Understanding Math Rather Than Merely Learning It

There’s a line from the original Star Trek where Khan says, “Improve a mechanical device and you may double productivity, but improve man and you gain a thousandfold.” Joan Horvath and Rich Cameron have the same idea about improving education, particularly autodidacticism or self-learning. They share what they’ve learned about acquiring an intuitive understanding of difficult math at the Hackaday Superconference and you can watch the newly published video below.

The start of this was the pair’s collaboration on a book about 3D printing science projects. Joan has a traditional education from MIT and Rich is a self-taught guy. This gave them a unique perspective from both sides of the street. They started looking at calculus — a subject that scares a lot of people but is really integral (no pun intended) to a lot of serious science and engineering.

You probably know that Newton and Leibniz struck on the fundamentals of calculus about the same time. The original papers, however, were decidedly different. Newton’s approach was more physical and less mathematical. Leibniz used formal logic and algebra. Although both share credit, the Leibniz notation won out and is what we use today.

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An Abstract Kind of Clock: The Chinese Remainder Clock

Hackaday likes clocks, a lot. Speaking personally, from my desk I can count at least eight clocks, of which seven are working. There’s normal quartz movement analog clocks, fun automatic wristwatches, run-of-the-mill digital clocks, a calculator watch, and a very special and very broken Darth Vader digital clock/radio combo that will get fixed one day — most likely. Every clock is great, and one of life’s great struggles is to see how many you can amass before you die. The more unique the clock is, the better, and nothing (so far) tops [Antonella Perucca]’s Chinese Remainder Clock.

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FxSolver is a Math Notebook for Engineers

If you like to rely on the web to do your electronics and computer math, you’ll want to bookmark FxSolver. It has a wide collection of formulae from disciplines ranging from electronics, computer science, physics, chemistry, and mechanics. There are also the classic math formulations, too.

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Wolfram Alpha Electronic Tips

Electronics takes a lot of math. Once you’ve mastered all the algebra and calculus, though, it is sometimes a drag to go through the motions. It also can be error-prone. But these days, you have Wolfram Alpha which will do all the work for you and very easily. I use it all the time when I’m too lazy to solve an equation or do an integral by hand. But did you know it actually has some features specifically for electronics?

If you want to do a lot with electronics — or nearly any technical field — you are going to have to learn some math and you shouldn’t just rely on tools like Wolfram to skirt understanding the math. Unfortunately, schools often teach us that the point to math is to get a correct answer. For bookkeepers and at the very final stage of engineering, that may be true. But the real value to math for engineers and scientists is to develop intuition about things. If you increase a capacitor’s value does that make its reactance go up or down? Does a little change in load resistance make a corresponding small change in power consumption or is it a lot more? So you should understand why math works. But once you do, using a tool like Wolfram can free you to focus on the abstract questions instead of the detailed “grunt work.”

 

Tip #1: Split Personality

Wolfram can’t seem to decide if it is a symbolic math program or a search engine. Sometimes just putting a topic name in can lead to some interesting calculations. For example, look what happens when you enter the word opamp: Continue reading “Wolfram Alpha Electronic Tips”

Learn to Count in Seximal, a Position Above the Rest

Believe it or not, counting is not special. Quite a few animals have figured it out over the years. Tiny honeybees compare what is less and what is more, and their brains are smaller than a pinky nail. They even understand the concept of zero, which — as anyone who has had to teach a toddler knows — is rather difficult to grasp. No, counting is not special, but how we count is.

I don’t mean to toot our own horn, but humans are remarkable for having created numerous numeral systems, each specialized in their own ways. Ask almost anyone and they will at least have heard of binary. Hackaday readers are deeper into counting systems and most of us have used binary, octal, and hexadecimal, often in conjunction, but those are just the perfectly standard positional systems.

If you want to start getting weird, there’s balanced ternary and negabinary, and we still haven’t even left the positional systems. There’s a whole host of systems out there, each with their own strengths and weaknesses. I happen to think seximal is the best. To see why, we have to explore the different creations that arose throughout the ages. As long as we’ve had sheep, humans have been trying to count them, and the systems that resulted have been quite creative, if inefficient.

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Understanding Math vs Understanding Math

One of the things hard about engineering — electrical engineering, in particular — is that you can’t really visualize what’s important. Sure, you can see a resistor and an LED in your hands, but the real stuff that we care about — electron flow, space charge, and all that — is totally abstract. If you just tinker, you might avoid a lot of the inherent math (or maths for our UK friends), but if you decide to get serious, you’ll quickly find yourself in a numerical quicksand. The problem is, there’s mechanically understanding math, and intuitively understanding math. We recently came across a simple site that tries to help with the latter that deserves a look.

If you don’t know what we mean by that, consider a simple example. You can teach a kid that 5×3 is 15. But, hopefully, a teacher at some point in your academic career pointed out to you what the meaning of it was. That if you had five packages of three items, you have 15 items total. Or that if you have a room that is five feet on one side and three feet on the other, the square footage is 15 square feet.

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Solve 2D Math Equations Colorfully

Electronics can be seen as really just an application of physics, and you could in turn argue that physics is the application of math to the real world. Unfortunately, the way most of us were taught math was far from intuitive. Luckily, the Internet is full of amazing texts and videos that can help you get a better understanding for the “why” behind complex math topics. Case in point? [3Blue1Brown] has a video showing how to solve 2D equations using colors. If you watch enough, you’ll realize that the colors are just a clever way to represent vectors and, in fact, the method would apply to complex numbers.

Honestly, we don’t think you’d ever solve equations like this by hand — at least not with the colors. But the intuitive feel this video can give you for how things work is very valuable. In addition, if you were trying to implement an algorithm in software this would be tailor-made for it, although you wouldn’t really use colors there either we suppose.

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