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Old 2020-04-02, 14:21   #19
Dr Sardonicus
 
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Feb 2017
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In addition to ordered fields which are not complete, there are complete fields which are not ordered. The complex numbers are perhaps the best known example.

But there are also complete "non-Archimedian" fields. Their topology is defined by a "valuation" which takes non-negative real values, and satisfies

|x*y| = |x|*|y|, and |x + y| <= Max(|x|, |y|).

The basic examples are the p-adic completions of the rational numbers, where p is a prime number.

The p-adic valuation of a nonzero integer n is p-e, where pe is the exact power of p dividing n. The p-adic valuation of 0 is 0. The p-adic valuation of any non-multiple of p is 1, and integers divisible by high powers of p are "small."

Thus, in the 2-adic rationals (and its completion), it is perfectly correct to say that the "infinite sum" of powers of 2

1 + 2 + 4 + 8 + ... = -1
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