Sunday, November 23, 2008

Do the digits always add up to nine?

It used to make Harry's sister Lizzie crazy when she would notice that me and Patty M were adding up customer bills at the produce market in our heads. So when she was around and paying attention we would sometimes act like we were adding up the customer totals on paper. Pretty dumb because almost all the prices were in round quarters, so adding up the totals was easy and natural; and at that time I could effortlessly do calculations in my head out to five and six digits.

At any rate one time at Harry's I noticed that when you multiple 9 by 1, 2, 3, 4, etc., the sum of the digits in the answer always adds up to 9. Naturally I realized that this was trivial for numbers up to 9 x 10; but it went on after that even into results that were four and five digits long. This is not as hard as it seems. You don't do multiplication; you just keep adding 9 to each new result and then mentally check the new result to see if it's digits add up to 9. Way back then I could consciously sort of start a little part of my head to doing arithmetic while still going on with other things, like waiting on customers and adding up their bills and imagining female customers naked; almost as though all of me was concentrating on what I was supposed to be doing. That may sound odd, but all of us do something like it when we daydream while driving.

But back to the nines. As I recall, I discovered a case in the relatively low thousands where the digits didn't add up to 9; but now I can't remember that result. And, of course, I may have made a mistake when I calculated in my head to come up with that result. So it may be that the digits always add up to 9. But I doubt that because there doesn't appear to be a mathematical proof of it, at least in the first few hits listed by google when I search for "digits add up to nine." There seem to be some interesting math games there related to the properties of products of 9 and the properties of 9 digit numbers in that list; but my head hurts when I try to really dig into math games now; so I didn't inspect them further than to get the impression that none of say outright that the digits of a product of 9 always add up to 9.

I wonder how many customer bills I added up wrong during the period when part of my brain was endlessly calculating the products of 9 and I was very irritated to be distracted by anything else. See below if you have no clue what I'm talking about.

If you find an instance where the digits don't add up to 9 you get this post named after you.

9 2 x 9 = 18 --- 1+8 = 9
3 x 9 = 27 --- 2+7 = 9
4 x 9 = 36 --- 3+6 = 9
5 x 9 = 45 --- 4+5 = 9

9 x 14 = 126 --- 1+2+6 = 9

9 x 29 = 261 --- 2+6+1 = 9

9 x 57 = 51 3 --- 5+1+3 = 9

9 x 2374 = 21366 --- 2+1+3+6+6 = 18 --- 1+8 = 9

9 x 79647732 = 716829588 --- 7+1+6+8+2+9+5+8+8 = 54 --- 5+4 = 9

5 comments:

Anonymous said...

Your post about the number 9 reminded me that i've been meaning to look up that exact property for a while now... anyway, after wikipediaing "9" i discovered an even more amazing property of the number 9. Take any number, reverse it's digits, and take the difference of the two, and you wind up with a multiple of 9...

63 - 36 = 27
92 - 29 = 63
814 - 418 = 396

Anonymous said...

They do always add up to nine. There is a rather simple proof where you factor out 9 from all of the powers of ten. You are then left with two parts one of which is obviously divisible by 9 due to the factor. The other is forced to be divisible by 9 since a theorem states that if the whole is divisible by a number then each of the parts is (not vice versa) It is sort of hard to explain without writing out the entire proof, but rest assured (from a math teacher and math graduate) it does always add up!

: )
Casey

Sully said...

Casey,
Thanks, I think I understand but I would like to see the proof.
Do you happen to know the name of the proof or a link to it?
Sully

Anonymous said...

I'm sure you've found a formal proof for this already, but in case you haven't, I saw something related to this on wikipedia.

What you're doing is finding the digital root of a number. The wiki article above does state that the digital root of a multiple of nine is always nine, however, it doesn't really give a proof for it. It does give a formula to find the digital root though:

dr(n) = 1 + [n - 1(mod 9)]

With this formula, it is easy to verify that plugging in a multiple of nine will always give nine:

dr(9x) = 1 + [9x - 1(mod 9)] = 1 + 8 = 9

Mary Byerley said...
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