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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Through which careered an undulous pageantry
Of fiends and suns, darkness and boiling sea,
All held in ordered sway by beauty’s mood.”
—Allen Tate (1899–1979)
“Nature will not let us fret and fume. She does not like our benevolence or our learning much better than she likes our frauds and wars. When we come out of the caucus, or the bank, or the abolition-convention, or the temperance-meeting, or the transcendental club, into the fields and woods, she says to us, “so hot? my little Sir.””
—Ralph Waldo Emerson (1803–1882)