Can complex numbers be well ordered?

The complex numbers can be ordered. It cannot be made into an ordered field with the usual addition and multiplication.

Is a complex number always a real number?

Complex numbers are numbers that consist of two parts — a real number and an imaginary number. The standard format for complex numbers is a + bi, with the real number first and the imaginary number last. Because either part could be 0, technically any real number or imaginary number can be considered a complex number.

Do complex numbers have ordering?

TL;DR: The complex numbers are not an ordered field ; there is no ordering of the complex numbers that is compatible with addition and multiplication. If a structure is a field and has an ordering , two additional axioms need to hold for it to be an ordered field.

Why are complex numbers not ordered?

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Squares are necessarily non-negative in an ordered field. This implies that the complex numbers cannot be ordered since the square of the imaginary unit i is −1.

What is meant by well-ordered?

Definition of well-ordered 1 : having an orderly procedure or arrangement a well-ordered household. 2 : partially ordered with every subset containing a first element and exactly one of the relationships “greater than,” “less than,” or “equal to” holding for any given pair of elements.

Why is C not an ordered field?

Note that 0<1 and −1=−1+0<−1+1=0. – 1 = – 1 + 0 < – 1 + 1 = 0 . Because of theorem 2, no ring containing Z[i] ⁢ can be an ordered ring. It follows that C is not an ordered field.

Why C is not ordered?

Because of theorem 2, no ring containing Z[i] ⁢ can be an ordered ring. It follows that C is not an ordered field….Proof.