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… read more
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… read more
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… read more
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… read more
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… read more
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… read more
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… read more
The distance between two points is a concept that is, in every day use, clear and unambiguous. Mathematically, the normal (every day) distance between two points is called the Euclidean distance, but mathematicians generalize this definition. I show how, and I try to make you think by showing how circles look using some of these alternative definitions… read more
If you would have to guess the value of the infinite sum 1 + 1/2 + 1/3 + 1/4 + 1/5 + …, what would you do? The value of the individual terms clearly goes to zero. But does that mean that the total sum will be finite? Well, it doesn’t. This was already proven by Nicole Oresme in about 1350. The proof is actually… read more
Showing mathematical formulas on the web is not straightforward, as HTML (Hypertext Markup Language) does not contain built-in commands to typeset mathematics. Although MathML (Mathematical Markup Language) was designed to be an extension of the standard markup language for mathematics, it is not yet supported by all browsers, so it’s not quite ready for prime time! So we are stuck with… read more