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:

Turkey DrumStik

about 3 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

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

Nov 14, 2024 at 12:10pm
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

January 5, 2019 at 12:27am

#948917

Baragouin

Write about a time you learned something new. Use and highlight at least 3 words in your entry that are not part of your normal vocabulary and provide your readers with definitions for those words.

I try to learn something new every day.

Many people do, of course. Others actively resist learning, dismissing any new information as inaccurate or false. To be fair, it's not easy to change one's worldview, and new information tends to do that. New facts become baragouin to them: meaningless, irrelevant.

Not me - I want to delve deeper and learn more. When you're a kid, learning comes naturally, like breathing or running. My running days are far behind me, but I do try to keep that childlike openness to new ideas. I'm not talking about jnana , either; I mean real, objective facts.

As a counterpoint, though, I'm essentially lazy, so it's more of a hobby for me than an obsession.

I've told this story before, I know, but I don't remember where or when: in 7th grade, I had a science teacher who insisted that the Earth's gravity was induced by its rotation; in other words, if the planet were to stop rotating, we'd all float off like balloons. I think she was confused by reading science fiction where a space station rotated to create artificial gravity through centripetal force. (No, I'm not defining "centripetal." I'm an engineer. That's been part of my vocabulary for a very long time.) Nothing I said would change her assertion; after all, what knowledge could a 7th grader possibly have that a seasoned public school science teacher did not? Never mind that I was reading better science fiction.

Point is, I learned something that day, even though it was not what she intended to teach: that teachers can be wrong, not just about their opinions on whether or not I should goof off in class (they've always been wrong about that), but about actual, physical facts.

Ever since then, I try not to trust just one source for anything. That said, Wikipedia is not nearly as untrustworthy as some people make it out to be. Sure, errors creep in, but I remember one study that showed it was about as accurate as a printed encyclopedia. You could say that Wikipedia is a hodiernal encyclopedia, and much grander in scope.

So when you learn something, always keep in the back of your mind that it could be wrong. I don't care if the source is a teacher, the internet, or your parents. Or me, for that matter. We're all unreliable narrators. It's just as important to be willing to change your worldview in the face of new information as it is to want to learn new things.

For example, I used the new-to-me word "hodiernal" up there, but I can't be sure I used it correctly. Does "the present day" mean, specifically, 5 January 2019? Or the more general present-day with its technological and social milieu? This is not clear to me. The adjective seems to mostly be used in connection with tenses in writing, though, so I'll be sure to explore it further. When I'm not feeling quite so lazy.

(Incidentally, for anyone who wants to seek out new words and new vocabularizations, I found this

to be essential for creating this entry.)

3 Like • Comment • View 3 comments

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