A question about mirror numbers

[edgitarra]

I was trying to search the theorem or the official formula of something that I discovered accidentally when thinking about mirror numbers.
It seems like the difference between a number and its mirror is a multiple of 9.
Example:
36,63 -> 63-36=27=9x3
152,251 -> 251-152=99=9x11
1005,5001 -> 5001-1005=3996=9x444

I didn't manage to find anything related to it. It might be also possible because I wasn't using the correct keywords.
Is there somebody who knows about this, to arrive faster at information about it?
Thanks for your responses!
Ed

 

[obyvatel]

Hi edgitarra,
I do not know whether there is any official theorem associated with this. Using your term mirror numbers, in general we can say:
The difference between two mirror numbers expressed in base (r) is divisible by (r-1).

Taking the example of decimal (base 10) numbers, the difference is divisible by (10-1) = 9.

To prove this we need a little algebra and express the number in expanded notation.

To take an example, let us choose any 4 digit number where the digits are represented by symbols "abcd" The mirror number would then be "dcba"

abcd is written in expanded notation as 1000* a + 100*b + 10*c + d
dcba is similarly written as 1000*d + 100*c + 10*b + a

(abcd) - (dcba) becomes after regrouping terms

= (1000-1)* (a-d) + (100-10)*(b-c)
= 999 * (a-d) + 90 *(b-c)
= 9 * [ 111*(a-d) + 10* (b-c)]

which is divisible by 9.

For a general proof for any number expressed in any base, one way is to use the binomial theorem .

Any general term which accompanies the difference of digits (like 999 and 90 in our 4 digit example) can be factored to take the form of
10^n - 1
= (9+1)^n - 1
= 9 * (.........)

where ^ indicates exponent.

Replace 10 by a general base (r) and we will get the conclusion
The difference between two mirror numbers expressed in base (r) is divisible by (r-1).

 

[trendsetter37]

Hey Edgitarra,

Just to add to obyvatel wrote, I believe those are referred to Palindromic Numbers. If you use that as a search term in google or wolfram some pretty interesting stuff comes up.

_http://mathworld.wolfram.com/PalindromicNumber.html

and

_http://en.wikipedia.org/wiki/Palindromic_number

Note: that the second number to the right of 7 is its power that is being evaluated.

73 = 111
74 = 777
76 = 12321
79 = 1367631

The sum of all of those are also divisible by 9, 6, or maybe even more importantly because of our 3D reality; 3.

1, 2, 3 multiplied by 3 gives you 3, 6, 9. 123 also being the first primes may be fundamental? I remember the C's discussing the significance of these numbers as well.

"If you only knew the magnificence of the 3, 6 and 9, then you would have a key to the universe." -- tesla

I think tesla theorized that the number 9 represented the aether ergo it was omnipresent and everywhere. Maybe this is mathematical evidence for this? But I could easily see getting drowned by following some of these mathematical trails. However with that being said it may be the key to "re-discovering" some of the purported ancient science that at one time was known.