math

80 Articles

No Tension For Tensors?

July 9, 2025 by 11 Comments

We always enjoy [FloatHeadPhysics] explaining any math or physics topic. We don’t know if he’s acting or not, but he seems genuinely excited about every topic he covers, and it is infectious. He also has entertaining imaginary conversations with people like Feynman and Einstein. His recent video on tensors begins by showing the vector form of Ohm’s law, making it even more interesting. Check out the video below.

If you ever thought you could use fewer numbers for many tensor calculations, [FloatHeadPhysics] had the same idea. Luckily, imaginary Feynman explains why this isn’t right, and the answer shows the basic nature of why people use tensors.

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Kids Vs Computers: Chisanbop Remembered

July 9, 2025 by 14 Comments

If you are a certain age, you probably remember the ads and publicity around Chisanbop — the supposed ancient art of Korean finger math. Was it Korean? Sort of. Was it faster than a calculator? Sort of. [Chris Staecker] offers a great look at Chisanbop, not just how to do it, but also how it became such a significant cultural phenomenon. Take a look at the video below. Long, but worth it.

Technically, the idea is fairly simple. Your right-hand thumb is worth 5, and each finger is worth 1. So to identify 8, you hold down your thumb and the first three digits. The left hand has the same arrangement, but everything is worth ten times the right hand, so the thumb is 50, and each digit is worth 10.

With a little work, it is easy to count and add using this method. Subtraction is just the reverse. As you might expect, multiplication is just repeated addition. But the real story here isn’t how to do Chisanbop. It is more the story of how a Korean immigrant’s system went viral decades before the advent of social media.

You can argue that this is a shortcut that hurts math understanding. Or, you could argue the reverse. However, the truth is that this was around the time the calculator became widely available. Math education would shift from focusing on getting the right answer to understanding the underlying concepts. In a world where adding ten 6-digit numbers is easy with a $5 device, being able to do it with your fingers isn’t necessarily a valuable skill.

If you enjoy unconventional math methods, you may appreciate peasant multiplication.

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Illustrated scheme of Sam Ben Yaakovs concept

Leakage Control For Coupled Coils

June 5, 2025 by 3 Comments

Think of a circuit model that lets you move magnetic leakage around like sliders on a synth, without changing the external behavior of your coupled inductors. [Sam Ben-Yaakov] walks you through just that in his video ‘Versatile Coupled Inductor Circuit Model and Examples of Its Use’.

The core idea is as follows. Coupled inductors can be modeled in dozens of ways, but this one adds a twist: a tunable parameter 𝑥 between k and 1 (where k is the coupling coefficient). This fourth degree of freedom doesn’t change L₁, L₂ or mutual inductance M (they remain invariant) but it lets you shuffle leakage where you want it, giving practical flexibility in designing or simulating transformers, converters, or filters with asymmetric behavior.

If you need leakage on one side only, set 𝑥=k. Prefer symmetrical split? Set 𝑥=1. It’s like parametric EQ, but magnetic. And: the maths holds up. As [Sam Ben-Yaakov] derives and confirms that for any 𝑥 in the range, external characteristics remain identical.

It’s especially useful when testing edge cases, or explaining inductive quirks that don’t behave quite like ideal transformers should. A good model to stash in your toolbox.

As we’ve seen previously, [Sam Ben-Yaakov] is at home when it comes to concepts that need tinkering, trial and error, and a dash of visuals to convey. Continue reading “Leakage Control For Coupled Coils” →

Understanding Linear Regression

May 8, 2025 by 7 Comments

Although [Vitor Fróis] is explaining linear regression because it relates to machine learning, the post and, indeed, the topic have wide applications in many things that we do with electronics and computers. It is one way to use independent variables to predict dependent variables, and, in its simplest form, it is based on nothing more than a straight line.

You might remember from school that a straight line can be described by: y=mx+b. Here, m is the slope of the line and b is the y-intercept. Another way to think about it is that m is how fast the line goes up (or down, if m is negative), and b is where the line “starts” at x=0.

[Vitor] starts out with a great example: home prices (the dependent variable) and area (the independent variable). As you would guess, bigger houses tend to sell for more than smaller houses. But it isn’t an exact formula, because there are a lot of reasons a house might sell for more or less. If you plot it, you don’t get a nice line; you get a cloud of points that sort of group around some imaginary line.

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Deriving The Reactance Formulas

April 28, 2025 by 9 Comments

If you’ve dealt with reactance, you surely know the two equations for computing inductive and capacitive reactance. But unless you’ve really dug into it, you may only know the formula the way a school kid knows how to find the area of a circle. You have to have a bit of higher math to figure out why the equation is what it is. [Old Hack EE] wanted to figure out why the formulas are what they are, so he dug in and shared what he learned in a video you can see below.

The key to understanding this is simple. The reactance describes the voltage over the current through the element, just like resistance. The difference is that a resistance is just a single number. A reactance is a curve that gives you a different value at different frequencies. That’s because current and voltage are out of phase through a reactance, so it isn’t as easy as just dividing.

If you know calculus, the video will make a lot of sense. If you don’t know calculus, you might have a few moments of panic, but you can make it. If you think of frequency in Hertz as cycles per second, all the 2π you find in these equations convert Hz to “radian frequency” since one cycle per second is really 360 degrees of the sine wave in one second. There are 2π radians in a circle, so it makes sense.

We love developing intuition about things that seem fundamental but have a lot of depth to them that we usually ignore. If you need a refresher or a jump start on calculus, it isn’t as hard as you probably think. Engineers usually use vectors or imaginary numbers to deal with reactance, and we’ve talked about that too, if you want to learn more.

Math On A Checkerboard

January 3, 2025 by 6 Comments

The word “algorithm” can sometimes seem like a word designed to scare people away from math classes, much like the words “calculus”, “Fourier transform”, or “engineering exam”. But in reality it’s just a method for solving a specific problem, and we use them all the time whether or not we realize it. Taking a deep dive into some of the ways we solve problems, especially math problems, often leads to some surprising consequences as well like this set of algorithms for performing various calculations using nothing but a checkerboard.

This is actually a demonstration of a method called location arithmetic first described by [John Napier] in 1617. It breaks numbers into their binary equivalent and then uses those representations to perform multiplication, division, or to take the square root. Each operation is performed by sliding markers around the board to form certain shapes as required by the algorithms; with the shapes created the result can be viewed directly. This method solves a number of problems with other methods of performing math by hand, eliminating other methods like trial-and-error. The video’s creator [Wrath of Math] demonstrates all of these capabilities and the proper method of performing the algorithms in the video linked below as well.

While not a “hack” in the traditional sense, it’s important to be aware of algorithms like this as they can inform a lot of the way the world works on a fundamental level. Taking that knowledge into another arena like computer programming can often yield some interesting results. One famous example is the magic number found in the code for the video game Quake, but we’ve also seen algorithms like this used to create art as well.

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Pi’s Evil Twin Goes For Infinity

December 24, 2024 by 37 Comments

Most people know about the numerical constant pi (or π, if you prefer). But did you know that pi has an evil twin represented by the symbol ϖ? As [John Carlos Baez] explains, it and its related functions are related to the lemniscate as pi relates to circles. What’s a lemniscate? That’s the proper name for the infinity sign (∞).

[John] shows how many of the same formulas for pi also work for the lemniscate constant (the name for ϖ). Some  (as John calls them) “mutant” trig functions use the pi-like constant.

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