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 2 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 4 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.

December 27, 2022 at 12:04am

#1042228

Origins

I'm still learning new stuff all the time. This is a good thing. Though I'm not taking anything from Cracked at face value.

5 Phrases No One Would Use If They Knew Where They Came From

"No One" is a bold statement, for instance.

We once told you that the famous quote about learning who rules over you by finding out who you’re not allowed to criticize actually originated in 1993 from a white supremacist sex criminal.

That doesn't require any digging to disprove. You're not "allowed" to criticize quadriplegics for being unable to move, for example, and I doubt they rule over you. I mean, you technically could criticize them, but you'd have to be a real dick.

And we revealed that "the definition of insanity is doing the same thing and expecting a different result” came not from Einstein but from Narcotics Anonymous and from a mystery novel about women’s tennis.

"If you want your quote to have weight, attribute it to Einstein." -George Washington

With the following phrases, you don’t need to have any idea where they came from to use them. But once you do know, ouch, you’ll never think of them the same way again.

Now, that phrasing, I can get behind.

5. ‘Yelling Fire in A Crowded Theater’ Comes from A Terrible Court Decision

When people say they’re being silenced, their opponents quickly point out that freedom of speech isn’t absolute (these same opponents, when their own side is silenced, will call this censorship deeply discriminatory and unfair). For example, they note you can’t yell “fire” in a crowded theater. The logic here is that the First Amendment has some exceptions, so this latest thing might fall under that exception, maybe.

I actually knew this one, kind of, though as they say, the devil's in the details. They're too long to paste here, so you might want to read it. If not, consider the (almost) closing argument:

The actual decision that originated that argument was already bad. It was so bad that the judge who wrote that phrase, Oliver Wendell Holmes, changed his mind about speech protections in less than a year.

4. ‘Vox Populi’ Is Really About Why the Majority Voice Shouldn’t Win

“Vox populi, vox Dei,” posted Elon Musk recently, when a poll told him to bring back Trump’s Twitter account. The Latin phrase translates as “The voice of the people is the voice of God.” When the people speak, you must pay heed, because their word is law.

Again, even the slightest bit of thought could contradict this one. If we have "inalienable rights" at all, the whole idea is that those are rights that can't be taken away with a simple majority vote. To use a possibly extreme example, let's say that the people decide, by majority vote, that all Canadians living in the US should be executed. You can take such a vote, but that doesn't mean that the sentence will be carried out; that would be a violation of the Canadians' human rights.

And I'm not even going to get into the theological implications.

The earliest recorded use of “vox populi, vox Dei” is actually a letter from the year 798 that says listening blindly to the majority view is a bad idea. “And those people should not be listened to who keep saying the voice of the people is the voice of God,” wrote the scholar Alcuin of York, “since the riotousness of the crowd is always very close to madness.”

3. Snake Oil Is Actually Effective, Hence the Phrase

When someone claims he can solve traffic using single-lane tunnels, or says you’ll double your money by buying Shiba Inu coins, you might say they’re slinging snake oil. That’s highly insulting — to snake oil.

I have to admit, this one was new to me, so I had to verify it elsewhere.

Wikipedia is only slightly more trustworthy than Cracked, but feel free to follow the links in the footnotes there.

These salesmen weren’t quacks because they sold snake oil. They were quacks because they didn’t sell snake oil.

In any case, even if snake oil is "effective," it's hardly the cure-all some of these hucksters made it out to be. That would be like saying that because aspirin cures headaches, it can also cure cancer. It did turn out to offer some protection from heart attacks, though, which is why science is important: you can investigate these claims and support or disprove them.

2. The Guy Who Coined the Term ‘Racism’ Had Some Strange Ideas About Race

Racism may have always existed. People were even racist in some Star Wars stories, and those happened a long time ago. But before 1902, if people had a word for the concept, it was “racialism.” The first person to call it “racism” was an army officer, Richard Henry Pratt.

And that's not even getting into droid slavery.

“A great general has said that the only good Indian is a dead one,” said Pratt. “In a sense, I agree with the sentiment, but only in this: That all the Indian there is in the race should be dead. Kill the Indian in him, and save the man.” That was from a speech where he spoke against racism.

I'm not a big fan of judging people of the past by today's moral standards, but that seems evil even by the standards of a century ago.

1. ‘Spare the Rod, Spoil the Child’ Comes from A 1600s Sex Poem

I'm not even quoting anything from this section. In any case, beating one's children for any reason fell out of fashion sometime in the last century. After I'd already become an adult, of course.

One has to be careful with stuff like this, though. Some phrases have been given a fauxtymology. For instance, I've heard people assert with quite a bit of confidence that "rule of thumb" came from an old law against beating your wife with any stick bigger around than your thumb. I even got screeched at once for saying it, because of its association with misogyny. But there's no truth

to that assertion at all. Most likely it just came from the association of the thumb with an inch.

Language changes over time, and it has a way of fulfilling its own prophecies. But I still think it's important to know where these things come from.

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