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 9 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 11 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.

February 2, 2024 at 9:48am

#1063315

Rats!

Despite the significance of the Great Groundhog today, we're going to talk about another rodent. The Platonic ideal of rodent, the Ruler of Rodentia:

In Defense of the Rat
They’re less of a pest than you think—actually, they have quite a few benefits.

Now, if someone had said that, say, mosquitos were "less of a pest than you think," I'd know they were bullshitting. But rats are cute, and don't we give cute critters the benefit of the doubt?

There was a time when we human beings used to put animals on trial for their alleged crimes against us.

Gosh, I wonder what would happen if they put us on trial for our alleged crimes against them?

The earliest of these prosecutions in the Western tradition of law appears to be a case against moles in the Valle d’Aosta, Italy, in 824 AD...

Another clear example of anti-rodent prejudice.

In the centuries between, a killer pig was dressed in human clothing and hanged in Falaise, France...

Okay, I can kind of see the pig wearing human clothing (just imagine Alex Jones), but how the hell do you hang a hog? They ain't got no neck.

...Marseille put dolphins on trial for crimes unknown...

That one's easy. The prince heir apparent of France was known as le dauphin. Someone mispronounced the definite article, and they thought they were trying les dauphins.

...and a rooster—in what must have been a case of mistaken identity—was burned at the stake in Basel, Switzerland, for the witchery of laying an egg while male.

Maybe he was just living his true life.

In 1522, “some rats of the diocese” of Autun, France, were charged with criminally eating and destroying barley crops. A skilled legal tactician, one Barthélemy de Chasseneuz, was assigned to defend the rats.

How come most of these are French?

Well. The article is fairly long, but I found it enlightening. Especially the part about how humans were a more proximate cause of The Plague than rats were: "In the case of the notorious plague in Europe, the event that forever marked rats as public enemy number one, the animals may be almost entirely innocent."

But there's also a long discussion about rats' intelligence and demonstrated capacity for empathy—not only with other rats, but with humans.

I'm not saying you should venture into the NYC subway tunnels and pick up a rat. There really are hazards with wild rats—just like there are with raccoons, and everyone thinks trash pandas are just the cutest things (mainly because they are). But I've known several rats in my life, albeit briefly (rodents don't tend to live very long lives), and, yeah... they're not all vermin.

Now, if I could only convince my cats to stop leaving them for me as gifts, that'd be great.

3 Like • Comment

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