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:
“The human mind is so complex and things are so tangled up with each other that, to explain a blade of straw, one would have to take to pieces an entire universe.... A definition is a sack of flour compressed into a thimble.”
—Rémy De Gourmont (1858–1915)
“I had but three chairs in my house; one for solitude, two for friendship; three for society. When visitors came in larger and unexpected numbers there was but the third chair for them all, but they generally economized the room by standing up.”
—Henry David Thoreau (1817–1862)
“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)