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.

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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 4, 2024 at 9:03am

#1065530

Facts Simile

Here's a scientist's take on battling nuttery. From Ars Technica:

The key to fighting pseudoscience isn’t mockery—it’s empathy
Evidence shows that shoving data in peoples’ faces doesn’t work to change minds.

Empathy? Well, then, my efforts are doomed to failure.

As a scientist heavily engaged in science communication, I’ve seen it all.

Somehow, I doubt that is true. This article is less than two months old, and I'll bet that this author saw even more in the scant 7 weeks since it was published. But okay. I'm being pedantic. Believe it or not, scientists have a poetic license, too.

People have come to my public talks to argue with me that the Big Bang never happened.

There have indeed been whisperings in the physics world about alternative origins, mainly based on data collected from the JWST. But it's one thing for trained scientists to use available data to come up with new hypotheses, and quite another for non-scientists to assert something without real evidence. Yes, that group includes me.

People have sent me handwritten letters explaining how dark matter means that ghosts are real.

I had a long discussion with myself in here, quite recently, about the spectrum between "real" and "not real." Ghosts lie somewhere on that spectrum. I have no doubt that people have experienced... whatever... and called that whatever "ghosts." (I also have no doubt that there are hoaxers out there, as documented in Scooby-Doo.) The leap there is from "experiencing something I can't explain" to "it must be the disembodied spirit of a dead person." Analogous to "lights in the sky = space aliens."

In any event, there is not one single shred of evidence that I'm aware of to link dark matter with unexplained haunt-like phenomena. Hell, the whole point of dark matter, as far as I understand the science, is that it only interacts gravitationally with ordinary matter.

People have asked me for my scientific opinion about homeopathy—and scoffed when they didn’t like my answer.

As Tim Minchin explained, funny how water "somehow forgets all the poo it's had in it."

People have told me, to my face, that what they just learned on a TV show proves that aliens built the pyramids and that I didn’t understand the science.

Sure; they were spaceship landing sites, as seen in the well-known documentary Stargate: SG-1.

Notably left out of this introduction is any discussion on flat-earthers, UFOs, astrology, psychic powers, cryptozoology, vaccine refusal, climate change "skepticism," or moon-landing hoaxers. Among myriad others. Fortunately, the author nods to these later.

But in all my years of working with the public, I’ve found a potential strategy. And that strategy doesn’t involve confronting pseudoscience head-on but rather empathizing with why people have pseudoscientific beliefs and finding ways to get them to understand and appreciate the scientific method.

I figured the answer to "why" is that people want certainty, which things like religion and pseudoscience can give them the illusion of. Science is the best tool we have of approaching the truth, but it's imperfect and it knows that.

But I'm not a very empathetic person. I try to be, but there are some things I just have to accept without understanding.

To get things started, let's figure out what we mean by “pseudoscience.” Unfortunately, there’s no universally agreed-upon definition for us to turn to, and the lines between science and pseudoscience can get a little blurry. For example, some people accuse super-theoretical investigations like string theory of veering into pseudoscience (I disagree, but that’s another story).

String Theory: The Universe is a big ball of string, and God is a cat.

Look. I'm allowed jokes, and that's one I'm inordinately proud of. The image of a cosmic feline batting around a stupendously huge ball of string is inherently amusing to me, and, let's face it, would explain a lot of the chaos in the universe.

Here in reality, though, string theory might well be a dead end. And that's okay. That doesn't make it pseudoscience, any more than the luminiferous ether was pseudoscience; it was just the best we could come up with when we had a more limited understanding of light.

And then there’s science that doesn’t live up to expectations. There are some bad scientists who create junk, lazy scientists who don’t do their homework, fraudulent scientists who tune their findings for a buck, and all manner of not-quite-good-enough scientific output. All of these blur the lines, too, even within disciplines that generally sit on firm foundations.

That's because scientists are, generally at least, human, and humans are subject to all kinds of fallacies, biases, desires, distractions, and yes, a certain level of darkness.

Perhaps the most obvious example of junk science was Andrew Wakefield's assertion that vaccines caused autism. While later debunked (and Wakefield defrocked), the damage had already been done. Not to mention what the whole kerfluffle said about the general public's attitude about those on the spectrum, which I imagine can be quite hurtful to those on the spectrum. (Okay, maybe I can exhibit empathy from time to time.)

An important part of the scientific method is to identify these mistakes and correct them. Unfortunately, it doesn't always happen quickly enough. And then you get branches that even I am wary of, such as nutrition science, which is notorious for going back and forth on things. (I think that's a case of things being so incredibly complicated that it's really difficult, if not philosophically impossible, to control for all possible variables.)

The word pseudoscience means “false science,” and that’s where my definition starts. Pseudoscience is a practice, a mode of investigation, that looks like science but misses the point. Or, as I like to phrase it, pseudoscience has the skin of science but misses its soul.

"Soul" is very close to the last word I'd expect a scientist to use to describe anything, but again... poetic license.

I won't continue to quote too much, but the next section makes clear that by "soul," the author is referring to the scientific method itself.

Many people around the world seek the advice of astrologers, whose practice was once considered a scientific discipline. And while astrology uses jargon and complicated mathematics, practitioners keep their methods secret and arcane; there is no community-wide accepted set of practices open to criticism and refinement.

The evolution of astrology into astronomy is a fascinating one, and I can't think of any discipline that better illustrates the history of science. I've noted before that Newton had what today would be called fringe beliefs, such as alchemy and astrology. His genius wasn't limited to being inspired by falling fruit, but that he showed the rest of the world a way to separate testable science from folklore and wishful thinking.

Just yesterday, I happened upon an article, which I shall not share, written by an astrologer. The argument in the article boiled down to lamenting that astronomers don't consult astrologers before making changes to how they classify things. It was a much-needed laugh, let me tell you.

So, after a while, the author finally gets to the question of why people believe pseudoscience. And, at least in part, it closely matches my guess, above:

Pseudoscience is seductive; it’s a counterbalance to the often cold, remote authority offered by scientists. It provides a “real” truth about the world that people may accept when scientific statements run counter to their personal or ideological beliefs.

In the spirit of the article, though, I'm not trying to interpret the wording here as confirmation of my pre-existing belief. And the article lists several other explanations, as well.

The soul of science is there to eliminate human bias as much as possible, to allow for nonintuitive answers to emerge that run counter to our expectations.

Which is precisely why I scoff at "common sense."

Humans tend to trust the word of their friends and family over distant scientists because that’s the way we’re wired. Humans tend to be swayed by a good story over a good data set.

In case you were wondering, that's where writing comes in.

As for suggestions on how to battle pseudoscience, that section's in there, as well.

Evidence has repeatedly shown that simply shoving data in peoples’ faces doesn’t work to change their minds. Neither does simply telling somebody they’re wrong and leaving it at that (to be honest, that strategy rarely works on me, either).

Nor me. As much as I try to keep an open mind about things, I can be just as stubborn as anyone when it comes to data that conflicts with my pre-existing beliefs. Like, recently, some article came out that asserted that alcohol is bad for you at any dosage. My first instinct was denial. And then, realizing my own hypocrisy (we all have hypocrisies), I concluded that, even if the science is sound (which is always in question), physical health isn't the last word on anything; you have to take into account quality of life, not just quantity. What use would it be to live to 100 if you have to give up everything that truly makes life worth living?

I don't doubt the facts, at least not any more than I doubt a lot of facts. Just the underlying assumptions.

I have a personal rule: Unless someone asks me directly for my opinion, I don’t offer it.

Shit, if I had that rule, this blog would be a tumbleweed wasteland.

Instead, I try to practice what’s known as radical empathy. This is empathy given to another person without any expectation of receiving it back in return. I try to see the world through someone else’s eyes and use that to find common ground.

Now that is, in my humble opinion, a thing worth striving for. If I could just remember to apply it in the moment.

I've railed on related topics in here, repeatedly. Most notably, in my "space aliens" rants. It's important, I think, to remember that people believe stuff because it brings them some benefit: comfort, peace, whatever. Something to help them sleep at night. And I think most, if not all, of us want the same benefits, ultimately; we just take different routes to the napping couch.

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