mathematics

34 Articles

Manually Computing Logarithms To Grok Calculators

July 24, 2024 by 20 Comments

Logarithms are everywhere in mathematics and derived fields, but we rarely think about how trigonometric functions, exponentials, square roots and others are calculated after we punch the numbers into a calculator of some description and hit ‘calculate’. How do we even know that the answer which it returns is remotely correct? This was the basic question that [Zachary Chartrand] set out to answer for [3Blue1Brown]’s Summer of Math Exposition 3 (SoME-3). Inspired by learning to script Python, he dug into how such calculations are implemented by the scripting language, which naturally led to the standard C library. Here he found an interesting implementation for the natural algorithm and the way geometric series convergence is sped up.

The short answer is that fundamental properties of these series are used to decrease the number of terms and thus calculations required to get a result. One example provided in the article reduces the naïve approach from 36 terms down to 12 with some optimization, while the versions used in the standard C library are even more optimized. This not only reduces the time needed, but also the memory required, both of which makes many types of calculations more feasible on less powerful systems.

Even if most of us are probably more than happy to just keep mashing that ‘calculate’ button and (rightfully) assume that the answer is correct, such a glimpse at the internals of the calculations involved definitely provides a measure of confidence and understanding, if not the utmost appreciation for those who did the hard work to make all of this possible.

Fourier, The Animated Series

June 5, 2024 by 7 Comments

We’ve seen many graphical and animated explainers for the Fourier series. We suppose it is because it is so much fun to create the little moving pictures, and, as a bonus, it really helps explain this important concept. Even if you already understand it, there’s something beautiful and elegant about watching a mathematical formula tracing out waveforms.

[Andrei Ciobanu] has added his own take to the body of animations out there — or, at least, part one of a series — and we were impressed with the scope of it. The post starts with the basics, but doesn’t shy away from more advanced math where needed. Don’t worry, it’s not all dull. There’s mathematical flowers, and even a brief mention of Pink Floyd.

The Fourier series is the basis for much of digital signal processing, allowing you to build a signal from the sum of many sinusoids. You can also go in reverse and break a signal up into its constituent waves.

We were impressed with [Andrei’s] sinusoid Tetris, and it appears here, too. We’ve seen many visualizers for this before, but each one is a little different.

Random Number Generation By Brain

January 24, 2024 by 29 Comments

If you want to start an argument in certain circles, claim to have a random number generation algorithm. Turns out that producing real random numbers is hard, which is why people often turn to strange methods and still, sometimes, don’t get it right. [Hillel Wayne] wanted to get a “good enough” method that could be done without a computer and found the answer in an old Usenet post from random number guru [George Marsaglia].

The algorithm is simple. Pick a two-digit number — ahem — at random. OK, so you still have to pick a starting number. To get the next number, take the top digit, add six, and then multiply by the bottom digit. So in C:  n1=(n/10+6)*(n%10). Then use the last digit as your random number from 0 to 9. Why does it work? To answer that, the post shows some Raku code to investigate the behavior.

In particular, where does the magic number 6 come into play? The computer program notes that not any number works well there. For example, if you used 4 instead of 6 and then started with 13, all your random digits would be 3. Not really all that random! However, 6 is just a handy number. If you don’t mind a little extra math, there are better choices, like 50.

If you think humans are good at picking random numbers, ask someone to pick a number between 1 and 4 and press them to do it quickly. Nearly always (nearly) they will pick 2. However, don’t be surprised when some people pick 141. Not everyone does well under pressure.

If you want super random numbers, try a lava lamp. Or grab some 555s and a few Nixie tubes.

50-Year-Old Program Gets Speed Boost

November 7, 2023 by 19 Comments

At first glance, getting a computer program to run faster than the first electronic computers might seem trivial. After all, most of us carry enormously powerful processors in our pockets every day as if that’s normal. But [Mark] isn’t trying to beat computers like the ENIAC with a mobile ARM processor or other modern device. He’s now programming with the successor to the original Intel integrated circuit processor, the 4040, but beating the ENIAC is still little more complicated than you might think with a processor from 1974.

For this project, the goal was to best the 70-hour time set by ENIAC for computing the first 2035 digits of pi. There are a number of algorithms for performing this calculation, but using a 4-bit processor and an extremely limited memory of only 1280 bytes makes a number of these methods impossible, especially with the self-imposed time limit. The limited instruction set is a potential bottleneck as well with these early processors. [Mark] decided to use [Fabrice Bellard]’s algorithm given these limitations. He goes into great detail about the mathematics behind this method before coding it in JavaScript. Generating assembly language from a working JavaScript was found to be fairly straightforward.

[Mark] is also doing a lot of work on the 4040 to get this program running as well, including upgrades to the 40xx tool stack, the compiler and linker, and an emulator he’s using to test his program before sending it to physical hardware. The project is remarkably well-documented, including all of the optimizations needed to get these antique processors running fast enough to beat the ENIAC. We won’t spoil the results for you, but as a hint to how it worked out, he started this project using the 4040 since his original attempt using a 4004 wasn’t quite fast enough.

Math Book Gets Real With Complex

October 22, 2023 by 22 Comments

The [Math Sorcerer] loves books. His latest acquisition is the famous Real and Complex Analysis, which is a very stout math book.  How stout? Well, there are several chapters on holomorphic functions, including how to do a Fourier transform on such a function. There’s also an appendix about Hausdorff’s maximality theorem. What are those? Beats us; read the book. You can also watch the short video review of the text below.

The author asserts right up front that the exponential function is “undoubtedly the most important function in mathematics.” Undoubtedly. [The Math Sorcerer’s] videos remind us of browsing a bookstore or a library. You don’t get a book summary as much as a preview of what’s in it, so you can decide if you want to read it.

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Native Alaskan Language Reshapes Mathematics

April 18, 2023 by 83 Comments

The languages we speak influence the way that we see the world, in ways most of us may never recognize. For example, researchers report seeing higher savings rates among people whose native language has limited capacity for a future tense, and one Aboriginal Australian language requires precise knowledge of cardinal directions in order to speak at all. And one Alaskan Inuit language called Iñupiaq is using its inherent visual nature to reshape the way children learn and use mathematics, among other things.

Arabic numerals are widespread and near universal in the modern world, but except perhaps for the number “1”, are simply symbols representing ideas. They require users to understand these quantities before being able to engage with the underlying mathematical structure of this base-10 system. But not only are there other bases, but other ways of writing numbers. In the case of the Iñupiaq language, which is a base-20 system, the characters for the numbers are expressed in a way in which information about the numbers themselves can be extracted from their visual representation.

This leads to some surprising consequences, largely that certain operations like addition and subtraction and even long division can be strikingly easy to do since the visual nature of the characters makes it obvious what each answer should be. Often the operations can be seen as being done to the characters themselves, instead of in the Arabic system where the idea of each number must be known before it can be manipulated in this way.

This project was originally started as a way to make sure that the Iñupiaq language and culture wasn’t completely lost after centuries of efforts to eradicate it and other native North American cultures. But now it may eventually get its own set of Unicode characters, meaning that it could easily be printed in textbooks and used in computer programming, opening up a lot of doors not only for native speakers of the language but for those looking to utilize its unique characteristics to help students understand mathematics rather than just learn it.

Understanding Wavelets

September 10, 2022 by 17 Comments

Mathematical transforms can be a great help in understanding signals. Imaging trying to look at a complex waveform and figuring out the frequency components without the Fourier transform. [Artem Kirsanov] calls the wavelet transform a “mathematical microscope” and his video gives you a great introduction to the topic. You can see the video below.

The video starts with a discussion about how the time domain and frequency domain have a dual relationship — not big news if you’ve dealt with Fourier transforms and — in fact — that’s the next topic in the video. However, there are limitations to the transformation — you lose time domain information in the process.

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