Complex operator $i$ and Exponents

Question

I am trying to understand the complex numbers and exponents.

I came across this question. I wonder how to explain the difference between $${2\cdot i} \text{ and } 2^i$$ as $i=\sqrt{-1}$

edit:

Rather than explaining the meaning of above two numbers in yet another equally difficult mathematical form.. i am more interested in knowing as to in which situation do we need to use one of the above numbers and in which other situation we would use the other number.

From many of the answers I now understand that bot of the above numbers are complex numbers but they are of different types in a way that one has a single value while the other has more than 1 values. So in which practical situation do we prefer to use one form and in which the other?

Answer 1

After thinking a lot about what was actually being asked (and your comments), I think I might have an answer explaining the differences between $2i$ and $2^i$.

(For the benefit of those unfamiliar with complex numbers, the expression $2i$ represents the complex number $\sqrt {-2}$, but it can also be represented as $$0+2i$$ in $a+bi$ (rectangular) form...as $$2 \angle \frac {\pi}{2}$$ in polar form (using $|z| \angle \arg (z)$) and as $$2e^{\frac {i\pi}{2}}$$ in exponential form (using $|z|e^{i \arg (z)}$) - in polar and exponential form, $|z| = \sqrt {a^2+b^2}$ and $\arg (z) = \arctan \frac {b}{a}$.)

We use the rectangular form of $2i$ when we have roots for a quadratic equation that have a negative discriminant $\Delta = b^2 - 4ac <0)$. (Rather than writing $\sqrt{-1}$ all the time, we just use $i$.)

We use the polar or exponential form to show the ease of multiplying, dividing, raising to powers or taking roots versus doing so in rectangular form. We can also express complex numbers in polar or exponential form for alternating-current circuits to show voltage lead or lag. In this case, $2i$ represented as $2 \angle \pi/2$ means the voltage is leading by $\pi/2$ radians.

The expression $2^i$ is a multi-valued function whose general value is $$2^i = e^{i \ln 2 + 2\pi k} =\cos (\ln 2 +2\pi k) + i \sin (\ln 2 + 2\pi k), k \in \mathbb {Z}.$$ The principal value ($k=0$) is $$2^i = e^{i \ln 2} =\cos (\ln 2) + i \sin (\ln 2)$$

You would probably use $2^i$ to illustrate the multiple values of a complex exponent and their relationship to natural logarithms. There is rotation involved when we have multiple values of $k$; for $k>0$ the rectangular values can vary extensively, but often we consider the principal value at $k=0.$

Answer 2

$$2^{x+iy}=(e^{\ln(2)})^{(x+iy)}=e^{x\ln(2)+i\cdot y\ln(2)}=2^x\cdot e^{i \cdot y\ln(2)}=2^x(\cos {(y\ln2)}+i\cdot \sin {(y\ln2)})$$
$$2^{x+iy}=2^x[\cos(\ln(2y))+i\cdot \sin(\ln(2y))]$$
Here, $2^i=2^{0+1i}$
So, $(x,y)=(0,1)$

It becomes: $$2^i=2^0[\cos(\ln(2)+i\sin(\ln(2)]=\cos(\ln2)+i\cdot \sin(\ln2)$$

Answer 3

$$2^i=\mathrm e^{i\ln 2}=\cos(\ln 2)+i\sin(\ln 2)\ne 2i. $$