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(no subject)
2012
unknownj

Have you ever noticed how the fraction 1/7 throws up numbers from the seven times table?

1 / 7 = 0.142857143

Just there, you've got 14, 42, 28, 7, 14.. It got me thinking many many years ago, until I finally decided to work out a proof for why that is.

Upon closer examination, the number is made up thus:

0.142857143

0.14
0.  28
0.    56
0.     112
0.       224
0.         448

If you sum each column, it gives you the full decimal. It goes on like that - effectively, the multiple of seven doubles each time. So to find out why, you have to go back and work out exactly what pattern you might want.

I decided that I wanted to find a way to get to the following decimal:

0. 01 02 04 08 16 32

So, I need a formula to generate that decimal. I also, ideally, want to be able to simplify it to a fraction. So:

(1 / x) = SIGMA: (2 ^ (i-1)) / 100 ^ i

And if I were to multiply both sides by 2/100, I'd get:

(2 / 100x) = SIGMA: (2 ^ i) / 100 ^ (i + 1)

Which just happens to represent the formula from the second term onwards. Which means that

(2 / 100x) + (1 / 100) = (1 / x)

Since the first part represents terms 2 to infinity, the second part represents term 1, and the third part represents terms 1 to infinity. Ergo

2/x + 1 = 100 / x
2 + x = 100
x = 98

And thus, in very roundabout (but wholly accurate) way, we have that the progression:

0. 01 02 04 08 16 32 ...

Can be expressed as 1/98. Which means that:

0. 02 04 08 16 32 ...

Can be expressed as 2/98, which happens to be 1/49. And if we were to multiply the whole thing by sevenm, we'd get that:

1/7 = 0. 14 28 ...

So the reason behind it is that the reciprocal of the square of seven produces a recognisable pattern, so the reciprocal of seven produces that pattern multiplied by seven, which is why it stands out..

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Well I'll be damned. That is rather interesting.
Anything in particular that brought out this brainstorm?

Oh, I wrote that ages ago, never got around to posting it.. it's something I've been kicking around for a while..

Have you ever noticed how the fraction 1/7 throws up numbers from the seven times table

Erm... no, but I am loving in now though! It's got chatup line potential!

When will I ever stop teaching you my genius, eh?

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