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 2 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 4 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 30, 2023 at 12:01am

#1043857

Psych Ops

Today in "you've got it all wrong," courtesy of Cracked...

5 Bits of Pop Psychology That Make Absolutely No Sense

Just to get this out of the way: something "not making sense" doesn't mean it's wrong; it could mean you're missing information. But there's stuff that doesn't make sense, and then this stuff, which has been proven wrong (or at least not shown to be right).

The fields of psychology and psychiatry are incredibly complex.

Oh, good, just right for this blog.

It’s not too surprising, given that “understanding human thought and behavior” seems more like a question you’d take to some wise man on a mountaintop than something you’d choose as a major.

You know why wise men live on mountaintops? Well, one, to hide from their wives. But also because when you climb the mountain and pass all the arduous tests and solve the unsolvable riddles and finally meet the guru, and you ask him a stupid question like that, he can kick you right off the cliff.

A lot of the ideas and advice dispensed by TikTok psychologists is obviously flawed, if not outright disproved.

This should go without saying, but apparently, I have to say it anyway: don't get your advice from DickDock.

So without further introduction (though the article does, indeed, provide further introduction), the circulating misinformation in question:

5. Smiling Makes You Happy

This one is the classic bugaboo of anybody with even a smidgen of clinical depression.

Vouch.

Making it worse is that the person who tells you this is usually the most carefree person you’ve ever met.

It would be wrong to punch them, but I understand the urge.

The roots of what is called the “facial feedback theory” comes all the way from Charles Darwin in the 1800s, and although Darwin’s got a pretty solid track record, psychology from the 1800s does not.

Okay, look: periodically, some outlet (usually affiliated with a group who wants to see the idea of evolution via natural selection go away) proclaims, "DARWIN WAS WRONG." You get the same thing with Einstein. People love to tear down other people who are more knowledgeable and influential than they are (I'm not immune from this, myself). Was Darwin wrong? I'm sure he was wrong about a lot of things, being, you know, human and all. Have some of his hypotheses been overturned? Sure. That's how science works. It's not like some other human pursuits, where the prophet's words are supposedly infallible for all time. Evolution is a solid theoretical framework built on a firm foundation. Psychology... well, it's a bit shakier.

Not only that, studies have found that if you’re not in a neutral state, but genuinely sad or angry, forcing a smile can make you feel worse. These studies also found that workers forced to smile all day were more likely to drink heavily after work.

As this article points out, the actual evidence is mixed, here. Given the uncertainties, I'd lean toward "stop making people smile when they don't feel like it, dammit." And yes, this includes service workers. Especially service workers.

In any event, this particular item is something I'd have guessed anyway, so it passes my personal "sense" test. This next one was maybe more surprising.

4. Brainstorming Is More Creative

Brainstorming: the persistent idea that a bunch of brains in a room and a whiteboard can produce more creative ideas than any of those brains alone. Unfortunately, research has found that this can’t always be the case, and for reasons that people who’ve sat through these kind of sessions probably felt at the time.

On the other hand, I'd wager that a brainstorming session is only useful if the people involved aren't just wishing they were somewhere else.

This section goes into exactly why brainstorming isn't all it's cracked up to be, and I won't replicate that here.

3. You Only Use 10 Percent of Your Brain

Seriously, people still believe that nonsense? Sigh... I guess because of anchoring bias. You learn something, and often you have to cling on to it in the face of evidence to the contrary. Like believing the last Presidential election was stolen. No amount of facts and evidence will get anyone to change their minds about that. Come to think of it, perhaps those people are only using 10 percent of their brains.

This one is another absolute chestnut of bullshit. There are even entire (bad) movie plots based around Bradley Cooper turning into a borderline superhero by turning all the lights on upstairs.

I don't remember that one offhand. Wasn't there one with a plot like that with Scarlett Johanssen?

If you’re saying to yourself right now, “Well, it’s EXAGGERATED maybe, but—,” allow me to refer you to neuroscientist Sandra Aamott, who tells Discover Magazine, “There is absolutely no room for doubt about this.”

Look, when a scientist says there's "no room for doubt?" Then you can have pretty high confidence, on the level of "the sun is bright" and "gravity is a thing."

2. The Power of Visualization

I’m sorry for The Secret lovers and vision-board crafters out there (on multiple levels), but the heavily touted “power of visualization” is not only a crock of bullshit, there’s evidence to support that it actually decreases your chance of success.

And they won't believe it, like I said above.

That’s because when you visualize yourself having achieved whatever your goal du jour is, you get a tiny sniff of the accomplishment of having done it, which can reduce your drive.

On the other hand, I can't imagine anything reducing my drive, short of death or coma.

What’s a lot more helpful, and a lot less fun (hence its lack of popularity), is specifically visualizing all the work necessary to achieve that goal.

Oddly enough, I was thinking about this sort of thing before I found this article. The context was cooking—it occurred to me that I have a habit of mentally going through all the steps for a recipe before actually starting. I don't "visualize" the resulting dish, or at least not longer than it takes for me to go "okay, yeah, I'm hungry," but mentally running through the steps helps me ensure I have all the stuff I need in the kitchen.

1. OCD Means Being Neat

This one is as pervasive as it is infuriating. Odds are some type-A friend or acquaintance of yours has said something like, “I’m completely OCD about my workspace.”

At least the incidence of using debunked Freudian terms ("anal") to describe it has decreased.

As psychology professor Stephen Ilardi explains in the Washington Post, most OCD sufferers are “plagued by a cascade of unbidden, disturbing thoughts, often in the form of harrowing images that they may feel compelled to ward off with time-consuming rituals. It’s a serious mental illness that typically causes great distress and functional impairment.”

I knew someone who was diagnosed with severe OCD, a single mom. This didn't manifest as her becoming some sort of neat freak; quite the opposite. Think of the worst hoarding situation you've ever witnessed or heard of. It was that bad. Shit piled everywhere (sometimes literal shit). There was even talk about getting the kid out of that situation, but honestly, I didn't pay enough attention to know if that was ever done or not.

I don't know enough about psychiatry to know how that sort of thing works. I got the impression that it was something like "if I disturb this pile, bad things will happen, so I'm just going to leave it alone."

From what I understand, she got help and is better now, but the article has it right: it's serious stuff, whether it manifests as neatfreakitude or hoarding or anything in between.

But while we're at it, can we also stop misusing "type-A?" Thanks.

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