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:
- a ≤ b implies a + c ≤ b + c;
- 0 ≤ a and 0 ≤ b implies 0 ≤ a × b.
Note that both (Q, +, ×, ≤) and (R, +, ×, ≤) are ordered fields, but ≤ cannot be defined in order to make (C, +, ×, ≤) an ordered field, because −1 is the square of i and would therefore be positive.
The non-strict inequalities ≤ and ≥ on real numbers are total orders. The strict inequalities < and > on real numbers are strict total orders.
Read more about this topic: Inequality (mathematics)
Famous quotes containing the words ordered and/or fields:
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—Johan Huizinga (1872–1945)
“The landscape was clothed in a mild and quiet light, in which the woods and fences checkered and partitioned it with new regularity, and rough and uneven fields stretched away with lawn-like smoothness to the horizon, and the clouds, finely distinct and picturesque, seemed a fit drapery to hang over fairyland. The world seemed decked for some holiday or prouder pageantry ... like a green lane into a country maze, at the season when fruit-trees are in blossom.”
—Henry David Thoreau (1817–1862)