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Complex Numbers
Complex Numbers
A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.
The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.
Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.
Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.
November 15, 2024 at 10:20am November 15, 2024 at 10:20am
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We all know I'm not religious. I'm also not spiritual, unless by "spirit" one means "distilled beverage." The closest I get is when I listen to music. But, of course, it has to be good music—which has nothing to do with whether it was created with religion in mind or not. So here's a Cracked article about songs you might not have realized were religious (and, somehow, Leonard Cohen isn't on the list).
For example, you might realize the band youâre listening to is named âCreed,â and the song is talking about going âhigher, to a place where blind men see.â Clearly, this about heaven. And yet the band actually insists they are not religious, as theyâre not trying to preach anything to anyone.
Songs (and other works of art) can be religious without being preachy. However: 1) A band naming itself "Creed" and claiming to not be religious sets off my bullshit detectors; 2) regardless of their inspiration, they suck. I like to change the lyrics of their most famous song to "With Legs Wide Open."
Or, you might hear that 1960s classic âSpirit in the Skyâ and note the lyric, âGotta have a friend in Jesus.â You then might conclude itâs a Christian song. Singer-songwriter Norman Greenbaum is actually Jewish, but penning that song was a smart move.
Insert Jewish joke about "business is business" here.
4 Imagine Dragonsâ âRadioactiveâ Is About Leaving Mormonism
If you were told that one of Imagine Dragonsâ songs is religious, maybe youâd go with âBeliever,â because it talks about being a believer. Maybe youâd go with âDemonsâ because it talks about demons. Maybe youâd go with âThunderâ because that oneâs so terrible that only divine intervention can explain its success.
Okay, that's funny. But... who?
The band formed at Brigham Young University, and when you go on to leave the Church of Latter-day Saints, the memory of your time there stays with you. We know for sure that one song of theirs is about that subject: âRadioactive.â
Yeah, so, kind of the opposite of religious, I guess? Hell if I know. I didn't watch the video they embedded in the article, so I still don't think I've ever heard any of their stuff.
3 Simon and Garfunkelâs âCeciliaâ Is St. Cecilia
The idea that âCeciliaâ is a song directed at a saint sounds absurd.
Yeah, it kind of does, especially since Paul Simon and Art Garfunkel are in the same tribe as Norman Greenbaum. But hey, Simon said it, and while he may have been trolling, it kind of works on a metaphorical level.
It doesn't help that the song is probably the duo's least awesome.
2 âDem Bonesâ Is a Spiritual About the Promised Land
Unlike the others, this isn't a relatively recent pop song, but, as the article notes, an older gospel tune. Which means it's not "secretly" religious at all, but I guess there's a secularized version for broader appeal.
Today, the song is most important during Halloween, because it gets children talking about spooky skeletons. Thatâs fine. When you get down to it, Halloween is the holiest night of the year.
And technically, anything involving an afterlife has religious connotations.
1 âThe Star-Spangled Bannerâ Is About the Greek Pantheon
This is the one I saved this article for, even though the header is misleading. "The Star-Spangled Banner" is the name of Key's poem, which, as we all know, was set to an existing tune.
If youâre a fan of trivia, you might already know that this melody came from a British drinking song.
What's weird to me is, I've known this for decades, but I never could be arsed to find out exactly what the drinking song was. Well, now I know...
The song is about a bunch of Greek gods arguing with each other. When the poet Anacreon urges humanity to drink and screw, the god Jupiter gets angry and considers intervening. Apollo disagrees, claiming Jupiter's mighty thunderbolts are nothing against the power of music.
And whaddaya know; Apollo was right.
The United States may not have been founded as a religious country, but if you ever feel the need to pledge allegiance to one nation under God, feel free to specify which god that it is.
Why, Dionysus, of course. |
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