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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 3 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

about 5 hours 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.

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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').

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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.

April 25, 2021 at 12:04am

#1009082

Secret

I mentioned before that I had at least one more thing about Leonard Cohen's song Hallelujah and that I'd get to it one day.

Today is that day. And oddly enough, the source is Cracked.

Music Mystery: What Was The 'Secret Chord' In Leonard Cohen's 'Hallelujah?'

Incidentally, free idea for Cracked (hey, one of them might read this, just like Halle Berry might come by my house in the catsuit carrying two growlers of craft beer): Start a dating app for comedians. Call it Crackd.

You're welcome. And while I did say "free" if you wanted to funnel me, say, 5% of the profits, I wouldn't complain. Much.

Okay, anyway, after that massive side of aside, back to the actual article.

According to Leonard Cohen and also, like, the Bible or whatever, there's a certain chord that God particularly likes, but everyone rudely refuses to tell us what it is.

Add that to the list. Hell, no one even knows God's actual name. Some actually refer to God as HaShem, which translates to "The Name."

In the unlikely event that you're still here reading this and you're not familiar with the lyrics from which this springs, here's the first verse of the song:

Now I've heard there was a secret chord
That David played, and it pleased the Lord
But you don't really care for music, do you?
It goes like this, the fourth, the fifth
The minor falls, the major lifts
The baffled king composing Hallelujah

It's entirely likely that the chord was meant to remain, you know, a secret, but plenty of people think Cohen left us clues as to what at least he thought the secret chord was.

I'd lay all kinds of money that it's not the mysterious chord that opens The Beatles' A Hard Day's Night. Maybe the one at the end of A Day in the Life? ...Nah.

Or maybe it's a B chord. Ever try to play a B chord on guitar without retuning or faking it? ...Okay, so I never got too far with guitar lessons.

In fact, Cohen's secret chord may not be a chord at all. It might be a metaphor for divine inspiration, and the subsequently described chord progression could be a code leading the listener to certain Bible verses, which are indeed eerily appropriate.

A metaphor? In a Leonard Cohen song?! Impossible!!!

That's sarcasm, by the way. You can tell by the plurality of punctuation marks.

But the biblical David was, in fact, a musician, and Cohen's song is more about bangin' than religion, so let's assume he's speaking literally because that's also a lot more fun and involves something called the Devil's Interval.

Unless he was a liar. Liar... lyre? Get it? Crap. Back to the professionals.

We may not be totally clear on how God jams, but it turns out "the devil's music" isn't just a term applied by grandmas to bands with haircuts they don't like. "The devil's interval" is a chord a half-step below a perfect fifth that sounds so dissonant to our ears that it was banned in churches in the Middle Ages because it was thought to induce sinful thoughts.

Bet the organists back then (or whatever, if this was before organs) snuck into the church at night just to play that banned chord.

According to one theory, if the devil loves this chord so much, God's fave must be the opposite: a perfectly harmonious major chord. Probably C major, which also happens to be the key of "Hallelujah."

C major is, coincidentally or not, also the easiest key on piano, because you don't have to mess with those annoying black keys.

But maybe we're thinking too hard: When Cohen recites the chord progression, he also plays it, so maybe the secret chord is just the chord he plays when he says "secret chord," which would be A minor.

And A minor uses the same keys as C major. Look, don't ask me; I've forgotten more music theory than I ever learned, but that little tidbit stuck with me. What's the difference, then? Go ask an actual musician.

Or maybe it's something totally different. Go onto any music theory subreddit and ask people what they think the secret chord is and you'll get 100 different, equally compelling and confusing answers.

Also coincidentally, that's what happens when you ask anything on reddit.

In any case, mostly I wanted to post about this because I don't have many opportunities to make a lyre pun, and I'll take one whenever I can get one. And also because, like I said, I'll never get tired of that song.

Mini-Contest Results!

Thanks for all the opening sentences / paragraphs from yesterday. Really, I liked all of them, from the flowery to the laconic. It was hard to choose just one favorite, but I suppose the poetic imagery in the one Kåre เลียม Enga

posted just barely edged out the others. So a MB to Kåre เลียม Enga will be on the way soon, and we'll do this again sometime.

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