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!

about 13 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 15 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').

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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.

May 14, 2020 at 12:03am

#983519

Man of Letters

What is this "letter" of which you speak?

PROMPT May 14th

What was the best/worst letter or email you ever received or wrote? Write about the situation surrounding that letter, and why it was so significant.

Fun Fact of the Day: the word "letter," which in English can, confusingly, mean either a single symbol of the alphabet or a collection of said symbols arranged into words, sentences and paragraphs and usually sent via courier or through something called the "mail" or "post" to someone, originally used the singular form for the former and the plural form for the latter. To make matters even more confusing, "lettered" means "educated," and "literature" seems to come from the same Latin root.

Language is weird.

My first professional job distinguished between a "letter" and a "memo" thus: if it fit on one page, it was a memo. Any more than that, and we had to use a letter format.

I can't remember the last time I wrote an actual letter. It was certainly something related to my job, but by the time I retired, pretty much everything was emailed. And I certainly can't remember writing any missive of substance, electronically or otherwise. It was mostly just boring reports and engineering explanations.

The process for becoming, and remaining, a professional civil engineer in the US is, in general, the following:

Graduate from high school

Four years of engineering school (yes, this is less than other professions such as law or medicine)

Work

Take the Engineer-in-Training exam (EIT)

Work some more

Take the Professional Engineering exam (PE)

Keep working

Take professional training classes (in most states)

Renew PE license

Repeat previous three steps until retirement or death

Generally only civil engineers have to maintain a PE license; other disciplines don't care as much. With said license, an engineer has the right and responsibility to approve plans, which basically says "this was done by me or under my supervision, reviewed by me, and adheres to the standards of practice of engineering drawings." You can then be sued if a dam breaks or a road washes out.

With that responsibility usually comes a higher pay scale. Not really high, mind you, but for me it was the difference between barely scraping by and being able to save a bit of most paychecks. So, naturally, I wanted to get my PE license.

I studied for the goddamned exam for at least three months, not to mention all the college and work and whatnot. I've always tested well, but that fucking exam about killed me. I was so sure I'd failed and would have to take it again that, after drinking for the rest of that weekend, I started studying to take it again.

But behold, three or four months later, I got the letter in the mail: "You're a licensed professional engineer in the Commonwealth of Virginia."

And then I drank some more, because I could finally afford it.

That's the great thing about drinking: it can make bad situations tolerable, and good situations better.

A whole lot of bad things happened in rapid succession around 2008-2009, including a lot of personal shit and a recession that pretty much halted all construction plans in my area, and I let my license lapse and retired, but by that time I'd fulfilled everything I'd set out to do professionally. So I call that a win. And that's why I'm calling that licensing letter the best one I ever received.

Ask me again tomorrow and it'll probably be the love letter I got from [name redacted] in 1986, or the one that said I was eligible to go to the Naval Academy, even though it was signed by that assclown Ronald "McDonald" Reagan, or, on the "worst" side, the one court summons I got that wasn't for jury duty. But that's a story for another time. Or, more likely, not at all. (I will say that it wasn't related to my engineering work, nor a criminal case.)

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