Stranger, thank you, thank you, thank you. Having read that Wikipedia article on Euler’s formula, I think I finally GET IT. :eek:
e[sup]pi i[/sup] = -1 + 0i
The 1 and the 0 in Euler’s identity make for a pretty poster, but the real significance of the constant 1 (or -1) is that it’s a coordinate on the complex plane. I swear I’ve tried to work this out in my head from a half dozen textbooks and Mathworld, but it never clicked until I saw that picture.
I doubt I would’ve seen that article had you not linked to it!
I’m totally geeking out over this. I’m calling my Dad, he’s been trying to explain it to me for decades.
Also my favorite, I think. It just pops up everywhere. I suppose I should be touting Manning’s Equation, since it’s so intertwined with how I make my living, but PV=nRT is just so right, so simple and to the point. (Note that this is the third vote for PV=nRT in this thread )
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I ran across that Engineering Toolbox website a few months back while trying to figure out how many watts I needed to maintain the temp of a forced air humidifier. It kicks some serious butt!
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Hey, no problem. It’s all a part of the service. I wish I could claim to have written the article, too, but no such luck.
This first made genuine sense to me when I was taking Machine Dynamics I. We were having to do all of these 2D vector problems and the author of the text recommended using polar vector notation. I’d seen Euler’s identity before and understood the basic relationship to trigonometry, and I’d dealt with natural exponents in Differential Equations and Linear Systems, and polar notation in my EE classes, but it just seemed to be kind of a historical thing rather than a method of practical use in mechanics. Then I realized I could express planar motion in one equation with natural exponents instead of mucking about with separate equations for the x and y components, and bam! I realized how effing incredible this was. It made complex problems almost trivial to solve.
And if you think I’m bad about this, don’t even get me started about quarternions. Unfortunately, outside of computer vector graphics and some esoteric areas of physics, the notation is rarely used or accepted. Still, very useful with time dependent 3D vectors.
Note to Euler’s fans: just started reading “Dr. Euler’s Fabulous Formula,” by Paul J. Nahin. He shares your sentiment. Upside to book: passionate and lively, unlike many books on the subject of math theory (lotsa actual math in this book, not just an historical account by any means). Downside to book: author’s schtick as snobby, elitist, overeducated jerkoff gets kinda old.
Also, a sort of mathematical “eye opener” for me was the day when I saw an algebraic representation of the golden ratio.
As a young 'un, I’d learned about the spirals, and the squares, and all that crap. Then one day, a high school teacher broke it down for me and reduced finding the ratio to solving
x[sup]2[/sup] - x - 1 = 0
It was sort of when I realized the power of mathematics to represent things in the world.
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