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 5 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 7 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.

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.

July 14, 2022 at 12:01am

#1035176

Vive La Révolution !

So. I'm cheating today. Rather than picking from my list at random, I'll be featuring an article from my queue about the French Revolution.

Obviously, because today is le quatorze juillet.

5 Misconceptions About the French Revolution

The French Revolution is a landmark period in European history, but people still get a lot of it wrong.

Are you kidding? We Americans can't even get our own history right. E.g.: The Alamo.

1. Misconception: Les Misérables takes place during the French Revolution.

Confession time: I have never read Les Mis. I have never seen the play. I have never seen any of the movies. It's not that I don't want to; it's just that I've never once sat down and gone, "You know what I should do? I should see Les Mis." Someday. But today is not that day, because as this article points out, it's not about la révolution at all.

After seeing the 2012 film adaptation of the musical Les Misérables in theaters, historian Julia Gossard caught a snippet of some other viewers’ conversation. “So, that was the French Revolution?” one woman asked. “And it was unsuccessful?”

Look, I'm not a historian. Nor have I, as noted above, ever been exposed to any version of Les Mis. And yet, somehow, I knew this.

The article goes on to explain the historical background of the story, and contains spoilers. Insofar as you can spoil a novel that was published 160 years ago.

2. Misconception: Rebels stormed the Bastille to free political prisoners.

This is, of course, what led me to link this article today.

When the Bastille was stormed on July 14, 1789, there were only seven inmates. One was a wayward relative sent by his family, four were serving time for forgery, and two had been committed due to insanity—not the political prisoners you might have imagined. But if the goal wasn’t to free prisoners, why attack a prison? The real reason, according to most historians, was for ammunition.

But why was... never mind. It's in the article.

3. Misconception: French reformists all wanted to end the monarchy.

Well, duh. That would be like saying that everyone who wanted taxation with representation in the Colonies wanted to revolt against King George III.

The economic crisis illuminated some of France’s long-standing issues, perhaps most notably the insidious effects of feudalism. Not only did the nobility and high-ranking clergy members own most of the land, but their positions came with a lot of perks and exempted them from a lot of taxes.

Hm... do I hear an echo?

But it wasn’t like “Death to the monarchy” and “Long live the monarchy” were your only two options. In fact, many political factions simply wanted a constitutional monarchy. And when the National Assembly created its constitution, that’s what it was for.

Well, it worked for England. France, however, has a long history of doing the exact opposite of what England was doing, just because.

4. Misconception: The guillotine was invented during the French Revolution.

Not only did Joseph-Ignace Guillotin not invent guillotines, he didn’t even design the ones used during the French Revolution. All he really did was suggest that France standardize executions.

I guess there weren't too many anti-death-penalty Revolutionaries.

Joseph-Ignace Guillotin was against the death penalty altogether, but apparently, he realized that France was nowhere near being ready to give it up.

Ironically, Guillotin was one of them.

“With my machine,” he explained, “I strike off your head in the twinkling of an eye and you won’t feel a thing.”

"My machine?" I thought you said... aw, forget it, Jacques, it's France.

Incidentally, I'm sure they thought it was humane, but it turns out that you're aware for possibly up to ten seconds after your body is separated from your head. Ten seconds of rolling around in a basket probably seemed like an eternity. So much for "humane."

5. Misconception: Marie Antoinette said “Let them eat cake.”

Hopefully, everyone already knew this. I'm sure it's a useful myth to tell your country, but it's still false.

Just for pedantry’s sake, the sentence in French is “Qu’ils mangent de la brioche,” which literally means “Let them eat brioche.” Brioche is a rich, buttery bread that’s way more extravagant than what poor peasants would’ve been eating—it’s not exactly cake, but it doesn’t really change the supposed sentiment.

Not unless Marie was about to open the pantries of Versailles and personally distribute said brioche to the revolting peasants.

But the correct French phrasing does shed a little light on how it got popularized. The earliest known record of “Qu’ils mangent de la brioche” comes from philosopher Jean-Jacques Rousseau’s Confessions, penned in the 1760s. In Book Six, he wrote, translated from French: “At length, I recollected the thoughtless saying of a great princess, who, on being informed that the country people had no bread, replied, ‘Then let them eat brioche!’”

Which, by itself, doesn't mean that Marie couldn't have said it. That horrid movie about her back in the noughties kept mum on the subject, too. If I recall it correctly. I'm not about to watch that pile of merde again to check my memory.

So anyway, in honor of Bastille day, there it is: the record (kind of) set straight.

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