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.

 

According to this site, the 10's complement of a negative number is done the same way and padded with 9's: https://cs.senecac.on.ca/~david.ward/ics124/notes/decimal_complements.html

So, for -417, the answer would be 9583

As a check add +423 and -417:

0423
9583
0006

John

 

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.

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.