We will now discuss the most important aspect of Real Numbers. One that allows a very powerful study of objects like functions, more importantly continous functions between such a number system. And even before that objects like sequences and series!
Analysis
Sep 19, 2025
In the last article, we finished constructing rationals as a quotient between integers. We made sure to capture the equivalence between different quotients seen as pair of integers. We parted from mere counting and deficit to something more intricate. In particular, compared to integers, this allowed an extra operation on these numbers - namely reciprocals (and thus quotients). And there began a proliferation of numbers from Integers to Rationals. We saw there is a rational between every two rationals! This also made writing rationals into a *linear* order not so straightforward. We ended the article mentioning -- although rationals are pretty wild and dense, numbers like $x$ such that $x^2 = 2$, are absent! But these numbers are very crucial starting from the very foundations of elementary geometry.
Analysis
Sep 13, 2025
Last time we constructed integers basically as an extension to Natural numbers although we proceeded abstractly using an auxillary operation between a pair of naturals. We realised it as the familiar subtraction towards the end. The same theme will be continued further.
Analysis
Sep 7, 2025
Last time, we saw how to construct natural numbers from the ground up. Starting from $0$ and the increment operation and then establishing some intuitive axioms. In particular the principle of induction is powerful and goes a long way, other axioms lead to many symmetries/properties between the new operations we defined -- addition and multiplication which occur as fundamental examples of recursive definitions. We also saw that the natural numbers are completely
ordered.
Analysis
Aug 30, 2025
Math is not about numbers, or heavy calculations. It's essentially a way to think.
Analysis
Aug 24, 2025
Terence Tao defines limit points the following way –
Analysis
Jul 20, 2025
This semester, I took a course on the geometric theory of curves and sufaces, which are basically
now refered to as the classical aspects of (differential) geometry. This is a post to put things in
perspective. I want to pick out the essentials of the course, and form a narrative around this subject.
Mathematics
Apr 27, 2024
Back when I was in second and third years here at NISER, I found the idea of Quantum Field Theory quite
exciting
and was looking forward to get to know how all the fundamental stuff is calculated. I think much of the
longing
came from the book Surely you are joking Mr. Feynman. I wished for someone to just talk about it
and pass the feels. Of course David Tong did, and I was waiting to see the calculations myself. Of course, I
was
no way ready until this semester to understand such calculations, but
you get it. I was excited, and also patient.
And today we finished calculating the anomalous magnetic moment!
Yes, for which Schwinger was awarded the 1965 Nobel Prize, and is one of the most accurate theoretical and
experimental agreement in the history of phyiscs. I am elated and I want to celebrate this with a series of articles
on some fundamental QFT results, that could probably keep the excitement high for
the studfents in their junior years. I want to do this atleast for my third year self.
Physics
Apr 10, 2024
The study of atoms is a mysterious one, especially when one begins to study this in high school, hearing the
abstract concepts of wavefunctions and probability. In a first course on Quantum Mechanics one finally tackles this
atom rigorously - as the most simple spherically symmetric system and also as a rare example to solve exactly. In this article we uncover some more secrets about this atom.
High resolution spectrometers observed a fine splitting over the degeneracy one is familiar with from high school. What follows is a story about this more richer and broken energy structure.
Physics
Sep 5, 2023