Thursday, July 3, 2008

Rule of 9

Do you remember when you were in grade school and someone offered to show you this magic trick? It goes something like this – pick a number between 1 and 10, then add *something* to it, then subtract, then multiply …blah blah blah, then multiply by 9 and add the sum of the digits. The person would promptly shout out the number you ended up with. Wow! Hopefully you realized it wasn't magic and there was a mathematical trick to it.

It's actually quite a strange phenomenon that I've dubbed the "Rule of 9." Basically if you multiply any number 1-10 by 9, then add the two digits, the result is always 9. So all the adding and subtracting in the trick is pointless; only the last two steps where you multiply a single digit number by 9 and add the digits is fruitful. I think there is some mathematical principle here, but I really don't know it. I haven't researched the topic thoroughly mostly because I have no idea what the principle is called so I can search for it. I implore the reader to enlighten me if they know more about it than I do.

I messed around with this and created the following table.

It shows that there is significant structure beyond the single digit numbers. Notice that at 11 the sum of the digits is 18. At 21 and 22 the sum is 18, and so on. The pattern follows in that manner until all 10 numbers give 18 as the sum of the digits. Then it starts over with 27 being the number that shows up rather than 18. I'm sure I could express it in some mathematical fashion, but I've never been that good at this sort of random mathematics, and frankly I don't feel like trying at the moment. I suspect this pattern will continue forever and the digits of every number that appears always add to 9. I'm not sure what the significance of this is, if there is any. It's an interesting observation I decided to write about because it reinforces my feeling that these things are the way they are because universe is highly structured and the universe is highly structured because these things are the way they are. It follows the chicken or the egg conundrum. Some physicists have posed that we exist because the universe is the way it is, but the universe is the way it is because we exist. It's along those same lines.

And now a deep thought…

Hmm... What if the Heisenberg Uncertainty Principle (by measuring (observing) something we ultimately change it) has a limit? Could the structure in the universe have appeared because our influence has forced us beyond the transient state?


yNot said...

It is always true. The sum of the digits of any multiple of 9 is a multiple of 9. The sum of the digits of this new number is also a multiple of 9... until you end up with one digit, which is a multiple of 9, i.e. 9!

This works because

10^n = 1 (mod 9) for any n >= 0.

Check this out:

Deep thought:

- Heisenberg's uncertainty principle does not apply to the large scale structure of the universe (to the best of my knowledge), so it is limited by the physical scale of the object being measured. However, it might have had significant implications (us maybe) if someone was trying to measure something during the primordial state of the universe.

-Doesn't our influence on the universe force it to be in a transient state because of our unpredictable behavior?

Josh said...

Ahh! Thanks for the insight on both topics. I suspected uncertainty principle was invalid or at least transcendentally small for the large scale. I was suggesting more of the primordial state of the universe. The problem is, I don't think we were around in the primordial times. So perhaps the "sensing" of particles by other particles in random orientations and directions through the universal forces or light reflections caused alignment giving us what is now universal order. I'm not sure it completely makes sense but lots of times my deep thoughts don't.

yNot said...

The data provided by the background radiation measurements shows inhomogeneities in the density of the primordial "soup". This suggests that these tiny regions where "matter was slightly denser than average, will, in the eons that follow, attract other matter to form the structures we see today".

ref. - scroll to the bottom of the page

I think that the reason for this inhomogeneity is still unknown, except if the Heisenberg principle is invoked: some "agent" decides to measure the position and velocities of the particles thus dislocating them and altering their momentum resulting in a density gradient. This agent might be a kind of particle called "measuron"!