Fun with numbers

Let's walk away from current events for a moment -- sad, so sad -- and spend some time on the ideal plane, on the ideal plane of numbers, where what is, is, and what ain't, ain't, on the ideal plane of numbers where spin is spin, definable by revolutions per minute or subatomic quantum mechanics of the sort not available to lay understandings, rather than a technique whereby people say up is down and disease is health and hate is love and death is birth and no trees is yes trees and negative numbers are positive.

Yes, as we go into the weekend, let's leave all that behind, shall we? Let's cozy up to some warm, wonderful, nonjudgmental numbers, shall we?

* * *

Numbers can be magical. Now, as quite a few people know, 72 is a magic number, and a very useful one at that, particularly for people who, wanting to gain a better understanding of their financial health, use it to help them better perceive their financial health as it is to be expected to change over time.

Another magical number, with much less practical use than 72 and a whole lot more obscurity, but with a real wonderment built into it, is this one:

142857

142857 is a magical number because, when you multiply 142857 by any number between 2 and 6, the product you get is quite similar to the number itself, snipped into two pieces and with each of the two pieces doing a dosie-doe.

Witness:

142857
X 2
________
= 285714

and 285714 is a dosie-doe'ed, or spoonerized, 142857, with the first two digits cut off and appended to the back, so you take the 14 off the front and put it behind the remaining 2857 and you get 285714.

And continuing on, in order, take a look at these:

142857
X 3
________
= 428571

and 428571 is a spoonerized 142857, with the first digit cut off and appended to the back.

142857
X 4
_________
= 571428

and 571428 is a spoonerized 142857, with the last two digits cut off and prepended to the front.

142857
X 5
_________
= 714285


and 714285 is a spoonerized 142857, with the last digit cut off and prepended to the front.

142857
X 6
_________
= 857142

and 857142 is a spoonerized 142857, cleft in two, with the front and the back halves switcheroo'ed.

After 6, things get a little less rhythmic and a bit more complicated, but still remain cool:

142857
X 7
________
= 999999

which is a cool number by itself, and hints that another magical number -- one known to grade schoolers for its magic -- is playing a role here. We refer, of course, to the magicalest of magicals, the magical number 9.

From here things really get interesting. Witness:

142857
X 8
_________
= 1142856

which is, starting with the second digit, 14285, and then to get the 7, so as to round out the 142857 pattern, you have to add the next two digits together, the 6 and the 1, jumping from the last digit to the first. That makes sense, doesn't it, because now we're dealing with a seven-digit, rather than a six-digit, product, right?

So here one of the digits of the 142857 pattern encompasses two different digits, by spilling out the back of the product and having whatever's needed to make up the difference take up residence in the front of the product.

It's kinda like "carrying" when you do simple addition. Right? But kind of upside down and backwards from that.

Wild, huh?

So let's continue on:


142857
X 9
_________
= 1285713


which is, starting in the second-to-last digit in the number 1, 4 (i.e., the 3 in the last place of the product plus the 1 in the first place of the product) and then resuming with the standard pattern of 2, 8, 5 and then 7.

And, wouldn't you know it, but this seems to just work and work and work, but it gets complicateder and complicateder.

Here's a double digit example:

142857
X 26
_________
= 3714282

so, adding the 2 and the 3 residing like bookends at either end of the product, you get 5, and that means that, starting with the 1, you get 142857.

After that, the carrying sometimes gets way more complicated. Witness:

142857
X 37
_________
= 5285709

So there's a 2857 in there, and now we need to see a 14 somehow. So after that 2857 there's a 0 and then a 9 and then a 5, which equals . . . 14.

Here's another:

142857
X 47
___________
= 6714279

So there's a 7142 in there, which means we need an 85, to form 714285, which in turn is a spoonerization of the magic number 142857.

So here's how these harder ones work: the 7142 in the product is followed by a 7 and then a 9 and then a 6. To get the 85, you have to work backwards digit to digit to digit, and when you carry over a double digit sum, you have to carry it in the normal way.

So, starting with the leftovers and working backwards, 6 plus 9 is 15. Keep the 5 and carry the 1 to the other digit, which is 7, and 7 plus 1 is 8.

Voila, there's your 85.

I've looked at a lot of these, and have yet to find one where there's a breakdown. Anyone out there care to carry on the archeology? Anyone out there care to tell me how this stuff works, at a deep level?

Hmmm . . .

* * *

Now the point here is twofold:

1. What we're doing here is the mathematical equivalent of taking a deep breath and doing some mental joy-calisthenics as we go about incorporating bad news from earlier this week, news that feels comparable to a death of a loved one (indeed, it might be . . . ). So marvel at the numbers, please, and know that, so far at least, no one has been able to change the miraculous properties of 72 or 142857, true unto themselves.

and

2. Numbers can serve you very well, both in play -- as is the case with 142857) and in some very important facets of your financial health -- which is the role of the majestic 72 and a few others, all of which are topics for another day.

Bright moments, all, and happy weekend. And deep breaths . . .