1/10 = 7/3 - 7/9 + 7/27 - 7/81 ...

You could express multiplication by 7 as multiplication by 9 and subtracting twice I guess.

But I think it's smarter to express division by 10 as repeated division by 9:

1/10 = 1/9 - 1/9 * 1/10 =>

1/10 = 1/9 - 1/81 + 1/729 - 1/6561 ...

You get much fewer terms, and don't need to worry about multiplication by 7.

If consideration is taken to the carry of the sum of the decimal part, this will produce exact results.

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In it's most general form, division by some number n in terms of division by some other number b can be expressed like this:

The relationship is derived by evaluating 1/n - 1/b and finding that

1/n - 1/b = (b-n)/nb = (b-n)(1/b)(1/n)

Then shifting over 1/b to the right side

1/n = 1/b + (b-n)(1/b)(1/n)

Then you simply replace the (1/n) in the right hand part of the equation with the right hand equation (over and over again)

1/n = 1/b + (b-n)(1/b)(1/b + (b-n)(1/b)(1/b + (b-n)(1/b)(1/b + (b-n)(1/b)([...]))))

If you clean that mess up, you get the pretty relationship above.