I'm writing this on Pi Day, but I think pi makes no sense as a fundamental mathematical constant, and it's used only because of force of habit.
The circle is defined as the set of points on the plane with a given distance to the center. That distance is the radius. So the "circle constant" should be defined using the radius, not diameter. It has been proposed that this constant would be denoted by the Greek letter tau. Tau is equal to 2*pi.
The full angle (360 degrees) in radians is on
Some time ago there was some controversy across the Internet whether 1+2+3+4+... = -1/12 or not. Here's what I think about this:
Clearly, this sum diverges to infinity, so you can't assign any finite value to it, using the standard math rules. But there is a way to associate this value with this sum, which uses some advanced concepts that I don't fully understand, like analytic continuation, Riemann zeta function, and Ramanujan summation.
So I did something different. First, I took the
It all started with a seemingly simple question I saw somewhere: Is it possible to divide a square into two triangles? You might think you just need to cut it through the diagonal and you're done. But, if you divide something, its every point must belong to one part or the other. So what about the points on the diagonal? To which triangle do they belong? This problem seems to me like something artificial and counterintuitive, and makes me think the rules of geometry should be modified. Normally
I'm going back to math as promised. Let's think about how and why the different kinds of numbers were created.
First there are "natural numbers". It's a concept that is easy to understand, because it can be used to count physical objects we see around us. We can also make some simple operations on them. We can add them - no problems here, because if we add two natural numbers, the result is also a natural number. But we can also subtract them, and that's where a problem arises: if we subtra
The title of this entry means we're still talking about the real numbers, but it also means we're talking about something called "continuum". But what is it? The answer is it's a kind of infinity, and it's different from the "countable infinity". (There are more kinds of infinity, but let's focus on these two.)
"Countable infinity" is, for example, the number of elements in the set of all integers. "Countable" means we can arrange the elements in a sequence, so for any integer there is a we
Let's talk about the different sets of numbers. First, we have integers, which are a simple and intuitive concept. It's a series of numbers going in both directions (positive and negative) to infinity. Rational numbers are not hard to understand either, as the name suggests they are ratios of integers. But what about the real numbers? Are they a valid mathematical concept? There is one fact that made me question it: most of them are "unreachable", which means we cannot create any formulas that "
