Ordering complex numbers - When is it right and when is it not? I know there are a lot of questions related to this topic. But, I have one specific doubt. If we order complex numbers, does that mean that we are wrong all the time?
If we say $4 + 3i < 5 + 7i$. I am wrong as I will get $-1 > \frac{1}{16}$ eventually which is incorrect.
How about $3i>2i$. Is that correct? I will get $3>2$ which is correct. 
 My main question is when is the ordering correct and when is that incorrect?
 A: It depends on the definition of <, > and others. Consider representation of complex number $a+bi$ as a pair $(a, b)$ in cartesian coordinate system. Then the question would be about ordering points in two-dimensional space.
A: Ordering on Complex Numbers isn't really defined because there is no natural definition.
You can define it any way you want. For example, you could define a+ib>c+id if a+b>c+d.
You can define a+ib>c+id to be true when $a^2+b^2>c^2+d^2$. Notice that under this definition -2=2. This doesn't mean that we did anything wrong. We have only changed the meaning of the > sign. Similarly, $-1>\frac{1}{16}$ is also true under this definition.
A: You can define it as anything you darned well please.
What do you want $z > w$ to mean?  
You can have it mean $|z| > |w|$ but then you will won't be able to compare $2+3i$ to $3 + 2i$.
You can have it mean if $z = a+bi$ and $w =c+di$ that $z > w$ means $a > c$ or if $a=c$ then $b > d$.  Then you can compare any two numbers but you don't have $z > 0; w > 0$ then $z*w > 0$ or that $z^2 \ge 0$.  SO no algebra or arithmetic will work.
The first $z > w$ means $|z| > |w|$ can give us arithmetic and algebra the the order isn't total.  ANd we can't say that that $z < w$ or $z > w$ of $z = w$.  It's very possible byt $z \not < w$ and $z \not > w$ and $z \ne w$.  (Example: $2+3i \ne 3+2i$ and $2 + 3i \not < 3+2i$ and $2+3i \not > 3+2i$.
But NOTHING you do will give you an order where:
i) every $a,b$ exactly one and only one of the following is true:  $a < b$ or $b < a$ or $a =b$.
ii)  $a<b$ and $b < c$ will mean $a < c$.
iii) If $a < b$ then $a + c < b+c$ always
iv) if $a < b$ and $c > 0$ then $ac < bc$.
It is impossible to come up with an order on $\mathbb C$ where all those are true.
(because iii) tells us that $x > 0 \iff -x < 0$ and iii) and iv) tells us that $x^2 \ge 0$ always, but with complex numbers we have $i^2 = -1\ge 0$ and $1^2 = 1 \ge 0$.)
But if those aren't all true you can make up what you want.  There is no "right" or "wrong" way because none of them make a complete ordered field and if you don't have a complete ordered field, no-one cares what you do.
You can say $3i > 2i$ if you want.  What do you do about $-3i$ and $2i$ or about $3 + 2i$ and $2 + 3i$?  You can do what you want but you will never have $x^2 \ge 0$ always.
