Complex Numbers (Book)

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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.

http://www.ddewey.net/mandelbrot/

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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 22 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

November 15, 2024 at 10:20am

#1079986

That's the Spirit

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).

4 Famous Songs That Are Secretly Religious
This was supposed to be a pop song. We didn’t know it was about Mormons

For example, you might realize the band you’re listening to is named “Creed,” and the song is talking about going “higher, to a place where blind men see.” Clearly, this about heaven. And yet the band actually insists they are not religious, as they’re 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 “Spirit in the Sky” and note the lyric, “Gotta have a friend in Jesus.” You then might conclude it’s 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 Dragons’ ‘Radioactive’ Is About Leaving Mormonism

If you were told that one of Imagine Dragons’ songs is religious, maybe you’d go with “Believer,” because it talks about being a believer. Maybe you’d go with “Demons” because it talks about demons. Maybe you’d go with “Thunder” because that one’s 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: “Radioactive.”

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 Garfunkel’s ‘Cecilia’ Is St. Cecilia

The idea that “Cecilia” 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 ‘Dem Bones’ 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. That’s 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 ‘The Star-Spangled Banner’ 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 you’re 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.

1 Like • Comment

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