>>6
.3 repeating and .3 repeating 4 are the exact same thing, assuming that the 3s repeat ad infinitum. In layman's terms, you never reach the infinitieth decimal point (the 4), so they are the same. Similarly, .0 repeating 1 = 0.
Because of the definition of real numbers, they are merely the limit of a cauchy sequence of rational numbers. Thus we must wonder if
lim{x[n]|x
[i]=sum{3/10^i}} = lim{x[n]|x
[i]=sum{3/10^i}}+lim{4/10^n} as n->infinity
Turns out, by simple maths, lim{4/10^n}->0 as n->infinity, so we're left with:
lim{x[n]|x
[i]=sum{3/10^i}} = lim{x[n]|x
[i]=sum{3/10^i}} as n->infinity
Replacing lim{x[n]|x
[i]=sum{3/10^i}} with, say, A, we have:
A=A as n->infinity, which is always true. Thus .3 repeating and .3 repeating 4 are the same real number. QED.
Of course this proof was thrown together at 1am when I ought to be studying for my law school finals, so there may be trivial holes. However, the gist of the proof is correct, and the holes are absolutely surmountable.