Is there a known name for the commutation property in the complex absolute value formula? The supposed property can be resumed as follows:

The absolute value of a complex number remain unaltered when real and imaginary parts of complex number commute, $e.i$:

$|a + ib| = |b + ia| = a^2 + b^2$

may this be related to the fact that the absolute value for the imaginary unit $i$ and its mapping $i^*$ need to have the same absolute value?

$|i| = |i^*| = 1$?


The mapping $a+bi \to b+ia$ is actually just a reflection over the line $\{x+xi| x\in\mathbb R\}$, and because this line goes through the origin (through $0$), the distance from any point to zero remains the same after performing the reflection.

For any two points, $z_1$ and $z_2$, no reflection can change the distance between them, meaning that if $z_1$ maps to $z_1^*$ and $z_2$ maps to $z_2^*$, then the distance between $z_1$ and $z_2$ is the same as the distance between $z_1^*$ and $z_2^*$.

This means that in your case, because $0$ maps to $0^*$, you know that for any point $z$, the norm of $z$ (which is the distance between $z$ and $0$) must be the same as the distance between $z^*$ and $0^*$ (which is the norm of $z^*$).

  • $\begingroup$ As reflection is an axis operation, we can put it like this: No axis transformation can modified the distance between two points? How is that? $\endgroup$ – israel.sincro Nov 30 '16 at 12:35
  • $\begingroup$ What do you mean by axis operation? Yes, the reflection keep distances. The main reason here is that the reflection axes goes thru the origin. $\endgroup$ – MarianD Nov 30 '16 at 12:41
  • $\begingroup$ Its correct to relate property $Abs(a,b) = Abs(b,a)$ where $Abs(a,b) = |a+ib|$ to the reflection of a complex number over line $x+ix$, In that case the answer to my original question is the answer to why complex space must be invariant to this operation? $\endgroup$ – israel.sincro Nov 30 '16 at 13:06

No, there is no name for it because it is not a commutation property.
Commutation means changing the order of operations, not exchanging 2 different things.

  • $\begingroup$ well, you can call the operation as you like. Maybe i do mistake the definition for commutation, but formula express all i just asked. thanks $\endgroup$ – israel.sincro Nov 30 '16 at 12:32
  • $\begingroup$ I'm sorry that I didn't satisfy you but reread your first sentence in your question, please. $\endgroup$ – MarianD Nov 30 '16 at 12:36
  • $\begingroup$ I do appreciate your help. The thing is i don't even have a proper way to express myself. Any help is again appreciated. $\endgroup$ – israel.sincro Nov 30 '16 at 12:39
  • $\begingroup$ However if absolute value is expressed as an operation like $Abs(a,b) = |a + ib|$, can the commutation property of the operation be written as: $Abs(b,a)$? $\endgroup$ – israel.sincro Nov 30 '16 at 12:40
  • $\begingroup$ It is an unary operation so there is no way to change order of operands. Exchanging real and imaginary part of complex number may be a function (not changing the absolute value of its argument, as you correctly wrote). $\endgroup$ – MarianD Nov 30 '16 at 12:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.