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 8 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 10 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.

March 21, 2024 at 10:06am

#1066678

Socra-Tease

Here's one from The New Yorker that, when I first read it, I thought: This has comedic potential.

When Philosophers Become Therapists
The philosophical-counselling movement aims to apply heady, logical insights to daily life.

I mean, think about it.

"Doc, I'm distressed because I can't find meaning in life."

"That's because life has no meaning, so stop looking for it."

"Wow! That's deep! Thank you."

"Sure. That'll be $180."

"Here's $200."

"Thanks!"

"Can I have change?"

"You already did."

Around five years ago, David—a pseudonym—realized that he was fighting with his girlfriend all the time. On their first date, he had told her that he hoped to have sex with a thousand women before he died.

All the TNY pieces I've seen start out with anecdotes and then, maybe, bit by halting bit, go around and around in circles until they get to something more general. Sometimes they never even get there, which I call "The New Yorker School of Not Getting to the Fucking Point." This one contains a whole lot of anecdotes. My point, which you'll note I'm getting to very quickly, is that I'm going to skip a lot of it here. In brief, he worked with a philosophical therapist named Lydia Amir.

I might have missed it, but I don't think the article notes whether he had sex with her or not.

Between meetings, Amir sometimes gave him reading assignments: Schopenhauer, Nietzsche, Hume.

Oh yes, definitely, if I'm having an existential crisis, those are absolutely the philosophers I want to read.

Amir is one of a small but growing number of philosophers who provide some form of individual counselling. In the United States, two professional associations for philosophical counsellors, the National Philosophical Counseling Association (N.P.C.A.) and the American Philosophical Practitioners Association (A.P.P.A.), list dozens of philosophers who can help you with your problems.

Well. Philosophy tends to pay jack shit. Therapy, however, can bring in the bucks.

Philosophy is both a natural and a strange resource for helping people resolve the problems of life. Ancient philosophical traditions such as Stoicism and Buddhism focussed on practical ethics and techniques for alleviating suffering, but much modern philosophy seems to aim to express suffering, rather than reduce it.

I wouldn't call the aim of those philosophies to be "alleviating suffering," but there's always the possibility that I don't really understand Stoicism or Buddhism.

Some think that the practice of philosophical counselling should be more standardized. Others worry that philosopher-counsellors will miss serious mental-health issues. The two major American professional organizations stress that philosophical counselling can’t address certain severe psychiatric disorders, and urge counsellors to refer clients to mental-health providers when their issues do not fit a philosophical scope of practice.

Well, that's somewhat relieving. It's one thing to seek someone else's point of view, or be able to talk things through with a neutral party. It's another to have legitimate mental health issues that need more medical treatment.

So, anecdotes (and jokes) aside, this kind of thing obviously exists (in the same sense that anything can be said to truly exist), and it mildly amuses me. Does it work? All the TNY anecdotes in the world won't convince me one way or the other; I want to see a study.

I'm sure a philosophical therapist would have something to say about my need for evidence-based science. But, you know. Whatever.

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