A geometrically flavored approach — When I studied mathematics I often thought about famous unsolved problems before going to sleep. I clearly remember one evening when I was thinking about how to visualize the Goldbach conjecture using circles. Needless to say, I didn’t get much sleep that night! The Goldbach conjecture is one of mathematics’s…

The genius who revolutionized mathematics with almost alien results never seen before and never seen since! — Can you evaluate the following expression?

The beauty of π, the magic of φ, the mystery of γ, and the powers of e — What is your favorite number? At the end of this article, I am going to reveal my favorite number but first I will give you my take on the most important mathematical constants in the World. We will see what makes them so special and historically important by going through…

A quick rundown of the hardest problems in the world. — In the year 1900, 23 problems in mathematics were published by German mathematician David Hilbert. These were all unsolved at the time. Hilbert hoped that these problems would have a great influence on 20th-century mathematics and indeed they had. A solution to any one of these problems would make you…

Surprising and extraordinary results involving our beloved circle constant — π is the biggest superstar of all numbers. It somehow encapsulates the notion of “roundness” and is, to us mere mortals, unreachable in that there will always be infinitely many digits hidden from us following absolutely no pattern whatsoever. As many of you know, π is defined as the ratio…

The topology of a mathematics book — What does the structure of a math book really look like? What theorems are the most important ones? Reading and trying to understand a mathematics book is a fun and usually exciting experience. Still, sometimes it feels like when you are deeply buried in the Wikipedia jungle, constantly being sent…

A million-dollar sudoku — Back in the 60s and 70s, people began to realize that it wasn’t always enough to make a program that could run error-free on their computers while solving a problem. They quickly realized that the runtime of the computer program was an essential factor; in the extreme case, a program…

A world where you can divide by zero — Something is going on with infinity that we don’t fully understand. Physicists have long been assigning finite values to infinite divergent series (infinite sums if you wish) such as 1 + 4 + 9 + 16 + ⋅⋅⋅ ≃ 0 and 1 + 2 + 3 + ⋅⋅⋅ ≃ -1/12…

Explaining the periodicity of the exponential function by a nice piece of infinite geometry — Why is the natural exponential function periodic with imaginary period 2πi? How come there are infinitely many (complex) logarithms of a number? If I told you that it has to do with points on an infinite circle, you would probably think that I am a nutcase! …