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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

October 23, 2023 at 10:00am

#1057890

Testing My Patience

"Sorry, we can't hire you. Your natal chart shows Mercury retrograde in Pisces, and Venus square Jupiter, with Neptune ascendant."

Deprogramming the Cult of the Workplace Personality Test
Introverted, extroverted, thinking or feeling? There’s really only one trait the personality-test industry is ever assigning to you: Gullible AF.

Wait, did we forget to actually ask you any questions? That doesn’t matter. Your responses wouldn’t have affected the accuracy of your results, and the $4 billion personality-assessment industry has already decided the outcome for you in any case: As far as it’s concerned, you’re all of the above, your path in life has been determined and there’s not a damn thing you can do to change that.

I've been likening that crap to astrology for a while, now. Thing is, yes, in my view, your path in life has been determined... but no one has any way of knowing what it is: no star charts, no genetic assessment, no divine revelation, no personality test, no Tarot reading. The future can go places that the past never dreamed of; you can't change what you can't predict, and if you could, that would lead to paradox. The difference between past and future is this: the past leaves evidence (memories, scars, bloodstains, ash, omelets), and the future does not.

As a rough analogy, we've gotten pretty good at predicting the general weather, short-term. While not 100% accurate, most of the time, if it says it's going to rain tonight, it's best to bring an umbrella. This comes from centuries of study, decades of computer modeling, and our very human trait of being able to project past trends into future probabilities. What we can't do, what we will never be able to do, is to predict precisely where and when the first raindrop will hit the ground. And yet, that raindrop was always going to land at that exact location at that precise moment.

On being handed his results, John raised an eyebrow at the fact that, according to the test publisher’s blurb at the top, it had color-coded his personality according to the “four humors,” the bodily fluids — yellow bile, black bile, blood and phlegm — that Ancient Greek medical theory held as responsible for regulating both people’s physical health and the underlying aspects of character, or “temperaments.”

As utterly silly as that test is, though (I'd be tempted to believe that the publisher was trolling), it, and other personality tests, don't purport to predict the future. They do, as far as I've heard, claim to predict how someone would react in a given situation—as if that's fixed, independent of other environmental factors, and completely discounting our ability to learn from our mistakes.

Did he really want to work for a company where teams were structured according to human-resources thinking from a time when motor function was attributed to animal spirits that roamed through the muscles and lived in the brain? “‘Where’s David?’” says John, mentally sketching a regular day at the office. “‘Oh, he’s in Meeting Room 2 at the moment, with the leeches on him.”

Turns out leeches do have some legitimate, though limited, medical use... but it's not because of unbalanced humors.

The most well-known hot-take taxonomy of all, the Myers-Briggs Type Indicator (MBTI), is famously the product of a homeschooled Philadelphia novelist, Isabel Briggs Myers, feverishly working in the 1940s and 1950s to graft her own notions about personality onto a typology proposed by her intellectual hero, Carl Jung.

Let's be fair, here: "homeschooled" is irrelevant. "Philadelphia" is twice as irrelevant. "Novelist" is misleading: Carl Sagan was a novelist, but he was also a brilliant scientist and communicator of nonfiction. That other Carl, Jung, is worthy of study. But the problem there is not any of that, but the "graft her own notions about personality..." which is kind of like when a pharmaceutical company salts their testing results to get the outcome they want, such as "this drug is entirely nonaddictive and totally doesn't cause liver damage."

In her jargon, you were either a “thinking” person or a “feeling” person (your nailed-down psyche wasn’t allowed to straddle both), you were either “introverted” or “extroverted” and you went about the world either “sensing” or “intuiting” it, and either rationally “judging” or irrationally “perceiving” things.

It is the binary nature of these purported attributes that should be a massive red flag. Most of us do "straddle both" of any opposing qualities.

Meanwhile, throughout the long decades since World War II, while this and similar systems have been refined, promoted and steadily entrenched as indispensable weapons of hiring and firing, pretty much the entire community of professional academic psychologists has been quietly coughing into its hand and saying “bullshit.”

For reasons, I'm sure, beyond merely the "binary" objection I have; but then, I don't exactly have the qualifications to be an expert on this sort of thing, either. Having grown up on a farm, I can generally smell bullshit, but that doesn't mean I always know where it lies.

(I'm pretty damn proud of that "lies" pun, though.)

But, as the article points out, actual experts are calling bullshit.

In any case, the article may be of interest whether you love, hate, or fall somewhere on the love/hate spectrum with regard to these sorts of assessments.

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