The complex numbers cannot be ordered

Builds on Ordered field i squared is minus one

Statement

There is no total order $\le$ on $\mathbb{C}$ that makes $\mathbb{C}$ , with its usual addition and multiplication, an ordered field.

Remarks

Proof: in an ordered field, each square is nonnegative, and $1 = 1^2 > 0$ . Then $-1 < 0$ . But $-1 = i^2$ is a square, by i squared is minus one, a contradiction. There are total orders on the set $\mathbb{C}$ , for example the lexicographic order. They are not compatible with the field operations. Thus inequalities such as $z < w$ make sense only for real numbers $z$ and $w$ .

Reading path

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Complex numbers object
Imaginary unit object
Ordered field definition
i squared is minus one statement
The complex numbers cannot be ordered statement

Where this leads

2 pages build on this one.

Tag	`01CJRJKDAA`
URI	`m2://01CJRJKDAA`
Kind	statement, theorem
Writing	W0 stub
MSC	12J15
Status	This copy of the site has no build data. Run `m2 status 01CJRJKDAA`.
Uses	Ordered field, i squared is minus one

The complex numbers cannot be ordered

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

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