NOTES ON STATISTICS, PROBABILITY and MATHEMATICS

Plotting in the Complex Plane:

The code in R can be found here.

The \(x\) coordinate is the real part, and the \(y\) coordinate, the imaginary part.

Translation can be achieved by adding a complex number to every point. Scaling is done by multiplying by a scalar.

Horizontal and vertical flips require separating the real from the imaginary components before multplying either the real part by \(-1\) to achieve a horizontal flip; or the imaginary part by \(-1\) for a vertical flip:

Inverting is accomplished simply multiplying by the scalar \(-1\), and the way to picture it is by realizding that the \(-1\) changes the the sign of the real and imaginary components.

To rotate \(90\) degrees we multiply by \(i\):

\(i \times (x + y*i) = -y +x*i.\)

Rotating by another angle requires some background:

Argument of a complex angle: angle between the vectorial representation of the complex number and \(1 + 0i.\) This correponds to the angle, measured in radians, between the x axis and the arrow representing the complex number.

Euler’s formula codes every point in the unit circle as:

\(\Large e^{\theta\,i}\) with \(\large \theta\) being the argument.

The trick is to divide the circle \(2\,\pi\) into an arbritary number of equally spaced points, for instance, \(200\): \(2\,\pi/200\). In this way, every point can be rotated a given amount of these divisions of the circle (e.g. \(n\)) as follows (Euler’s formula):

\(\Huge e^{\,\huge i\times\,n \,\times \frac{2\pi}{200}}= \Large \cos(n \,\times \frac{2\pi}{200}) + i\,\sin(n \,\times \frac{2\pi}{200}).\)

The code in R is:

exp(#_of_"slices" * 1i * (2*pi / total_number_slices)) * complex_#s

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NOTE: These are tentative notes on different topics for personal use - expect mistakes and misunderstandings.