The link between complex arithmetic and matrix transformations. Complex multiplication has the effect of rotating the position of a number in the complex plane. In this article, we will see an intuitive explanation for this effect. Multiplying two complex numbers The left hand graph below shows two complex numbers, a = 2 + 2i…

Numerical methods meet complex derivatives meet fractals! The Newton Fractal is a fractal based on the Newton-Raphson method for finding the roots of polynomials and other functions. When this method is applied to functions of complex numbers, interesting fractal patterns are created. This fractal wasn’t discovered by Newton, it is…

You probably know of the three main trigonometry functions — sine, cosine and tangent. You might also know of some extra trig functions called secant, cosecant and cotangent. And you might have seen that the inverse trig functions are sometimes referred to as arcsine, arccosine, etc. But where do these…

When we differentiate a function f(x) we obtain its derivative f’(x). The derivative is a function that tells us the slope of the curve for any value of x. In this article we will see how to differentiate a function from first principles. …

Differentiation is a mathematical method for finding the slope of a curve at any point. In this article, we will look at the slope of a straight line, and see how this concept can be applied to a curve. We will also look at some of the techniques that can…

The exponential function for an irrational value of x How can we find the value of a to the power where x is an irrational number? For example, what is 2 to the power π? The normal definition of a to the power n, when n is a positive integer…

There are many proofs of Pythagoras’ theorem, but perhaps one of the most elegant is Einstein’s proof. Here is a video on the topic: https://youtu.be/FT7lK25L5ZY The proof itself is quite simple, but there are a couple of important elements of the proof that are worth looking at first. We don’t need to draw squares to prove Pythagoras’ theorem For a…

From first principles A Maclaurin series allows us to calculate the approximate value of a function f(x) as a polynomial:

Recursion is a common technique that is often associated with functional programming. The basic idea is this — given a difficult problem, try to find procedure that turns the original problem into a simpler version of the same problem. …