# Finding Two Distinct Square Roots of I [closed]

I am working through the book Linear Algebra Done Right. I am stumped on this exercise question.

I used ChatGPT and understand up to here:

I just don't understand how I deduce that the real part=0 and the imaginary=1? Is this a fundamental concept I'm unaware of? Something else?

• Do you understand what "Equate real and imaginary parts" means? Commented Jun 21 at 13:04
• I just figured it out. Since the statement is set to i, a^2-b^2 must be 0, while 2ab must be 1. Commented Jun 21 at 13:08
• This is the first time I have seen ChatGPT successfully complete a proof or derivation with no errors. Well done to it, I guess... Commented Jun 21 at 13:24
• Don't use ChatGPT for anything except personal amusement. It is not a search engine, it is not a valid reference. It is a synthetic text extruder that works by deciding what is the likely next word on the basis of what the previous words and prompt were. This is a classic example. You want two square roots of the identity linear transformation. It is giving you two complex square roots of the imaginary number $i$. Commented Jun 21 at 19:55
• @ChristianE.Ramirez Except that it is extruding a proof to the wrong question. But what can you expect of a stochastic parrot? Commented Jun 21 at 19:57

In the definition $$z = a+bi, \; a, b \in \mathbb R$$.
And since $$\mathbb R+\mathbb R =\mathbb R$$, means that there is no sum of real numbers that is an imaginary number this means that only $$2abi$$ can contribute to the imaginary part of $$z$$ and only $$a^2-b^2$$ to the real one. Notice that $$i = 0+i$$, the real part is zero: so we get a system of equations: $$\cases{a^2-b^2 = 0\\2abi=i}\implies\cases{a^2-b^2=0\\b = \frac1{2a}}$$ Substituting $$c = a^2$$: $$c-0.25c^{-1}=0\implies c^2 -0.25=0\implies c=\pm0.5\implies b=a = \pm\frac{\sqrt{2}}2$$ So $$z = \pm\frac{\sqrt{2}}2(1+i)$$
$$a$$ and $$b$$ are real. So the complex number $$a^2-b^2+2abi$$ can be expressed as a purely real component $$a^2-b^2$$ plus a purely imaginary component $$2abi$$.
Also, the complex number $$i$$ can be expressed as a purely real component $$0$$ plus a purely imaginary component $$i$$.
But these two complex numbers are equal. Therefore $$a^2-b^2=0$$ and $$ab=\frac12$$.