The rule of base minus one, aka the rules of nine and three

[Feynmanfan]

I was curious about why any number with digits totalling a number divisible by 9 is itself divisible by 9 and couldn't find the answer on the web, though I'm sure it's out there somewhere. Forced to actually think about it, or else remain forever mildly intrigued, I noticed that when you add nine to any number the total of the digits will always remain the same, increase by nine or decrease by nine, from which the rule of nine follows. I also noticed that this applies in any base: the rule of nine is really the rule of base number minus one.

Similarly, adding three to any number always increases or decreases the total of the digits by a number divisible by three, from which the rule of three follows. This also appears to be a general rule, which is to say, the rules of nine and three are actually the rule of base -1 and all it's factors. So in base seven there are the rules of two and three, in base nine the rules of eight, four and two, and so on. All odd base numbers have the rule of two, and can end in an even or an odd digit.

I have satisfied my curiosity though I know this is far from a rigorous proof. Perhaps someone would like to add some rigor to this conjecture and prove the general case.

 

[Bob Brown MSEE]

proof

click here

 

[JeffM]

I was curious about why any number with digits totalling a number divisible by 9 is itself divisible by 9 and couldn't find the answer on the web, though I'm sure it's out there somewhere. Forced to actually think about it, or else remain forever mildly intrigued, I noticed that when you add nine to any number the total of the digits will always remain the same, increase by nine or decrease by nine, from which the rule of nine follows. I also noticed that this applies in any base: the rule of nine is really the rule of base number minus one.

Similarly, adding three to any number always increases or decreases the total of the digits by a number divisible by three, from which the rule of three follows. This also appears to be a general rule, which is to say, the rules of nine and three are actually the rule of base -1 and all it's factors. So in base seven there are the rules of two and three, in base nine the rules of four and two, and so on. All odd base numbers have the rule of two, and can end in an even or an odd digit.

I have satisfied my curiosity though I know this is far from a rigorous proof. Perhaps someone would like to add some rigor to this outline and prove the general case.

This is even less than an outline. It identifies the crucial piece of a formal proof.

It is your proposition that if a is an integer the decimal digits of which sum to a multiple of 9, then the next higher integer with that property is a + 9.

The other general proposition needed is that if b is a multiple of 9, the next higher integer with that property is b + 9.

All that is left is to demonstrate that some integer has both properties. Zero is such an integer. Given that, the set of all integers that are a multiple of 9 must equal the set of all integers whose decimal digits sum to a multiple of 9.

 

[Feynmanfan]

click here

Thank you for that. As I suspected, I don't really understand the proof of the rule of nine, since it uses modular arithmetic, indeed, I don't even understand the question as it is stated in the link. A further link from that page - https://math.stackexchange.com/ques...r-bases-radix-and-doing-check-sums-for-binary gets a little closer to the general rule I've suggested:

I was showing my son how to cast out nines the other day. He noted that based on the way it worked, we should be able to cast out 7s when we work with octal. We tested this in several bases and it seems to work. [/FONT]

Clever boy! Though he probably isn't a boy any longer sine the post is seven years old. But that's not very close to saying that for any base, base minus one and all it's factors follow the rule. I never learnt to cast out nines, or what a checksum is and what, if anything, they have to do with what I call the rule of nine. I am and intend to remain a mathematical ignoramus by the standards of this forum, so noticing that the rule of base-1 is apparently a general one is "peak me", mathematically, so I took the opportunity to air it here.

I'm surprised that it isn't well known, since it's the only interesting thing I've ever heard about the base concept, and as such it could be useful to teachers, but apparently very few are aware of it.