In this article, I present the simple idea of exponentiation by squaring. This idea saves computation time in determining the value of large integer powers by “splitting” the exponentiation in a clever way into a series of squaring operations. The technique is based on the fact that…
Mathematics is everywhere. In this article, I present some of the mathematics behind “Catching Them All”. Say that a rare Pokémon pops up in your Sightings, and that you walk in a straight line until it disappears again. What have you then learned about the position of that Pokémon? Well, since the…
Recently, there has been much progress towards proving the twin prime conjecture, one of the great open questions in number theory. There are infinitely many primes, a fact that was already known by the ancient Greek. A short proof, although not the…
The term “exponential growth” strongly implies quick growth. However, something that grows exponentially will seem to grow quite slowly at first. In practice, there is a good chance that you won’t notice it at all until it suddenly “pops into view”. Take the adoption of…
Today is 3/14 in month/day notation, and \(\pi\approx 3.14\). My daughter Noor (8) and I felt like releasing yet another image of \(\pi\) into the world, in addition to all the existing images. We made this using Hama beads. Today is a good occasion…
We all have to look things up once in a while, and I can tell you right now that Wikipedia is the best source for practical information on mathematics. But, there are some others out there that are also worth knowing. Most of these sites are encyclopedic, and not meant for browsing top down, so I’ve used the entry for eigenvector as an example for…
The concept of a fractal dimension is based on the idea of measuring things with differently sized rulers, as already mentioned in the article on the Mandelbrot Mandelbrot set. Mathematically (but very loosely), the fractal dimension is defined as: if I cover an object with…
The Mandelbrot set is named after Benoît Mandelbrot, a French American mathematician. The set is a part of the complex plane. It is created by iterating the complex quadratic polynomial \(f_c(z)=z^2+c\). For each point \(c\) of the complex plane…
If you think about a set with an infinite number of elements, you might conclude that there is not much to say about the number of elements anymore. Infinite is infinite, right? Well, it turns out that not every infinity has the same “value”, and that there’s actually quite a lot to say about it. Take the natural numbers…
Euler’s identity, \(e^{i\pi}+1=0\), must be the most beautiful formula in the whole of mathematics. The expression establishes an amazing relation between five important numbers…