# Study Session 1: Complex Numbers

Recall: $$i=\sqrt{-1}$$ and $$i^2=-1$$

Use the quadratic formula to find roots for $$z^2-2z+5=0$$

(Recall that the roots for a quadratic equation, $$ax^2+bx+c=0$$, can be found from quadratic formula, $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$)

We formulate complex numbers with a real and imaginary part, $$z=x+iy$$.

The real part is given by $$Re(z)=x$$. the imaginary part is given by $$Im(z)=y$$.

For the quadratic equation, $$z^2-2z+5=0$$, What is $$z$$? What are $$x$$ and $$y$$?

The complex conjugate of a complex number is simply that number with the sign of the imaginary part reversed, that is, if $$z=x+iy$$, then the complex conjugate is $$z^*=x-iy$$.

What is the value of $$zz^*$$

We often conceptualize complex numbers in a graphical manner, on a complex plane. The real part is the $$x$$-axis, and the imaginary part is the $$y$$-axis.

What is the length of the vector, $$r$$, sown in the figure?

We refer to the angle, $$\theta$$, shown in the figure above, as the phase angle. We can find it from the legs of the triangle, that is $$\theta =\tan{\frac{y}{x}}$$.

If $$z=2+2i$$, what is the length of this vector and what is its phase angle?

An incredibly useful formulation of $$z$$ can be found by transforming from $$x$$ and $$y$$, to $$r$$ and $$\theta$$, using Euer's formula, $$e^{i\theta}=\cos{\theta}+i\sin{\theta}$$. Then $$x=r\cos{\theta}$$ and $$y=r\sin{\theta}$$.

Express $$z=x+iy$$ in terms of $$x=r\cos{\theta}$$ and $$y=r\sin{\theta}$$.