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Complex numbers have never been so intuitive to me
Imaginary numbers always confused me. Like understanding e, most explanations fell into one of two categories: Gee, what a great way to encourage math in…
Most cited
Posts that commenters keep linking to, ranked by how many different people have cited them in Hacker News comments. 417 posts.
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An Intuitive Guide to Linear Algebra (2012)
Despite two linear algebra classes, my knowledge consisted of "Matrices, determinants, eigen something something". Why? Well, let's try this course format:
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Show HN: Interactive Essay on Signals, Sampling and the Fourier Transform
This text is designed to accompany your study of introductory digital signal processing. 1 It’s an eccentric piece of not-so-rigorous literature with a preoccupation for explaining things using interactive visualizations, animations and sound. 2 My goal is to explain the Discrete Fourier Transform using a miniature…
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Numbering should start at zero (1982)
To denote the subsequence of natural numbers 2, 3, ..., 12 without the pernicious three dots, four conventions are open to us
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Understanding LSTM Networks (2015)
Humans don’t start their thinking from scratch every second. As you read this essay, you understand each word based on your understanding of previous words. You don’t throw everything away and start thinking from scratch again. Your thoughts have persistence.
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Mining Bitcoin with pencil and paper: 0.67 hashes per day
This article is now available in Japanese: 紙と鉛筆でビットコインをマイニング：1日に0.67ハッシュ and Russian: Майним Bitcoin с помощью бумаги и ручки . I decided...
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Quake's Fast Inverse Square Root
I'm no graphics expert, but appreciate why square roots are useful. The Pythagorean theorem computes distance between points, and dividing by distance helps…
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Yacc is dead: An update
The draft of " Yacc is Dead " that David Darais and I posted on arXiv received some attention, so we'd like to share updates and feedback.
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Maze-generation algorithms, with JS demos
Algorithms, circle mazes, hex grids, masking, weaving, braiding, 3D and 4D grids, spheres, and more!
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Bicycle
Interactive article explaining the physics of a bicycle.
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The Dunning-Kruger effect is autocorrelation
Do unskilled people actually underestimate their incompetence?
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Feynman vs. The Abacus
A Japanese man came into the restaurant. I had seen him before, wandering around; he was trying to sell abacuses. He started to talk to the waiters, and challenged them: He said he could add numbers faster than any of them could do.
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Scott Aaronson explains his $200k bet (8 Signs A Claimed P≠NP Proof Is Wrong)
As of this writing, Vinay Deolalikar still hasn’t retracted his P≠NP claim, but a clear consensus has emerged that the proof, as it stands, is fatally flawed. The first reason is that weR…
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Crit-bit Trees: the best way to store sets
A crit-bit tree can be used as a high-speed associative array. For example, an array mapping 54 to 1, 19 to 2, 99 to 3, 85 to 4, and 88 to 5 can be stored as a crit-bit tree containing 54=1, 19=2, 99=3, 85=4, and 88=5. The smallest string in the crit-bit tree larger than 85= is 85=4.
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52 Factorial
They seem harmless enough, 52 thin slices of laminated cardboard with colorful designs printed on their sides. Yet, as another illustration of the mantra that complexity begins from the most simple systems, the number of variations that these 52 cards can produce is virtually endless. The richness of most playing card…
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Energy Is Not Conserved (2010)
I’ve been meaning to link to this post at the arXiv blog , which is a great source of quirky and interesting new papers. In this case they are pointing to a speculative but interesting paper by Martin Perl and Holger Mueller, which suggests an experimental search for gradients in dark energy by way of atom…
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Math For Programmers
Probability theory might be the most useful area for programmers, but it is also a blurry field, difficult to learn right. You might check this book for details, and there is also an free, online draft version available. The Logic of Science
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The 8000th Busy Beaver number eludes ZF set theory (2016)
I’ve supervised a lot of great student projects in my nine years at MIT, but my inner nerdy teenager has never been as personally delighted by a project as it is right now. To…
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Let’s talk about PAKE
The first rule of PAKE is: nobody ever wants to talk about PAKE. The second rule of PAKE is that this is a shame, because PAKE — which stands for Password Authenticated Key Exchange — is actually o…
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A fast alternative to the modulo reduction (2016)
Suppose you want to pick an integer at random in a set of N elements. Your computer has functions to generate random 32-bit integers, how do you transform such numbers into indexes no larger than N? Suppose you have a hash table with a capacity N. Again, you need to transform your hash values (typically 32-bit ……
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Regular Expression Matching in the Wild
The first two articles in this series, “ Regular Expression Matching Can Be Simple And Fast ” and “ Regular Expression Matching: the Virtual Machine Approach ,” introduced the foundation of DFA-based and NFA-based regular expression matching. Both were based on toy implementations optimized for teaching the basic…
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Viktor T. Toth
This page contains a JavaScript calculator of Hawking radiation and other parameters of a Schwarzschild black hole.
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The Patriot Missile Failure
On February 25, 1991, during the Gulf War, an American Patriot Missile battery in Dharan, Saudi Arabia, failed to track and intercept an incoming Iraqi Scud missile. The Scud struck an American Army barracks, killing 28 soldiers and injuring around 100 other people. A report of the General Accounting office…
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The Three Projections of Doctor Futamura
Introduction The Three Projections of Futamura are a sequence of applications of a programming technique called 'partial evaluation' or 'sp...
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Chicago undergraduate mathematics bibliography
Somehow I became the canonical undergraduate source for bibliographical references, so I thought I would leave a list behind before I graduated. I list the books I have found useful in my wanderings through mathematics (in a few cases, those I found especially un useful), and give short descriptions and comparisons…
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1/0 = 0
Disclaimer 2024-12-02 Since this rant is going viral again (again (again)), some disclaimers: I wrote this six years ago to “”“win”“” an ugly Twitter fight. I’m glad I anonymized the inciting tweet and hope I’m the only person still getting grief over this. The core interesting bit— why it’s mathematically valid to…