Intereting Posts

If $d=\gcd\,(f(0),f(1),f(2),\cdots,f(n))$ then $d|f(x)$ for all $x \in \mathbb{Z}$
Dense and locally compact subset of a Hausdorff space is open
If $G$ is a groupe such that $|G|=p^m k$, does $G$ has a subgroup of order $p^n$ with $n<m$.
Computing RSA Algorithm
Proving that $64$ divides $3^{2n+2}+56n+55$ by induction
Characterizing the continuum using only the notion of midpoint
Why the principle of counting does not match with our common sense
$A \subseteq B$ if and only if $B' \subseteq A'$?
Existence of solution of $\frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^2-R(x,t)$
What is the Riemann-Zeta function?
Reading the mind of Prof. John Coates (motive behind his statement)
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
Coker of powers of an endomorphism
Size of a union of two sets
Proof of the Début theorem

I’ve always had this doubt.

It’s perfectly reasonable to say that, for example, 9 is bigger than 2.

But does it ever make sense to compare a real number and a complex/imaginary one?

For example, could one say that $5+2i> 3$ because the real part of $5+2i

$ is bigger than the real part of $3$? Or is it just a senseless statement?

- Factorization of polynomials over $\mathbb{C}$
- question related with cauch'ys inequality
- Proving an inequality: $|1-e^{i\theta}|\le|\theta|$
- Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”
- Equation of line in form of determinant
- Law of exponent for complex numbers

Can it be stated that, say, $20000i$ is bigger than $6$ or does the fact that one is imaginary and the other is natural make it impossible to compare their ‘sizes’?

It would seem that the ‘sizes’ of numbers of any type (real, rational, integer, natural, irrational) can be compared, but once imaginary and complex numbers come into the picture, it becomes a bit counter-intuitive for me.

So, does it ever make sense to talk about a real number being ‘more than’ or ‘less than’ a complex/imaginary one?

- Find all reals $a, b$ for which $a^b$ is also real
- Equation of ellipse, hyperbola, parabola in complex form
- Prove that $\sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2$ for odd $n$
- Simple Proof of the Euler Identity $\exp{i\theta}=\cos{\theta}+i\sin{\theta}$
- Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $\in\mathbb{R}$
- Find all solutions to the following equation: $x^3=-8i$
- How to extend this extension of tetration?
- Non-Euclidean Geometrical Algebra for Real times Real?
- Proving that if $|W(-\ln z)| < 1$ then $z^{z^{z^{z^…}}}$ is convergent
- Sum of real numbers that multiply to 1

You can put (partial) orders on the complex numbers. One choice is to compare the real parts and ignore the complex ones. Another is to use the lexicographic order, comparing the real parts and then comparing the imaginary ones if the real parts are equal. Another is to use the modulus. There are many more. The distinction with the order on the reals (or subsets of the reals) is that the order relation is compatible with addition and multiplication. You can’t do that in the complex numbers. The simple proof is to ask whether $i$ is greater or less than $0$. In either case, $i^2=-1$ should be greater than zero.

To compare two complex numbers, we usually look at their modulus: if $z = x+iy$, then the modulus of $z$ is $|z| := \sqrt{x^2 + y^2}$. Regarding $z$ as a point in the complex plane, the modulus of $z$ is the distance to the origin. We can now compare two complex numbers such as $5+2i$ and $3$: notice that $|5+2i| = \sqrt{29}$ and $|3| = 3$, so in this sense, $5+2i$ is `larger’ (better to think: farther away from the origin) than $3$.

Since $\mathbb{R}\subset\mathbb{C}$, every $x\in\mathbb{R}$ can be written as $x + i\cdot 0$. Now if we prescribe the lexicographical (dictionary) ordering, we can compare them.

Let $z,w\in\mathbb{C}$ and $z = x+iy$ and $w=a+bi$.

Then the lexicographical ordering is $z < w$ if $x<a$ or $x=a$ and $y<b$, $z = w$ if $x=a$ and $y=b$, and $z>w$ otherwise.

Order is easy and non ambiguous in $\mathbb{R}$, because it is unidimensional. $\mathbb{C}$ on the other hand is generally seen as a plane. So you will easily define pre-order on it, that means transitive and reflexive relations, that do have sense such as the examples of Ross Millikan’s answer.

But except for the lexicographic order, they are not true order relation because they are not anti-symetric : you can have $a < b$ and $b < a$ without $a = b$.

And the lexicographic order is not *natural* because it is not compatible with the current topology : $x+iy$ and $x +\epsilon + iy$ are topologically near, but if $\epsilon>0$, we get $x+iy < x+i(y+bigNumber) < x+epsilon+iy$ which is not *natural* because $x+iy$ and $x+i(y+bigNumber)$ are not topologicaly near.

**Observation:** Many of the properties of the real number system $\mathbb{R}$ hold in the complex number system $\mathbb{C}$, but there are some rather interesting differences as well–one of them is the concept of *order*. The concept of order used in $\mathbb{R}$ does not carry over to $\mathbb{C}$. That is, we cannot compare two complex numbers $z_1=a_1+ib_1,b_1\neq0$, $z_2=a_2+ib_2,b_2\neq0$, by means of inequalities. Statements such as $z_1<z_2$ or $z_2\geq z_1$ have no meaning in $\mathbb{C}$ except in the special case when the two numbers $z_1$ and $z_2$ are real. Thus, if you see a statement such as $z_1=\alpha z_2, \alpha>0$, it is implicit from the use of the inequality $\alpha>0$ that the symbol $\alpha$ represents a real number.

A number system is said to be an *ordered system* provided it contains a subset $P$ with the following two properties:

- For any nonzero number $x$ in the system, either $x$ or $-x$ is (but not both) in $P$.
- If $x$ and $y$ are numbers in $P$, then both $xy$ and $x+y$ are in $P$.

**Question:** In the real number system the set $P$ is the set of *positive* numbers. In the real number system we say $x$ is greater than $y$, written $x>y$, if and only if $x-y$ is in $P$. Can you see why the complex number system has no such subset $P$?

**Answer:** By the conditions given for an ordered system, if $i\in P$, then $i\cdot i=-1\in P$. Thus, we have that $(-1)\cdot i=-i\in P$, which is a contradiction ($i$ and $-i$ cannot both be in $P$). If $-i\in P$, then $(-i)(-i)=-1\in P$. Thus, $(-1)(-i)=i\in P$, and this is also a contradiction. Consequently, no such subset $P$ exists.

If $s>t$ then we have $s-t>0$. If $s$ and $t$ are complex, and $s-t=u$ (u<>0), then we need $u>0$. However as u lies on a circle with +ve radius $r, u$ is always greater than $0$. This means that $-u=t-s>0$, implying $t>s$, a contradiction, so there is no order over the complex numbers.

Simple answer. No. The complex numbers can not be an ordered field. [if $a \ge 0$ then $a^2 = a*a \ge 0$. If $a < 0$ then $a^2 = a*a > 0$ so $a^2 \ge 0$ for all $a$ so $1 = 1^2 > 0$ and $-1 < 0$. If $\mathbb C$ were an ordered field, $i^2 > 0$ so $-1 > 0$. Impossible. $\mathbb C$ can not be an ordered field.]

But $\mathbb C$ can be ordered without holding to the field axioms. One simple way is the “dictionary” ordering. $a + bi > c + di$ if $a > c$ or $a = c$ and $b > d$. This is consistent with the order on R. But we can’t do much with it. It doesn’t follow that if $z < w$ and $v > 0$ that $zv < wv$. It doesn’t follow *at all*.

Or we can have partial orders. $z > w$ if $|z| > |w|$. But this isn’t total order. $z < w$, $z > w$, $z = w$ are not exhaustive and mutually exclusive; we can have cases where none of the three apply.

- Confusion in proof of theorem ($2.7$) in Rudin's Real and complex analysis
- Does constant modulus on boundary of annulus imply constant function?
- Generalization of $f(\overline{S}) \subset \overline{f(S)} \iff f$ continuous
- Calculating the length of the semi-major axis from the general equation of an ellipse
- What is the difference between stationary point and critical point in Calculus?
- Limit superior of a sequence is equal to the supremum of limit points of the sequence?
- Why Dirac's Delta is not an ordinary function?
- How many elements does $\mathcal{P}(A)$ have?
- Show $a_{n+1}=\sqrt{2+a_n},a_1=\sqrt2$ is monotone increasing and bounded by $2$; what is the limit?
- Examples of Cohen-Macaulay integral domains
- computing a trignometric limit
- Does $\sum _{k=2} ^\infty \frac{(-1)^k}{\sqrt{k}+(-1)^k}$ converge conditionally?
- Global maxima/minima of $f(x,y,z) = x+y+z$ in $A$
- Generators of the multiplicative group modulo $2^k$
- Topology of convergence in measure