The conceptual step that took humans from their pre-conceived “classical” notions of the world to the “quantum” notion was the realization that measurements don’t commute. This means, as an example, that if you measure the position of a particle exactly, you cannot simultaneously ascribe to it an infinitely precise momentum. This is not simply a … Read More “A story of commutators” »

This article concerns a new paper I just submitted. It concerns a peculiar feature of quantum mechanics (and also of classical mechanics). The feature is this. The Laws of Physics appear indifferent to the direction of time. If you play a video of two balls colliding elastically with each other, you could play the video … Read More “Schrodinger’s Cat Lives again!” »

I just finished a bit of summer reading – in particular, three books with very similar scope. The first is by a well-known physical chemist and author – it is called “Four Laws that drive the Universe“. The second is by a well-known quantum information expert, and called “Decoding reality: The universe as quantum information“. … Read More “Three pieces – and some puzzles” »

This article is based on a brilliant essay in Quanta magazine, behind which lies a lovely story of a mathematical discovery. I take the essay a little further and describe an algorithm we posted on arxiv to increase the speed of the calculation even further (though not as fast as the fastest method there is). You … Read More “Tales of Karatsuba” »

This article was inspired by a very nice article in Quanta magazine (by Natalie Wolchover) about the connection between error-correction codes and space-time. I thought the quantum mechanics concepts were glossed over, so decided to expand on it a little. Electrical engineers and digital-signal-processing engineers study error correction for a simple reason – the same … Read More “Error Correcting Codes and the Quantum version” »

Another week, another Manjul Bhargava delight. Arithmetic is usually taught in base 10. We have 10 unique symbols (0,1,2,3,4,5,6,7,8,9) and a place value system with the lowest value being , the next being , the next being and so on. So a number like . So far so good, but you could do this with … Read More “Modulo arithmetic & cards” »

This post, aimed at people with some knowledge of Maxwell’s equations, is aimed at connecting a bunch of concepts that are all central to how we understand the universe today. Nearly every word in the title has the status of being a buzz-word, but for good reason – they help organize the ideas well. Some … Read More “Gauge invariance, Global and Local Symmetry” »

This trick uses the representation of numbers in a different base. Solution in a post in a day!

Continuing our saga, trying to be intellectually honest, while a little prurient (Look It Up!, to adopt a recent political slogan), let’s look at the ridiculous “measured” correlation in point 3 of this public post. Let’s call it the PPP-GDP correlation! The scatter graph with data is displayed below Does it make sense? As in … Read More “p-‘s and q-‘s redux” »

In the practice of statistical inference, the concept of p-value (as well as something that needs to exist, but doesn’t yet, called q-value), is very useful. So is a really important concept you need to understand if you want to fool people (or prevent yourself from being fooled!) – it’s called p-hacking. The first (p-value) … Read More “Minding your p-‘s and q-‘s” »

Another Wednesday, another session of Manjul Bhargava’s entertaining and instructive class at the National Museum of Mathematics, in New York City. This time, the topic was that of rhythmic combinations and their connection to mathematics. As the sentence itself suggests, combinations of rhythms lead to combinatorial arithmetic – the notions of Fibonacci numbers and Pascal’s … Read More “Math, Rhythmic patterns & A Card Trick” »

I spent a pleasant evening at the National Museum of Mathematics this week – the first session of a semester long program of lecture demonstrations about mathematics and magic. The instructor is Manjul Bhargava, the famous Princeton mathematician. I thought the ideas were worth discussing in a more public forum, so I resolved to demonstrate … Read More “Of Baby Hummers and clock arithmetic with Aryabhata and Archimedes” »

Albert Einstein is well known to be one of the most creative scientists of the last couple of centuries. He produced fascinating theories that really burnished this reputation. But he also had several ideas (trying to undermine, for instance, ideas about quantum mechanics) that didn’t work – often the exact way in which they did … Read More “Gedankenexperiments #1” »

There is a well-known paper by the famous scientist and Nobel laureate C. V. Raman about the harmonic drums of India – the mridangam and the tabla. While the paper was written in the 1930s, it is quite detailed and refreshing in its clear description of how these instruments work. This post attempts to popularize … Read More “The Indian musical drums” »

I was listening to an academic talk and someone mentioned the “Rule of 72”. Apparently invented by Einstein, it is a simple numerical approximation that helps you understand the power of compound interest. This, according to legend, became popular when interest rates offered on deposits by the Swiss National Bank dropped to in the 1930s. … Read More “The Rule of 72 – and what does the Swiss National Bank have to do with it” »