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:
“Then he rang the bell and ordered a ham sandwich. When the maid placed the plate on the table, he deliberately looked away but as soon as the door had shut, he grabbed the sandwich with both hands, immediately soiled his fingers and chin with the hanging margin of fat and, grunting greedily, began to much.”
—Vladimir Nabokov (1899–1977)
“I will never accept that I got a free ride. It wasn’t free at all. My ancestors were brought here against their will. They were made to work and help build the country. I worked in the cotton fields from the age of seven. I worked in the laundry for twenty- three years. I worked for the national organization for nine years. I just retired from city government after twelve-and-a- half years.”
—Johnnie Tillmon (b. 1926)