How can I find the square roots of the following complex number?

Question

How do I find the square roots of a complex number defined as followed: $z=(1,1)$ ?

I am a bit confused the way the number is defined. Shouldn't a complex number be defined like this: $z=a+ib$ ?

Answer 1

My guess is that is asked in the context of identifying $\mathbb C$ with $\mathbb R^2$, in which case $(1,1)=1+i$. So, use the fact that $1+i=\sqrt2\left(\cos\left(\frac\pi4\right)+\sin\left(\frac\pi4\right)i\right)$.

Answer 2

We you define a complex number in the form: $$z:(1,1)$$ this is the same as: $$z=1+i=\sqrt2(\cos(\pi/4)+i\sin(\pi/4))$$ To compute the square root, we define: $$w=r(\cos(\theta)+i\sin(\theta))\leftrightarrow w^2=r^2(\cos(2\theta)+i\sin(2\theta))$$ We impose $w^2=z$ and we have: $$r=\sqrt[4]{2}$$ and: $$\theta=\pi/8+k\pi, k=0,1\leftrightarrow \theta=\pi/8 \vee \theta=9\pi/8$$ Substituing the values of $r$ and $\theta$ we have done.