Here we learn an ingenious application of a divide and conquer algorithm combined with linear algebra to multiply large numbers, fast. And an introduction to big-O notation! In fact this result is so counter-intuitive that the great mathematician Kolmogorov conjectured an algorithm this fast did not exist. Part 0: Long Multiplication is Slow Multiplication. (And an […]
I love patterns and take great joy in searching for them. During my 11th grade, we came across the question: Our professor suggested that there is no general formula for the summation in the numerator and we ended up solving the problem using integration. However, this left me on a search for a general formula […]
Geometry Visualizing Mathematical Equations and Operations Capture the Essence of an Equation and Solve it Visually by Using Mathematical Graphs Series The Mind-Boggling Properties of the Alternating Harmonic Series Riemann’s simple algorithm to approximate computationally any real number Series The Basel Problem: 1+1/2² + 1/3² +… = π²/6 Solving the Basel problem, using an elementary […]
Previously, we were investigating circles centred at the origin. Our main goal was to understand how many rational points circles with different radiuses could contain. We call the point on the coordinate plane rational if both its coordinates are the rational numbers. In the first article, we introduced the method for getting an infinite amount of rational points […]
Superfluidity is a property of some fluids to flow apparently without any viscosity (with constant kinetic energy). Examples of superfluids include helium-3 (or ³He) and helium-4 (or ⁴He). For temperatures below 2.17 K, helium-4 becomes a superfluid. Helium-3 becomes a superfluid only below 0.0025 K. Also, when superfluids are stirred, they form vortices that “rotate indefinitely” (see Fig. 1). The appearance of the […]
In this article, I want to introduce you to a simple problem with an easy-to-apply, yet awfully unintuitive solution. It is one kind of envelope problem and goes like this There are two envelopes with some different amounts x and y of money in them. The envelopes look exactly the same and are randomly shuffled before they […]
To determine the position of a particle can be a very challenging task. The obvious way is to look at it and see where it is at that time. However, as one can expect, measuring the position of a particle at every instance of time is very difficult (impossible). (A particle could have a deterministic […]
Have you wondered why there are 60 seconds/minutes in a minute/hour? Or why we decided that a circle is exactly 360 degrees? For the curious cats, it’s likely a common musing that’s piqued your interest at one point or another — our numbering system consistently wraps around ten digits(0–9), except for a few, one-off concepts, why is […]
Unless you’ve studied some topology, you probably haven’t heard of a “compact space.” This is a type of space (I’m leaving that term vague for now) that has a lot of really nice properties. In the early 1900’s, people wanted to understand how to generalize many of the most useful theorems from Calculus to more […]
The discovery of the form of the tautochrone, “the curve for which the time taken by an object sliding (without friction) in a uniform gravitational field to its lowest point is independent of its starting point,” was made in 1659 by the Dutch physicist, mathematician, astronomer, and inventor Christiaan Huygens. Huygens is widely regarded as one of […]
Math and religion are the same in that they both aspire to a sort of grace. They are conscious endeavours to turn abstractions to actuality. They work to uncover truths: truths about the universe, and truths about us. And so religion and mathematics are vehicles in which we turn to, in which some of us […]
There are various philosophers, logicians and scientists who’ve argued that Nature (or the world) is fuzzy, inconsistent and/or uncertain. It’s hard to grasp what the words “fuzzy”, “inconsistent” and “uncertain” could even mean when applied to Nature (or to any given x in the world). This is the case even of things “at the tiniest scales”. It’s […]
When I was in grad school, I noticed a funny fact about the zero sets of analytic and harmonic functions. I showed it to several people, and they all shrugged it off as uninteresting. I guess I’m weird like that, so I’m going to share it with you all in hopes someone else finds it […]
Have you ever wondered what restrictions symmetries inside bigger symmetries have to obey? In this article, we will dive into the beginning of subgroup theory by proving a result first discovered in the 18’th century by the brilliant mathematician Joseph Louis Lagrange. In fact, he is so popular that the French call him French and the […]