Can magnitude of complex numbers raised to some power

In summary, the conversation discusses whether the equation |z23| = |z|23 is necessarily possible for a complex number z. It is noted that in polar form, the equation can be proven to be true.
  • #1
StephanJ
4
0
Hey People just a general question

Is the following necessarily possible?

|z23|=|z|23

Where z is a complex number. I can't think of a reason why not but then again complex numbers have some subtle behaviours.

Thanks
 
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  • #2
Welcome to Physics Forums.
StephanJ said:
Hey People just a general question

Is the following necessarily possible?

|z23|=|z|23

Where z is a complex number. I can't think of a reason why not but then again complex numbers have some subtle behaviours.

Thanks
If we take a look in polar form, we have

[tex]|z^{n}| = ||z|^ne^{in\cdot\text{arg}z}| = |z|^n|e^{in\cdot\text{arg}z}| = |z|^n[/tex]
 
  • #3
Haha I can't believe I didn't try polar form. Thanks for that
 

FAQ: Can magnitude of complex numbers raised to some power

What is the magnitude of a complex number raised to a power?

The magnitude of a complex number raised to a power is equal to the absolute value of the complex number raised to the same power. This can be calculated by taking the square root of the sum of the squares of the real and imaginary parts of the complex number raised to the power.

How does the magnitude of a complex number change when raised to a power?

The magnitude of a complex number can either increase or decrease when raised to a power, depending on the value of the power and the original magnitude of the complex number. If the power is a positive integer, the magnitude will increase. If the power is a negative integer, the magnitude will decrease. If the power is a fraction, the magnitude will change in a more complex manner.

Can the magnitude of a complex number be negative?

No, the magnitude of a complex number is always a positive real number. This is because the magnitude is calculated using the absolute value of the complex number, which ignores the sign of the real and imaginary parts.

Is the magnitude of a complex number raised to a power equal to the magnitude of the complex number multiplied by itself that many times?

No, the magnitude of a complex number raised to a power is not equal to the magnitude of the complex number multiplied by itself that many times. This is because the magnitude takes into account the real and imaginary parts of the complex number, while multiplying by itself only affects the real part. However, if the power is an integer, the magnitude of a complex number raised to that power will be equal to the magnitude of the complex number multiplied by itself that many times.

How can the magnitude of a complex number raised to a power be used in applications?

The magnitude of a complex number raised to a power can be used in various applications, such as calculating the amplitude of a signal in electrical engineering or determining the distance between two points in the complex plane. It can also be used in solving equations involving complex numbers, as it helps in simplifying the expression and finding the roots of the equation.

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