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 22 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 23 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.

January 24, 2020 at 12:03am

#974073

Famous Blue Raincoat

PROMPT January 24

Yesterday celebrated National Handwriting Day in the USA. How often do you still hand write anything substantial? Do you think the decline in children learning cursive writing will be a hindrance to their generation?

The decline in the ancient and venerable art of flint-knapping is worrisome. All these kids running around today with their fancy bronze swords - they'll never know the joys of chipping off a perfect flake of stone, coaxing from the rock an exquisite hand-axe or arrowhead...

It's too bad that kids these days will never know the finer points of team-driving. With the advent of the "auto-mobile," what beauty will be lost in the decline of the unification of Man and Equine...

Schools these days just aren't teaching FORTRAN the way they used to. Why, back in my day, we'd have to punch holes in cards, each one perfectly formed to deliver a useful program to the mainframe. You just don't get that precision with compilers of C++ and the knowledge that, if you make a mistake, you can just edit the code like it's some sort of word processing program...

Yeah, I'm probably going to have to turn in my Old Guy Card.

Some years ago, I read about a dude who, when prompted to sign his credit card receipts or pay pad, instead of laboriously writing his name in cursive like a normal person, would sketch a male organ in the space. This worked for an entirely unreasonable amount of time, until someone actually compared this schmuck's johnson to the signature on the back of his card and found that they didn't match.

The moral of this story is: if you're going to be a dick, go all the way.

I relate this for two reasons: first, a friend of mine pointed out that my blog suffers from an appalling lack of dick jokes [Narrator: this is a lie; no one has said that to him, ever], and second, to illustrate that even that last bastion of cursive writing, the signature, isn't strictly necessary. My own Hancock (see what I did there?) is an impenetrable scrawl that bears almost exactly zero resemblance to the flowing curves of the handwriting we were taught in elementary school, and which, to be a lot more honest than I was at the beginning of this paragraph, I sucked at (snort - I did it again).

By the time I went to engineering school, I'd abandoned cursive in favor of the block-caps preferred by those in my profession-to-be. It was actually faster for me to take lecture notes in block caps than it was to cramp my hand into the unnatural and entirely too artistic knots required for script.

This ended up serving me better than even I expected; when I got my first job in the field, it was as a drafter. In the days before AutoCAD, we hand-drew plans in ink on mylar or vellum, and my capacity for crafting clear, bold, and consistent letters, freehand, only helped me when it came time to label the drawings.

Now even that skill has been rendered obsolete for me, first by the aforementioned drafting software, and then by my retirement. (Do I mourn the lack of skill in using traditional drafting tools? No. Plan readability is improved, and that's what matters.) I still write in block caps on those occasions when I actually have to hand-write something, which in practice is usually only when ordering a beer and popcorn from my seat at the drafthouse cinema (talking is prohibited, so we write our orders; I am sure that my block lettering is much easier to read in a dim theater than any cursive would be).

Is something lost in the process of cursive's obsolescence? Sure, and change is often difficult for some people to accept. But as in the examples with which I opened this entry, it's not like we don't have things to replace it. One "feature" of cursive is its variability among different individuals, combined with its supposed consistency as regards a single individual. That feature is one reason the "signature" came to be a thing in the first place. Traditionally, I've been informed, the illiterate would mark an X, which as I understand it was supposed to be witnessed by someone who could read and write - hence the modern practice of marking the signature lines on forms with that cryptic letter. But I digress; I was writing - I mean, typing - about individual differences. See, as with Dick Boy above, there's no real reason why a person's signature, or mark, couldn't be a particularly individualistic sketch; such would be just as easy, or difficult, to forge as a signature is. (Oddly enough, I got really good at faking my parents' very different signatures, although I never did master my own.)

The other interesting thing about cursive was its use in the personal letter. I know a lot of people believe that there's something special about receiving a missive from a friend or lover, written in their personal style by their own hand. I'm inclined to agree with that point of view, though I'd assert that such individuality, today, finds its expression in the particular way someone spells (or fails to spell) words in a text message, and their idiosyncratic preferences in type and density of emojis. Just as I have difficulty interpreting emoji, though, I often struggled to figure out what a cursive-writer was actually trying to say.

So - no, there's no hindrance to kids-these-days to remove cursive from the curriculum. It's not a life skill, even if it was once. It's an art, and like any art or skill, people can learn it on their own if they're so inclined. I know I wouldn't be, but as far as any art is concerned, my skills are indifferent at best.

Also, as today's musical illustration hopefully demonstrates, it's not the formation of the characters that matters so much as the actual content. Again, this may be my personal bias showing - I always did focus on function over form - but I can't imagine that Leonard Cohen's actual letter, if it actually existed, could display in the sweeping curls of its words any more emotional punch than is contained in the words themselves.

It's four in the morning, the end of December
I'm writing you now just to see if you're better
New York is cold, but I like where I'm living
There's music on Clinton Street all through the evening

I hear that you're building your little house deep in the desert
You're living for nothing now, I hope you're keeping some kind of record

Yes, and Jane came by with a lock of your hair
She said that you gave it to her
That night that you planned to go clear
Did you ever go clear?

Ah, the last time we saw you you looked so much older
Your famous blue raincoat was torn at the shoulder
You'd been to the station to meet every train, and
You came home without Lili Marlene

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