Complex Numbers and Inequalities
The set of complex numbers with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that becomes an ordered field. To make an ordered field, it would have to satisfy the following two properties:
- if a ≤ b then a + c ≤ b + c
- if 0 ≤ a and 0 ≤ b then 0 ≤ a b
Because ≤ is a total order, for any number a, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ ). In either case 0 ≤ a2; this means that and ; so and, which means ; contradiction.
However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b then a + c ≤ b + c"). Sometimes the lexicographical order definition is used:
- a ≤ b if < or ( and ≤ )
It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.
Read more about this topic: Inequality (mathematics)
Famous quotes containing the words complex, numbers and/or inequalities:
“It’s a complex fate, being an American, and one of the responsibilities it entails is fighting against a superstitious valuation of Europe.”
—Henry James (1843–1916)
“Old age equalizes—we are aware that what is happening to us has happened to untold numbers from the beginning of time. When we are young we act as if we were the first young people in the world.”
—Eric Hoffer (1902–1983)
“In many places the road was in that condition called repaired, having just been whittled into the required semicylindrical form with the shovel and scraper, with all the softest inequalities in the middle, like a hog’s back with the bristles up.”
—Henry David Thoreau (1817–1862)