Aleph 0

I will start with a somewhat provocative statement: The set of ALL positive integers has the same size as the set of EVEN positive integers. At first, this sentence seems obviously wrong - I mean, one set is a proper subset of the other. How can they have the same size? But it is true. And hiding behind this statement is a lot of very beautiful math. In this post, we’ll explore what that math is.…

When I was in undergrad, I was endlessly enamored with topology. I loved how it took intuitive ideas like “wiggle room” and “nearness”, and turned them into precise math. At the heart of the subject is one notion: the open set. So in this issue, I’ll explain open sets with pictures! Armed with that, we’ll explain the formal definition of a topological space. an open set looks like a potato Here’s…

In math, it’s very common to see two objects that look superficially different on the outside, but are actually the same on the inside. We’ll look at two such examples in this post: - the 2-D plane - the set of real-valued continuous functions on [0,1] These two objects look different but actually have a deep underlying similarity. Rather than studying each object in isolation, we’ll define a gen…

in search of a thesis topic … In the early 1900s, a French graduate student named Henri Lebesgue was looking for a thesis topic. He asked a very simple sounding question: How do you measure the size of a set of points? For an interval, like say [0,1], that’s easy. The “size” should be the length (in this case, 1). But what about stranger sets, like the rational numbers between 0 and 1? his big id…

When I was in high school, my math teacher asked the class a question which I still think about quite often. Can you show that, at any time, there are two different points on the Equator that have the same temperature? I thought: is this even math class? Am I in the right room? Perhaps I’d walked into a science class by accident. But no, this was indeed a math class! In this issue, I want to expl…

When I was a baby undergrad, I was endlessly fascinated by the word group theory. It sounded so innocent, but whenever I opened a book about it, I was confronted with some very scary looking math. But group theory is not all that scary! So in this post, I’d like to explain what a group is (with some pictures). This is the explanation I wish I’d heard when I was first learning it. group theory is …

I was flipping through my old files and I found this flyer from an old conference. It was a 3-day conference in analytic number theory at the Fields Institute. A lot of the talks there were about the Riemann Hypothesis, one of the biggest open problems in number theory. I learned something very surprising: there has actually been quite a lot of progress towards the Riemann Hypothesis! But for som…

The standard undergrad math curriculum has loads of material to cover: analysis, topology, algebra, differential geometry, PDEs, … the list goes on and on. But we never specifically practice problem solving! In this issue, I’ll share my favorite books to practice problem solving specifically. My two favorite problem books - The Art and Craft of Problem Solving by Paul Zeitz. This is honestly one …

Hello friends, Some exciting news! A few days ago, I had the pleasure of being featured as a guest creator on 3Blue1Brown. The video was about a topic that I have been thinking about for a while: AlphaGeometry. This is Google DeepMind's AI model that solved IMO-level geometry problems at a silver-medal level. Understandably, the AlphaGeometry model generated quite a lot of buzz. So it seemed wort…

A few days ago, I came across a cute problem about systems of equations. At first, I thought it would be simple. But when I tried looking into it more closely, I realized that it was actually quite tricky. Today, I want to share that problem with you: Problem of the Week Find all real solutions of the system If you have a solution to this challenge problem, submit it here for a chance to be featu…

We’ve all learned about prime numbers in grade school. A prime number is a number which is divisible by 1 and itself. A basic question you can ask about primes is: how many primes are there? The answer is: There are infinitely many primes. This theorem quite ancient: it was proven in Euclid’s Elements (among other places). Let’s add a twist … You might think that we got off easy. But if you take …

Hey there, it’s Adithya. There are 48 hours left to enroll in the Aleph 0 Math Membership. You can click here to sign-up: What’s in the membership? We’re kicking things off with a 6-month deep dive into Real Analysis (Advanced Calculus), covering limits, continuity, derivatives, and integrals. Here’s what’s included: - Structured video lessons + problem sets, released each week - Weekly live call…

Before we begin, a quick announcement: I’m launching a math membership! We’re kicking things off with a 6-month deep dive into Real Analysis (Advanced Calculus), covering limits, continuity, derivatives, and integrals. Here’s what’s included: - Structured video lessons + problem sets, released each week - Weekly live calls to work through challenging problems and discuss key ideas - An exclusive …

Hey, this is Adithya. I’m launching a new kind of online math space: a membership for people who want to learn math seriously alongside like-minded learners. The membership is now open! Signup here: What’s in the membership? We’re kicking things off with a 6-month deep dive into Real Analysis (Advanced Calculus), covering limits, continuity, derivatives, and integrals. Here’s what’s included: - S…

Hey there, it’s Adithya. I’ve got something exciting to share: I’m launching a new kind of online math space: a math membership for people who want to learn math seriously alongside like-minded learners. What's in the membership? We’ll start with a 3-month dive into Real Analysis (Advanced Calculus): limits, continuity, derivatives, and integrals. Here's what the membership will feature: - Weekly…

A while back, I made a YouTube video explaining how to self-study pure math: To date, it’s the most viewed video on the channel. Since then, I’ve received a lot of emails asking for more detailed learning resources. So in the next few issues, I wanted to collate this information and share it with you. Here are the topics I’ll be covering in this guide: - Linear Algebra - Point Set Topology - Grou…

One of my favorite theorems is the Prime Number Theorem. To motivative it, let’s start with an easier question: How many primes are there? Alright, this is an easy question to answer. It’s been known since ancient times that there are infinitely many primes. But I’m really asking a more quantitative question: how many primes are there, quantitatively speaking? Here’s a good place to start. It's a…

Here is this week’s challenge problem. It looks easy but is actually quite tricky! Problem of the Week Find the number of ordered pairs (x,y) of positive integers that satisfy If you have a solution to the above challenge problem, submit it here for a chance to be featured in the next issue of this newsletter. Solution from last week Last week, we had the following problem: Does the sequence of s…