Question relating to complex numbers

Question

Say we have a non-zero complex number $z = Ae^{ix}$, where $A$ is any nonzero integer, and $x$ is a real number.

Let's say we know that $\Re(z) = 0$ and $\Im(z) = 2$, is it correct to say $A = 2$ or $A = -2$?

I know it's a probably simple question, but just want to make sure. Thanks.

Answer 1

The number you are trying to express is $2i.$ We can write it as $$2i = 2e^{i\pi/2} $$ or $$ 2i = -2e^{3\pi i/2}$$ Since $e^{i\pi/2}=i$ and $e^{3\pi i/2}=-i.$ We can also represent $i$ as $e^{\pi i /2 + 2\pi i n}$ for any integer $n$ and $ -i = e^{3\pi i/2 + 2\pi i n}$ and plug those in in the same way. There are many ways of writing a complex number, although not all can be written as $Ae^{ix}$ for $A$ a nonzero integer and $x\in\mathbb R$ like the one you selected can.

However, there is a convention to write complex numbers in polar form $Ae^{ix}$ where $A$ is a non-negative real and $x$ is real (usually taken to be between zero and $2\pi$). So $2e^{i\pi/2}$ is expressed in standard polar notation whereas $-2e^{3\pi i/2}$ is not. Though they're both well-defined expressions that equal $2i.$

Answer 2

What you have written is actually Euler's form of representation of a complex number $z=re^{i\theta}$ where $\theta$ is the argument of the complex number. It is the same as $z=r(\cos\theta + i\sin\theta)$. Let us write the complex number as $z=a+ib$. According to your question, $\Re(z)=0$ .i.e. $a=o$. $\Im(z)=2$, i.e. $b=2$.

$$ z = (a^2 +b^2)^{1/2} \left( {a \over \sqrt{a^2+b^2}} + {b \over \sqrt{a^2+b^2}} \right) $$

The first term in square brackets forms the 'A or R' as you mentioned. As you can see it's gotta be positive because both $a$ and $b$ are real right? The next bit forms $\cos\theta$ and $\sin\theta$ respectively.

Hope this helps!