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..

reizarAnything in particular that brought out this brainstorm?

unknownjprobeHave you ever noticed how the fraction 1/7 throws up numbers from the seven times tableErm... no, but I am loving in now though! It's got chatup line potential!

unknownj