Exponential law and complex numbers

AI Thread Summary
The discussion centers on the failure of the exponential law a^{mn} = (a^m)^n when applied to complex numbers, highlighting that this rule holds true only for real numbers. The user initially misapplied the law, leading to the incorrect conclusion that e^{2\pi i/5} equals 1, while it actually approximates 0.309 + 0.951i. It is noted that the exponential law can work in the complex plane, but requires an understanding of branch cuts and the multivalued nature of complex roots. The importance of considering the magnitude of complex numbers is emphasized, clarifying that the two methods of calculation are indeed equivalent. Understanding these concepts is essential for navigating the complexities of complex exponentials.
Positronized
Messages
16
Reaction score
0
I was playing around with complex exponentials and came to this result:

$\begin{eqnarray*}<br /> e^{\frac{2\pi i}{5}}&amp;=&amp;e^\left(\frac{2}{5}\right)\left(\pi i\right)\\<br /> &amp;=&amp;\left(e^{\pi i}\right)^{\frac{2}{5}}\\<br /> &amp;=&amp;\left(-1\right)^{\frac{2}{5}}\\<br /> &amp;=&amp;1\end{eqnarray*}$

But obviously e^{\frac{2\pi i}{5}}=\mathrm{cos}\frac{2\pi}{5}+i \mathrm{sin}\frac{2\pi}{5}\approx 0.309+0.951i\neq 1

So after some research I found that the exponential law a^{mn}=\left(a^{m}\right)^{n} is only true when a,m,n\in\mathbb{R} and not otherwise.

My question now is WHY does the index law fail for imaginary base/exponents?

Thanks!

*PS how can I get rid of that (0) appearing after the eqnarray in my $\LaTeX$ code above?? :P
 
Last edited:
Mathematics news on Phys.org
Because dealing only in real numbers, questions like "the fifth root of 1" is easy, its just 1. However with complex numbers we know that there are 5 solutions, and 1 is the only real one, the rest are imaginary. I think you will find that exp( 2*pi*i /5) is one of the roots =]
 
Positronized said:
So after some research I found that the exponential law a^{mn}=\left(a^{m}\right)^{n} is only true when a,m,n\in\mathbb{R} and not otherwise.
In fact, the rule does work for all a,m,n\in\mathbb{C} but only by extending the complex plane using an infinite number of branch cuts and planes. The logarithm rule works in this way as well. The rule will fail if you are only using one "copy" of the complex plane.

The rule does work in your example as well. (-1)^2 = 1, and one of the fifth roots of 1 is indeed e^{\frac{2\pi i}{5}}. In fact, in the branced complex plane, this is the only fifth root of the 1 in question, as in the branched plane, roots are no longer multivalued.

It's all very confusing at first, but you'll get used to it.
 
Positronized, the awnser to your question is simple. The first expression you wrote, is equivalent to the second one. The only error you did was not to take the "magnitude" of your complexe number, this is the awnser you are looking for. All you do is:

Magnitude = ((Real)^2 + (Img)^2)^0.5

So, in your case, you find:

Magnitude = (Cos(2*pi/5)^2 + Sin(2*pi/5)^2)^0.5 = 1^0.5 = 1 hence the two methods you used are equivilent.

I suggest you go read wiki for any basic questions on complexe numbers.
 
Last edited:
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Back
Top