Fractions and Addresses

Now suppose the address of x is a1a2 ... an(an+1 ... an+m)infinity.

Then x = fa1(fa2(...fan(y)...)), where y is the fixed point of fan+1fan+2...fan+m.

We have seen that y = k/(2m - 1).

Arguing similarly, we see x = y/2n + a1/2 + a2/4 + ... + an/2n = (y + q)/2n.

While it is not obvious that together with the constant and repeating addresses seen earlier, these values of x exhaust all the rational numbers, nonetheless, this is true.

Trying a few examples should make this plausible.

2/9 = (2*7)/(9*7) = (2*7)/(64 - 1) = (2*7)/(26 - 1)

7/12 = 7/(4*3) = 1/(4*3) + 6/(4*3) = 1/12 + 1/2

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