Can We Assume Equality of Complex Numbers Based on Their Norm?

[Bachelier]

This question might be elementary:
If the norm of two complex numbers is equal, can we deduce that the two complex numbers are equal.
I know in ℝ we can just look at this as an absolute value, but what about ℂ?

So mainly:

let |z| = |w|*|r| can we say → z = w*r ?

Thanks

 

[mfb]

|1|=|1|*|-1|
It can be violated with real numbers, and every real number is a complex number.

The other direction is true, of course.

 

[JonDrew]

Yes because square roots are multiplicative, but your statement above is false on the following grounds.

assuming w and r are complex you would have to say that:
|z| = |w|*|r| we can say → -z = w*r

 

[HallsofIvy]

That's a strange question. You appear to know that if |a|= |b| for real numbers, then it does NOT follow that a= b but are you thinking that with complex numbers we might not have that ambiguity? In fact, for complex numbers the situation is much worse!

In the real numbers, if |a|= 1 then a can be either 1 or -1. In the complex numbers there are an infinite number of possible values for a. There exist an infinite number of complex numbers, a, such that |a|= 1.

 

[rustynail]

Complex numbers, much like vectors, are quantities defined by both a modulus (norm) AND an argument (direction). An infinity of complex numbers share the same norm, but have different arguments. The other way around is also true.

Therefore, |z| = |w| does not imply z = w.

 

[jbunniii]

This question might be elementary:
If the norm of two complex numbers is equal, can we deduce that the two complex numbers are equal.
I know in ℝ we can just look at this as an absolute value, but what about ℂ?

Draw a circle in the complex plane, centered at 0, with radius R. Every point on that circle has norm equal to R. Thus, except for the R = 0 case, there are infinitely many points with the same norm.

 

[Bachelier]

That's a strange question. You appear to know that if |a|= |b| for real numbers, then it does NOT follow that a= b but are you thinking that with complex numbers we might not have that ambiguity? In fact, for complex numbers the situation is much worse!

In the real numbers, if |a|= 1 then a can be either 1 or -1. In the complex numbers there are an infinite number of possible values for a. There exist an infinite number of complex numbers, a, such that |a|= 1.

lol...indeed it is a strange question. I just needed to brush up on my linear algebra a little and apply the vector normalization to get an equality. I don't know why this idea crossed my mind.

Thank you all for the great clarifications.