- #1

brotherbobby

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- Homework Statement
- ##\textbf{Evaluate : }\,\boldsymbol{\sqrt{i} + \sqrt{-i}=?}##

- Relevant Equations
- 1. ##e^{i\theta} = \cos\theta+i\sin\theta##

2. ##\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}##

**Statement of the problem :**We have to find what is ##\sqrt{i} + \sqrt{-i}##

**First Attempt (Euler's Formula) :**I use the Euler's formula (see Relevant Equations 1) above which yields ##i = e^{i\frac{\pi}{2}}##. Likewise ##-i = e^{i\left(-\frac{\pi}{2}\right)}##.

Now I evaluate

where, in the last step, I used Relevant

**Second Attempt (geometry) :**I use the fact from

**attempt 1**above wherein I showed that ##i = e^{i\frac{\pi}{2}}## and ##-i = e^{i\left(-\frac{\pi}{2}\right)}## to plot the two numbers into the complex plane as shown to the right. From the image it is evident that the imaginary parts of the two numbers vanish when put together. Their real parts however "add up" to yield

So far, I have been able to establish by slightly different methods that the answer to the problem ##\textbf{Evaluate : }\,\boldsymbol{\sqrt{i} + \sqrt{-i}}## is ##\boldsymbol{\sqrt{2}}##. So the answer is a

**real number**.

But when I use the method of surds for the same problem, I run into trouble!

**Third Attempt (Algebra, the method of surds) :**

**Doubt :**Where am I mistaken? I suppose in taking the square room of the expression in the last but one step. But we are aware that in algebra when we have ##\sqrt{f(x)}##, the conditions are that (1) ##f(x)\ge 0## and (2) ##\sqrt{f(x)}\ge 0##. Acccordingly, I took the positive square roots in both cases. Of course for the second term ##(i-1)^2 = (1-i)^2##, and hence the positive square root in those cases are ##i-1## and ##1-i##. Why take one and not the other? Is this something to do with principal values?

A hint or comment would be most welcome.

**My math is not formatted. I am not sure what the problem is.**

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