# Imaginary isn't real, right?

It might be obvious that $$2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { ... } } } } } }$$ equals 4. So what about $$i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { ... } } } } } }$$? The answer might be $$-1$$, but I'm not sure as $$i$$ is not a real number. Can anyone help?

Note by Steven Jim
10 months, 1 week ago

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I don't know if it's absolutely correct, but I am posting it.

If we write $$i$$ as $$e^{i\pi/2}$$, then the given series becomes:

\begin{align} & e^{i\pi/2} \sqrt{e^{i\pi/2}\sqrt{e^{i\pi/2}\sqrt{e^{i\pi/2}\sqrt{e^{i\pi/2}} \cdots}}} \\ &= e^{i\pi \left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8} \cdots \right)} \\ & =e^{i\pi \left( \frac{1/2}{1-1/2} \right)} \\ & =\boxed{e^{i\pi}=-1} \end{align}

Edit: Sorry for the initial error, I wrote $$i=e^{i\pi}$$, which was absolutely incorrect. It has been corrected now to $$e^{i\pi/2}$$

- 10 months, 1 week ago

Thanks for fixing. I've been thinking hard ;)

- 10 months, 1 week ago

Sorry for the inconvenience caused due to it. It was an absolute brainfade.

- 10 months, 1 week ago

Eh, it's all okay. Don't blame yourself.

- 10 months, 1 week ago

Let $$x=i\sqrt{i\sqrt{i\sqrt{i\sqrt{i\sqrt{i\sqrt{i\sqrt{......}}}}}}}\\\Rightarrow x=i\sqrt{x}\\\Rightarrow x^2=i^2\cdot x\\\Rightarrow x^2=-x\\\therefore x=i\sqrt{i\sqrt{i\sqrt{i\sqrt{i\sqrt{i\sqrt{i\sqrt{......}}}}}}}=\boxed{-1}$$

I am little confused about the 3rd line where I take square in both side. But I think it can be $$-1$$.

- 5 months, 3 weeks ago

Consider that sqrt(-1) = i not sqrt(i) = -1

- 10 months, 1 week ago

I don't really understand what you wanted to mention. Can you please explain clearer?

- 10 months, 1 week ago

Comment deleted 10 months ago

I haven't mentioned that $$\sqrt{i}=-1$$.

- 10 months, 1 week ago

Comment deleted 10 months ago

But Steven never gave any solution, he was asking for one. He never said $$\sqrt{i}= -1$$.

- 10 months, 1 week ago

What solution? I only asked a question!

- 10 months, 1 week ago

I'm terribly sorry, my browser has done something strange and my original reply was meant to be posted elsewhere. Please ignore it :(

- 10 months, 1 week ago

Eh, don't be stressed. We all have bad times XD

- 10 months, 1 week ago

No it can't be

- 10 months, 1 week ago