10's Complement

Discussion in 'Homework Help' started by dalam, Dec 31, 2014.

  1. dalam

    Thread Starter Member

    Aug 9, 2014
    How can I find 10's complement of -417?
    I know that 10's complement of any number, for eg. 423 is (999-423)+1=577.
    But how to do the same with negative number. I am a bit confused.
  2. jpanhalt

    AAC Fanatic!

    Jan 18, 2008
    dalam likes this.
  3. WBahn


    Mar 31, 2012
    Your problem is that you are getting confused by commonly used, but improper, terminology.

    10's complement, like 2's complement, is not an operation but a representation. So saying, "take the 10's complement" is inaccurate and misleading. Also, saying "10's complement" is not sufficient to establish what number base you are talking about (consider that in base-2 we commonly use 1's complement and 2's complement, but in base-3 we would use either 2's complement or 3's complement, so if we just say 2's complement then the reader has to infer that we are talking base-2 and not base-3). The correct terminology, BTW, is "radix complement" and "diminished radix complement".

    I'm going to assume that you are talking about the base-10 radix complement notation.

    In base-b radix complement notation, the additive inverse of the non-negative integer x is represented as (b^N)-x where N is the number of digits in the representation.

    So if you want the 10's complement representation of -417, you need to specify the base (we'll assume b=10) and the number of digits (you are implying N=3, but is that really the case?).

    Thus the way that we represent -417 is as the additive inverse of 417 which is (10^3)-417.

    If we have a negative number to start with, then what we actually have to work with from the start is the 10's complement representation. For -423 that would be 577. That's our starting point and that IS -423; it is NOT 577 because, in a three digit base-10 radix complement notation we can only represent values from -500 to 499 and the representation for -423 just looks like 577 to our eyes because of what we are using to working with, which is a signed decimal notation having an extra symbol for the minus sign, but it isn't.

    Bottom line: The phrase "the 10's complement of X" means "the representation of the value X using base-10 radix complement notation". Thus the 10's complement of 423 is 423 while the 10's complement of -423 is 577. When someone says, "take the 10's complement of X", what they SHOULD say is "find the 10's complement representation of -X".

    So if X=423 then the 10's complement representation of -423 is 577.

    If X=-423 then the 10's complement representation of 423 is 423.
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