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Scattershot Private
Joined: 05 Feb 2004 Posts: 4

Posted: Fri Feb 06, 2004 11:31 am Post subject: 10's complement of a positive decimal integer 



Hi Chris,
This is an exercise taken from a book that I am currently studying from. I don't understand this question very well, among others, what the d is supposed to represent. It also doesn't provide the answer in the back of the book. Could you please shed some light on this?
And explain how I can calculate the first question, I'll try the second myself.
The 10's complement of a positive decimal integer n is 10 to the power of k minus n, where k is the number of digits in the decimal representation of n. It can be calculated in the following way:
1. All the zeros at the righthand end of the number remain as zeros in the answer.
2. The rightmost nonzero digit d of the number is replaced by 10  d in the answer.
3. Each other digit d is replaced by 9  d.
Find the 10's complements of the following decimal numbers using the rules given above, and check your answers by evaluating 10 to the power of k minus n on a calculator:
(a) 3296
(b) 10350 

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chris Dark Lord of the Sith
Joined: 10 May 2003 Posts: 6267 Location: Outer Space

Posted: Fri Feb 06, 2004 12:49 pm Post subject: Re: 10's complement of a positive decimal integer 



Scattershot wrote:  Hi Chris,
what the d is supposed to represent.

It tells you in 2. what d is: the rightmost nonzero digit d of the number. In other words, d is the nonzero coefficient of the smallest power of 10 in the decimal representation of the number.
To take (a) as an example:
3296
Rule 1. does not apply, since we don't have any zeroes in that number.
Rule 2: we must find the rightmost nonzero digit. What are the nonzero digits of 3296?
3, 2, 9 and 6 of course!
What are their positions?
3 is the leftmost, 2 is to the right of 3 and 9 to the right of 2 and 6 to the right of 9. What is the "rightmost" digist of those?
6 of course. What does rule 2 say? "The rightmost nonzero digit d of the number is replaced by 10  d in the answer.". So our d is 6 and we replace 6 with 106 = 4.
Now comes rule 3 into play: "Each other digit d is replaced by 9  d.". What are our "other digits"? What remained of course! That's 3, 2 and 9. Now take each one of them, let d be the digit and replace d by 9d:
 For d = 3: we replace the digit 3 with 9  3 = 6
 For d = 2: we replace the digit 2 with 9  2 = 7
 For d = 9: we replace the digit 9 with 9  9 = 0
So our 10's complement is 6704. Indeed 3296 + 6704 = 10000 = 10^5. _________________ Regards
Chris Karakas
www.karakasonline.de 

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dhwani Private
Joined: 28 Apr 2009 Posts: 1

Posted: Tue Apr 28, 2009 5:04 am Post subject: Re: 10's complement of a positive decimal integer 



Hi Chris,
Thanks a lot as the way you have explained 10's complement is really simple. _________________ Regards
Dhwani 

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