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!

a day 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

a day 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.

July 18, 2019 at 12:13am

#962792

Oh, Great.

Complete the following statement: To achieve greatness, one must...

...avoid complacency.

This is necessary, but not sufficient. There's luck, skill, talent, whatever. The vast, overwhelming majority of people will never achieve greatness. Yes, this in all likelihood means you. Me too, obviously. Ain't I a ray of sunshine? With high levels of ultraviolet?

Fortunately, the benefit is in the striving, not the achieving. Journey, not destination.

This leaves out the question of how exactly one defines greatness. I would argue that the greatest businessman of all time was Steve Jobs. Supporting documentation: He and a couple of friends built a computer out of spare parts in a garage, and he parlayed that into a vast business empire; at the time of his death, Apple was the most valuable company in the world (defined as having the largest market capitalization). And yet, by all accounts, he was a shitty boss, an even worse father, and prone to the kind of cognitive bias that leads people to believe bullshit (hence his death).

Did Steve Jobs "achieve greatness?" By some measures, certainly. By others, no.

Now, I'm no expert, clearly, as evidenced by the fact that I'm not great. But it seems to me that you have to qualify what you mean by it. And I think that, like the ever-elusive "happiness," you can't get there by wanting it. I mean, you can wake up one morning and affirm to yourself: "I shall achieve greatness!" You can do this all you like, and you'll die in obscurity - though maybe you've lived a good, full life; that's irrelevant to the discussion.

Since it's de facto Moon Week, I'll provide another example: Neil Armstrong. He gets credit for being the first dude to put boots on the moon. Solid achievement. Not everyone could have done it. But there were at least two (Aldrin, Collins) and several more astronauts who also could have done it. They all had the same training. They likely all had similar aspirations. There was a vast network of support behind all of them; the "first" honor could have easily gone to another person. It was very nearly a Russian. That would not have diminished the achievement in the slightest, but the story would be different. Point is, that's just the most obvious example: not only must one work smart and hard in one's chosen field, but there's an element of luck, a need for support (even Einstein admitted he built on the achievements of others, or, in his words, stood on the shoulders of giants), and the kind of mindset most of us are just not cut out for.

And then there are those who achieve greatness without hardly trying, those who probably don't deserve it. Thomas Edison comes to mind. Dude cheated.

Everyone I've mentioned so far has been male, but gender is irrelevant to the topic. I could as easily talk about Rosalind Franklin, Hypatia, or Marie Curie. Our culture has a bias; we don't have to internalize it.

As Shakespeare noted: "Some are born great, some achieve greatness, and some have greatness thrust upon them." (That last bit is especially prescient when one considers Neil Armstrong, who quite literally had it thrust upon him in the form of five giant rocket engines.) Any way you look at it, though, most of us will never be in any of those three Shakespearean categories.

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