Tom Adams, post: 391189, member: 7285 wrote: Everyone probably already knows this, but if you accidentally flip the last two digits of a number, the difference is always a multiple of 9. 64 vs 46, 64-46=18
52-25=27 18 or 27 is a multiple of 9
The rule is even better than that - it works for any digits in the number, not just the last two.
For any single transposition error of adjacent digits, the offage will be divisible by nine. It works for decimals too, but might not work outside of base-10 numbering systems?
A lot of people don't know this one - I sure didn't until one day I overheard a banking friend* help a teller find a balancing error over the phone, "Yeah, Ok... How far are you off? Ahhh, yeah - re-run your checks again - you've got a transposition error in there somewhere."
When you hear that kind of wizardry you just HAVE to ask how its done.
*The kind of banker who could remember the days when there were more ash trays behind the teller line than computers.
321-123 = 198 = 9 x 22
321-213 = 108 = 9 x 12
321-231 = 90 = 9 x 10
6483756 -6843756 = 360000 = 9 x 40000
8th grade teacher (Mr. Barndt) called it "the rule of nines"
pretty handy, especially in the days before the calculators (or abacus if you are older than me)
The proof seemed simple to imagine.
If a and b are adjacent numbers where the a column represents 10 times more than the b column then to transpose the numbers will result in a difference of (b-a) x 10 and a-b.
Which is (b-a) x 10 + (a-b)
= (b-a) x 10 - 1 x (b-a)
= (b-a) x (10 - 1)
= 9 x (b - a)
The solution seems to scale by a factor of the column occupied by b. I.e. Add a zero for columns remaining after b
Actually it seems as though if you transpose any two digits in a number the difference is always divisible by 9. And the difference seems to be able to be given by the following rule: If we let c = b - a, then the difference is a string of c's which numbers the same as the column place difference of a and b, with a zero for every column after b, and then multiplied by 9, and you should have the answer.
E.g. 1234567 to 1734562
C = 7 - 2 = 5
Column Separation = 5
Diff = 55555 x 9
= 499995
Again. 86420 to 06428
C = 0 - 8 = -8
Col Sep = 4
Diff = -8888 x 9
= -79992
Visually it just looks like you take c x 9, then jam a bunch of nines after the first digit, equal to the number of other columns between a and b, then add a zero for each column following b at the vey end.
987654321 to 987154326 should be 1-6 = -5 x 9 = -45 with 4 9's jammed in between: -499995