How can we have "distance" without "order" in complex numbers? I'm taking an algebra class and we're examining the complex numbers. A lot of properties of $\mathbb{C}$ are new to me. I know that the complex numbers form a field, and I understand that they do not form an ordered field under the normal definition of order using $<$. 
However, we have been given a definition of a norm on $\mathbb{C}$: the distance between a number $z \in \mathbb{C}$ and the origin (or the "length" of the number) is: $\sqrt{x^2+y^2}$, assuming $z=x+iy$. I don't really understand this. How can we have a "length" for a number, if we don't have an order? If there is no order, how can the plane even be constructed? We don't know if $1i<2i<3i$. How can we graph these as points of the "y" axis on the plane, if we don't know which comes after the other? 
Perhaps I am confused about a more general question: how can a field that isn't ordered have a measure of "length", or have a "plane"?
 A: Are the points in a plane naturally ordered?
Can you measure the distance between two points in the Euclidean plane?
Note that the Argand Diagram models complex numbers as points in a plane, where distances are natural, but there is no obvious linear order.

You know, surely, how to represent the points in a plane with co-ordinates. Each of the co-ordinate axes is ordered, but the points in the plane are not. You haven't thought of ordering points in a plane before.
The complex numbers give us a way of multiplying points in the plane which is compatible with the natural (vector-style) addition (distributive law, inverses except for zero). It turns out that this is a useful and natural definition of multiplication, which extends our normal understanding and solves a whole host of problems. We think of complex numbers as abstract entities on the whole, but the plane representation is behind a lot of useful applications.
A: There is certainly a meaningful notion of order for the imaginary axis of the complex numbers and it is given by identifying it with the real line in the obvious manner. The same holds for any line in $\mathbb C$, just as in $\mathbb R ^2$. The problem - just as in the real plane - as user 'dbx' writes in the comments, is that these obvious orders on lines do not coalesce into a sensible order on the entire plane. This does not seem to bother you in the case of the real plane because you feel the axes are naturally ordered, but the picture is exactly the same for the complex numbers. Do not feel alienated by the imaginary unit $i$ - the pictures all stay the same.
The standard norm of a complex number comes from the Pythagorean theorem in the real plane, which lets you measure length by the difference between coordinates along orthogonal axes. It is these axes which have an obvious order underlying your intuition, not the entire plane. The order on the axes is what furnishes the notions of 'left, right, above, below'.
More fundamentally, the abstraction of "length of a vector" by the notion of a norm on a vector space does not depend in any way on an order on the set underlying the vector space or any subset thereof. The axioms for a norm comprise a list of synthetic properties you want "abstract length" to satisfy without concerning yourself with how to actually calculate it: You want


*

*the zero vector and only the zero vector to have zero length;

*the length of any vector is non-negative;

*the triangle inequality;

*nice behavior with "elongation" i.e multiplication by scalar.


So planning ahead, do not expect some order to underlie a general notion of norm.
This isn't so strange if you think about it: merely knowing one thing is larger than another says nothing about how large either thing is, and conversely, knowing a thing is longer than another says nothing about which is 'to the right' of which.
