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:
“I am aware that I have been on many a man’s premises, and might have been legally ordered off, but I am not aware that I have been in many men’s houses.”
—Henry David Thoreau (1817–1862)
“It matters little comparatively whether the fields fill the farmer’s barn. The true husbandman will cease from anxiety, as the squirrels manifest no concern whether the woods will bear chestnuts this year or not, and finish his labor with every day, relinquishing all claim to the produce of his fields, and sacrificing in his mind not only his first but his last fruits also.”
—Henry David Thoreau (1817–1862)