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/

Signature for Honorable Mentions in 2018 Quill Awards

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Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

October 4, 2020 at 12:04am

#994992

My Land

First off, I'd like to thank the judges and other people involved in running "30-Day Blogging Challenge ON HIATUS"

[13+]. Last month was a lot of fun to blog and read others' entries and responses, and I'm (almost) humbled to have been selected as September's winner.

And also congratulations to the other winners: Eric Wharton

and WakeUpAndLive~doingNaNo'24

, who, among other participants, had excellent entries last month.

Anyway, enough with the bragging. The problem with winning awards, like that one or last year's Quill, is now I'm under even more pressure to keep up whatever it is that I've been doing right. If you know what that is, tell me. Maybe the Merit Badges I've been giving out help. Or at least don't hurt.

Speaking of which, I'll announce yesterday's Mini-Contest results below. But today's blog is about music. Specifically, a song I'm sure everyone, at least other Americans, will hopefully know.

https://www.mentalfloss.com/article/585577/this-land-your-land-americas-best-kno...

This Land Is Your Land: The Story Behind America's Best-Known Protest Song

Few songs are more ingrained in the American psyche than "This Land Is Your Land," the greatest and best-known work by folk icon Woody Guthrie. For decades, it's been a staple of kindergarten classrooms "from California to the New York island," as the lyrics go. It's the musical equivalent of apple pie, though the flavor varies wildly depending on who's doing the singing.

From Wikipedia: An apple pie is a pie in which the principal filling ingredient is apple, originated in England.

I guess that doesn't completely negate the cliché "as American as apple pie," as a lot of our heritage (not to mention our most popular language) also originated in England. Still, let's not be under any illusion that its origins are on this side of the pond.

That said, I have no idea how familiar the people in other countries might be with this particular song, so if you don't know it, take a look at the article. It's fairly short and has pictures and videos, and I'm not going to reproduce it in full because the only thing more American than apple pie, bald eagles, personal assault weapons, and meddling in other countries' politics is copyright law.

On its most basic level, "This Land Is Your Land" is a song about inclusion and equality...

And already we see in this song, now about 80 years old, the seeds of the Great Divide that still pervades our country. Or, perhaps, the Divide has always been there and it's only fairly recently that its chasm has been fully realized.

But there's more to "This Land Is Your Land" than many people realize—two verses more, in fact.

I knew this, to no one's great surprise I'm sure. But one of my blind spots has always been assuming that if I know something, then so does everyone else. So here it is.

Throughout his travels in the late '30s, Guthrie was haunted by Kate Smith's hit recording of Irving Berlin's "God Bless America." Guthrie found Berlin's song to be jingoistic and out of touch with the reality facing many of his fellow citizens. So he set about writing a response.

I remember when I first learned of this aspect of the song's origin. It certainly wasn't in kindergarten or whenever it was we were first forced to sing the thing when I was a kid. I mean, back then, it was part of the patriotic indoctrination we all received: Stand for the National Anthem. Put hand to heart and recite the Pledge of Allegiance (no one tried to make us kids sing The Star-Spangled Banner, probably because hell, most adults don't have the required voice range, and kids trying to do it would drive up sales of earplugs and headache pills). Sing This Land is Your Land.

So I can excuse the general idea that some people have that it's just another Thing that we do to Promote Patriotism. But the point is, and the article delves into this later, it was only when Springsteen's first live album came out in the mid-80s that I caught a glimpse of the actual origins of the song: not so much patriotic as revolutionary.

Of course, in this country, what is revolutionary should be considered patriotic. But that's not always the case, as the article demonstrates.

Instead of doing a sarcastic parody, he wrote a song that pulls double-duty, celebrating America's natural splendor while criticizing the nation for falling short of its promise.

Which makes Guthrie a better man than I am. I'm all over the "sarcastic parody" idea.

Regardless of which verses are included, "This Land Is Your Land" is terrific for singing. That was by design. Guthrie likely stole the melody from the Carter Family's 1935 tune "Little Darling, Pal of Mine," which itself was patterned after an old gospel hymn titled "When the World's On Fire" (sometimes called "Oh, My Loving Brother").

Ah, yes, another great American tradition: theft of intellectual property. Hence the sometimes draconian copyright laws I mentioned above.

To be fair, though, I don't think an "old gospel hymn" can be copyrighted.

But regardless of Guthrie's intentions, "This Land Is Your Land" has come to mean different things to different people.

As should be the case with all great works of art.

Anyway, like I said, the article goes into much greater depth of its history and includes videos of different versions. I should also give a shout-out to the excellent cover performed by Counting Crows. Can't be arsed to link it tonight, but you can find it if you want.

And I would be remiss if I didn't mention the parody of This Land that circulated amongst my peers when I was young:

This land is MY land
It isn't YOUR land
I got a shotgun
And you ain't got one
If you don't get off
I'll blow your head off...
This land is private property.

The irony of subverting a subversive song is not lost on me, and I find it endlessly amusing.

Mini-Contest Results!

Great responses from yesterday, and they certainly all made me hungry! (Though part of that is that I can now use the left side of my mouth for chewing again, which is a big relief.)

Write_Mikey_Write! spoke of crepes from his local restaurant, which it sounds like I'm definitely going to have to visit next time I'm in his neck of the woods. Crepes, of course, are also a French thing. Probably more French than French toast.

WakeUpAndLive~doingNaNo'24 - Eggs Benedict is a quite popular breakfast or brunch meal here in the US, too, yes, complete with Hollandaise. Only our tradition is to serve it on a split English muffin - which, as with so many foods, is neither English nor a muffin. The closest you get to one in actual England is crumpets. Also, she provided a great resource in a different comment for how to pronounce the "ch" sound for challah and other words.

Andy~hating university spoke of potato rosti, and its resemblance to latkes. Potatoes, of course, aren't exclusively a breakfast food, and latkes certainly aren't. I had latkes with lox in, of all places, a Belgian restaurant (not in actual Belgium but in NYC), and it was delicious. Also, I don't believe in day drinking. There is only drinking. Position of the accursed daystar is irrelevant.

Lilli 🧿 ☕ - those sound delicious and now I wish more Middle Eastern restaurants (and Indian, etc.) would serve breakfast in the US.

Brandiwyn🎶 knows I'm lazy, so that recipe is intriguing to me. Probably not as delicious as some of the other entries, but balances that out with absolute ease of preparation.

Lazy Writer est 4/24/2008 - while I like oatmeal just fine, it's never going to be near the top of my "delicious breakfast" lists. It's more... I don't know... utilitarian? Sometimes you just need the good carbs.

As for the Merit Badge, I think it'll go to Lilli 🧿 ☕ today [though I will wait until at least next Friday to actually send it because I sent you one last week] - because those were foods I never heard of and now I definitely want to try them. But I wouldn't turn up my nose at any of the other foods -- far from it! Except for coffee. I know you guys love the stuff, but I just never got a taste for it.

Wouldn't mind a good English Breakfast Tea, though.

I'll do another mini-contest soon - thanks again for reading and commenting!

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