Complement of a number with any base b

In this post, a general method of finding complement with any arbitrary base is discussed.

Remember: It is recommended to go through below as follows:

• One’s Complement of an Integer
• 1’s and 2’s complement of a Binary Number

Steps to find (b-1)’s complement: To find (b-1)’s complement, 

• Subtract each digit of the number from the largest number in the number system with base .
• For example, if the number is a three-digit number in base 9, then subtract the number from 888 as 8 is the largest number in base 9 number system.
• The obtained result is the (b-1)’s (8's complement) complement.

Steps to find b’s complement: To find b’s complement, just add 1 to the calculated (b-1)’s complement. Now, this holds true for any base in the number system that exists. It can be tested with familiar bases that is the 1's and 2's complement.

Example: 

Let the number be 10111 base 2 (b=2)
Then, 1's complement will be 01000 (b-1)
2's complement will be 01001 (b)

Taking a number with Octal base:
Let the number be -456.
Then 7's complement will be 321
and 8's complement will be 322

Below is the implementation of the above idea as follows:  

41
42

Time Complexity: O(log n)

Auxiliary Space: O(1)