There is a very famous theorem which says that there are infinitely many prime numbers. For people who are new to this, a prime number is a number that doesn’t have any divisors except for 1 and itself. For example, 11 is a prime number because it doesn’t have any divisors apart from 1 and 11. On the other hand, 12 is not a prime number because it is divisible by 1, 2, 3, 4, 6, and 12. Now how do we know that there are infinitely many primes? As numbers get bigger, they tend to have more divisors. So may be at some point, all the numbers can possibly start being composite and they will have a lot of divisors, right? We can delve into a deep mathematical proof to prove this, but let’s take a different route. Let’s see if we can prove this with logic, shall we? Continue reading “How Do We Know That There Are Infinitely Many Prime Numbers?”
What Is Zeta Function Regularization?
There is a popular mathematical result which says that the sum of all natural numbers is -1/12. I have discussed it in detail here. This looks very unintuitive to a first time observer. In fact, most people would say that this is some kind of mathematical trickery. How can a bunch of positive numbers sum up to a negative fraction, right? Actually, there is a very real purpose to this whole thing of adding up all the natural numbers to get a negative fraction as the result. However, our general sense tells us that this shouldn’t be possible. The discussion in one of my previous blog posts was about the mathematics involved in this result. This discussion is more about the underlying fundamentals and where these results come from. So how do we explain this situation? Where is it used in real life? Continue reading “What Is Zeta Function Regularization?”
What Is A Singularity?
You must have heard the term “singularity” being used in various contexts. We see it in mathematics, physics, circuit design, mechanics, fluid dynamics, etc. The word “singular” means something that is extraordinary, unique, and strange. When we talk about singularity in mathematics, we usually refer to the uniqueness of mathematical objects. In particular, singularities refer to the points where the mathematical objects are not well-behaved i.e. we can’t define them for those points. In higher mathematics, we usually define functions to explain the behavior of any system. These functions are the mathematical objects under consideration, and we will see how to understand their singularities. But why do we care about singularities? Is it just a mathematical concept or do we ever see it in real life? Continue reading “What Is A Singularity?”
What Do You Know About Infinity?
What do you think of the term “infinity”? In layman’s terms, infinity is something that has no end, something that is bigger than all the things we know. This is what most people think! Many people who have not seen advanced mathematics do not know about the fact that there are many different types of infinity, some of which are bigger than others. Now how is that possible? Isn’t there just one single infinity which quantifies the essence of uncountability? Continue reading “What Do You Know About Infinity?”