# Commutation property in the complex absolute value formula

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?

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

## 2 Answers

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 $z^*$).

• 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? – israel.sincro Nov 30 '16 at 12:35
• 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. – MarianD Nov 30 '16 at 12:41
• 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? – 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.

• 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 – israel.sincro Nov 30 '16 at 12:32
• I'm sorry that I didn't satisfy you but reread your first sentence in your question, please. – MarianD Nov 30 '16 at 12:36
• I do appreciate your help. The thing is i don't even have a proper way to express myself. Any help is again appreciated. – israel.sincro Nov 30 '16 at 12:39
• 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: $Abs(b,a)$? – israel.sincro Nov 30 '16 at 12:40
• 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). – MarianD Nov 30 '16 at 12:46