Sum of digits without regard to place value?

[marcuss421]

I apologize for my lack of knowledge on the topic.
I recently started writing programs to solve Project Euler problems and it rekindled my interest in number theory. Especially as it relates to a peculiar relationship I found back in high school. I would like to learn more about number theory and I need some direction with regards to the names of some functions/operations.

Essentially at the heart of the relationship I found was the fact that every number I obtained as a result differed from its "counterpart" by a multiple of 9. Always. After some research I believe this to be "Mod 9".

The one of the operations that got me to this point however I can't for the life of my find a name for. I was essentially adding the digits of a number without regard to place value. e.g. 123 becomes 1+2+3 = 6. Is there a formal name for this operation? And can you point me in the direction of any existing work on integers and relationships that occur when you do these types of operations that discard place value and instead focus on the individual digits themselves? I've done a fair amount of googling and I can't seem to find any formal names or fields of study research.

 

[TylerH]

It's called being "congruent mod 9." But you need to define the term "counterpart."

You could define them, like so:
$$a := \sum_i d_i 10^i$$
$$b := \sum_i d_i$$

Then your conjecture could be written $a \equiv b \mbox{ mod 9}$, which is read "a is congruent to b modulus 9."

http://en.wikipedia.org/wiki/Modular_arithmetic

 

[micromass]

Hi marcuss421!

Not really an impressive sounding name, but isn't "digit sum" what you're looking for:

http://en.wikipedia.org/wiki/Digit_sum
http://mathworld.wolfram.com/DigitSum.html

 

[marcuss421]

@Tyler H
Pardon my ambiguity, but I prefer to remain veiled in my description of this relationship in hopes it may be something new. However, to define the term "counterpart" I can provide some clarity. Let's say I start with the number 1234, and then I do some manner of manipulation to this number, and arrive at 45. The "counterpart" would be for example 2341, which is related to 1234 in the sense that the digits have simply been shifted. When that same manner of manipulations that transforms 1234 into 45 is applied to 2341, the result becomes let's say 27. 45-27 = 18, hence the relationship being congruent mod 9. The moment I find out that I am not observing something new, I will reply/repost with a full explanation.

@micromass

Thank you! I was hindered in finding that by my assumption that the operation would have some fancy name.

Do you know of any other operations/studies involving the digits of numbers themselves? So far it looks like the existing topics include: Digit Sums, Digit Roots, Checksums, and various number sets that are some application of the aforementioned topics?

 

[micromass]

Check out http://mathworld.wolfram.com/topics/Digit-RelatedNumbers.html for some digit-related properties and functions on numbers...

 

[ramsey2879]

To prove the divisibility by nine test you rediscovered, simply note that
$$a-b = \sum_i d_i *(10^i - 1) = \sum_i d_i * 9s$$ Thus the difference between the two sums is divisible by 9. By the law of congruence http://mathworld.wolfram.com/Congruence.html , if the difference between two numbers is divisible by m, the two numbers are congruent modulus m, i.e have the same remainder when divided by m.

 

[HallsofIvy]

You might also want to look up "casting out nines".

 

[marcuss421]

New Question... Does anyone know if there's a term for "semi palindromic" between two numbers? Eg. 155 and 551. They aren't in and of themselves palindromic, but they are "mirrored". My interest in these numbers is that they are vastly different numerically, yet they have identical digit sums.