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:
“In spite of our worries to the contrary, children are still being born with the innate ability to learn spontaneously, and neither they nor their parents need the sixteen-page instructional manual that came with a rattle ordered for our baby boy!”
—Neil Kurshan (20th century)
“Come up from the fields father, here’s a letter from our Pete,
And come to the front door mother, here’s a letter from thy dear
son.”
—Walt Whitman (1819–1892)