# Study Session 1: Complex Numbers

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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 \(y=r\sin{\theta}\).