structural:reformatting, information:extraction
Ordered fields

If (F, +, ×) is a field and ≤ is a total order on F, then (F, +, ×, ≤) is called an ordered field if and only if:

Both (Q, +, ×, ≤) {\displaystyle (\mathbb {Q},+,\times,\leq)} and (R, +, ×, ≤) {\displaystyle (\mathbb {R},+,\times,\leq)} are ordered fields, but ≤ cannot be defined in order to make (C, +, ×, ≤) {\displaystyle (\mathbb {C},+,\times,\leq)} an ordered field, because −1 is the square of i and would therefore be positive.

Besides being an ordered field, R also has the Least-upper-bound property. In fact, R can be defined as the only ordered field with that quality.