# Vauled-ness of a complex number to an irrational power

• willybirkin
In summary, the conversation discusses the multi-valued nature of the complex number z\sqrt{2} and its corresponding equation z\sqrt{2}=ze\sqrt{2}. It is concluded that z\sqrt{2} is infinitely-valued and the equation is incorrect. Additional discussion is had about the conventions for representing multi-valued numbers.
willybirkin

## Homework Statement

For z complex:

a.) is z$\sqrt{2}$ a multi-valued function, if so how many values does it have?

b.) Claim: z$\sqrt{2}$=e$\sqrt{2}$ln(z)=e$\sqrt{2}$eln(z)=ze$\sqrt{2}$
Since $\sqrt{2}$ has 2 values, z$\sqrt{2}$ is 2 valued.

Is this correct? If not, correct it.

## The Attempt at a Solution

For part a I would intuitively assume that it is infinitely-valued since z1/n has n values and $\sqrt{2}$=1+4/10+1/100+4/1000... so its denominator is approaching infinity. But this isn't exactly mathematically sound reasoning since referring to the "denominator" of an irrational number doesn't make any sense.

For part b nothing really jumps out at me as being incorrect, except that it's conclusion disagrees with my belief for part a. It seems like all the steps are mathematically sound, unless there is some reason that you can't simplify eln(z) to z, in which case ln(z) being infinitely-valued would make the whole thing infinitely-valued

For part b:
What two values does $\sqrt{2}$ have ?

The positive and negative square roots of 2. Or, if you consider 2 to be complex with imaginary part 0 and real part 2, root(2)eik$\pi$ with k=0,1.

Last edited:
For the most part,
$$e^{ab} \neq e^a e^b$$

Also, whether or not $\sqrt{2}$ is a multi-valued number depends on what precisely you mean by the notation -- two different conventions both make sense. Does your textbook state its convention anywhere?

Wow, I can't believe I missed that. Well then, after some manipulation that equation ought to become r$\sqrt{2}$ei$\sqrt{2}$($\theta$+2k$\pi$) which unless I'm mistaken shouldn't ever be able to return to the original $\theta$, making it infinitely valued.

## What is the meaning of "valued-ness" in the context of a complex number to an irrational power?

"Valued-ness" refers to the numerical result obtained when a complex number is raised to an irrational power. This result is typically a complex number itself, and it represents the "value" of the original complex number when raised to the given power.

## How is a complex number to an irrational power calculated?

To calculate a complex number raised to an irrational power, you can use the formula:
(a+bi)^r = (a^2 + b^2)^(r/2) * (cos(r*theta) + i*sin(r*theta))
where a and b are the real and imaginary parts of the original complex number, r is the irrational power, and theta is the angle of the original complex number in the complex plane.

## What is the significance of raising a complex number to an irrational power?

Raising a complex number to an irrational power can result in a unique and interesting value, as it combines both the real and imaginary components of the complex number in a non-traditional way. This can be useful in solving certain mathematical problems or in representing complex phenomena in the real world.

## Are there any limitations to raising a complex number to an irrational power?

One limitation is that the result of raising a complex number to an irrational power may not always be a real number. Also, the calculation of a complex number to an irrational power may become increasingly complex as the power becomes more irrational, making it difficult to obtain an exact result.

## Can a complex number be raised to any irrational power?

Yes, a complex number can be raised to any irrational power, as long as the calculation is done using the appropriate formula. However, the result may not always be a real number and may be difficult to calculate exactly.

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