base complement of base 14 number -A13.
(4) Calculate base-1 complement of base 4 number -12300.
(5) Calculate base-1 complement of base 4 number -635.

Answers:

(1) 21100
(2) 801979
(3) 3CB
(4) 21033
(5) 253

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