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Hello,
If you want a mathematical answer, be clear that it is what you want. There is not even a sentence in your post!!
Given a rational p/q you can define its height to be max(log(|p|), log(|q|)). And given a vector, you can define its height as the maximum of the heights of its entries...
sage: A = AffineSpace(2,QQ)
sage: A((1/3,2/5))
(1/3, 2/5)
sage: A((1/3,2/5)).global_height()
1.60943791243410
sage: log(5.)
1.60943791243410
The definition extends to number fields.
Vincent
