Another proof:
Let study the sequence sn = 0.9999...9 , with n 9s.
Or s0= 0 and s(n+1) = sn + 9 / 10 ^n
lim sn = 0.99999.......
and lim sn = 1.
so ....
If I remember my meth correctly,
the number 0.9999...... does not exist.
This not a legal decimal sequence.
(Any decimal sequence finishing by 9999...... is forbidden to allow a
one to one mapping between real numbers and
decimal sequence.)
This kind of infinity is one of the reason equality is not devidable
on real numbers.
You can manipulate square root directly. For example by defining
numbers as a map from ratio to ratio.sqr
sqrt(5) is represented by {5 1}
sqrt(5) + 3. sqrt(2) by {5 1 , 2 3}
15 + sqrt(3) ----> {1 15, 3 1}
adding is just a reduce of one map into the other.
neg is a map.
multiplying is more complex. For each two pairs in the map
[a b] [c d], you check wether (ac) can be written as sqr(e).f
(For example 2 * 6 can be written as sqr(2)*3)
If it is the case, you return [f (* e b d)]
else you return [(* a c) (* b d)]
Dividing is more difficult.
I don't know if it is the most efficien way to do that, but it is the
easiest to code.
Best,
Nicolas.
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