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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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.

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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.

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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.

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
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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

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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.

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February 21, 2021 at 12:03am

#1004906

Philosophy of Science

This one's a little on the deep side, and I can't claim to understand all of it. But I'm putting it out here anyway in the hope that someone will get something out of it.

Questioning Truth, Reality and the Role of Science
In an era when untestable ideas such as the multiverse hold sway, Michela Massimi defends science from those who think it hopelessly unmoored from physical reality.

Incidentally, my browser's spellchecker doesn't recognize the legitimacy of "untestable." There's probably a metaphor in there somewhere, but I'm entirely too sober to tease it out right now.

It’s an interesting time to be making a case for philosophy in science. On the one hand, some scientists working on ideas such as string theory or the multiverse — ideas that reach far beyond our current means to test them — are forced to make a philosophical defense of research that can’t rely on traditional hypothesis testing. On the other hand, some physicists, such as Richard Feynman and Stephen Hawking, were notoriously dismissive of the value of the philosophy of science.

I expected better from Feynman, and Hawking always struck me as having something akin to a philosophical streak.

That value is asserted with gentle but firm assurance by Michela Massimi, the recent recipient of the Wilkins-Bernal-Medawar Medal, an award given annually by the U.K.’s Royal Society. Massimi’s prize speech, delivered earlier this week, defended both science and the philosophy of science from accusations of irrelevance. She argues that neither enterprise should be judged in purely utilitarian terms, and asserts that they should be allies in making the case for the social and intellectual value of the open-ended exploration of the physical world.

Personally, I think judging anything on "purely utilitarian terms" misses an important part of the human experience. Which is not to say I don't have a utilitarianist perspective on a lot of things; I just don't think that's the only viewpoint worthy of consideration.

The article goes on to interview Massimi, and while I won't quote most of it here, it's worth seeing both the questions and her answers. But this is one question that struck me as interesting -- not for the answer, appropriately enough, but for the question itself:

One criticism made is that science moves on, but philosophy stays with the same old questions. Has science motivated new philosophical questions?

Her answer is important from one perspective, but I'll offer another point of view: Yes, science motivates new philosophical questions. It does this all the time. These discussions even seep into popular culture. The very first true science fiction book, Shelley's Frankenstein, delves into the philosophical repercussions of applied science (albeit from a purely fictional perspective). The question isn't answered, of course -- we're still asking it, in stories, right up to the present day. And undoubtedly beyond.

Without science offering the possibility of creating something new in the world, there would be little reason to ask those questions on a serious level.

Later, a part of her answer to another question goes like:

Obviously it is not the job of philosophers to do science, or to give verdicts on one theory over another, or to tell scientists how they should go about their business. I suspect that some of the bad press against philosophers originates from the perception that they try to do these things. But I believe it is our job to contribute to public discourse on the value of science and to make sure that discussions about the role of evidence, the accuracy and reliability of scientific theories, and the effectiveness of methodological approaches are properly investigated.

Ethics is in the purview of philosophy. Perhaps not so much in physics, but other branches of science, particularly biology and related fields, absolutely has to consider ethics. Animal testing? Hell, human testing? The potential for cloning? The quote from the original Jurassic Park movie comes to mind: "Your scientists were so preoccupied with whether they could, they didn't stop to think if they should."

But this is, for me, the most important point she makes:

In this respect, I see philosophy of science as delivering on an important social function: making the general public more aware of the importance of science. I see philosophers of science as public intellectuals who speak up for science, and rectify common misconceptions or uninformed judgments that may feed into political lobbies, agendas and ultimately policy-making.

This article is from 2018, and since then I think at least one major thing has happened that demonstrates the value of science in the public sphere. While there is certainly research that is, at best, questionable -- nutrition science comes to mind -- I think one of the greatest issues facing us right now is general distrust of science and the dismissal of experts, in science or other fields, as no more useful than your uncle's deranged Failbook posts.

So yes, by all means, keep up the philosophy. I won't understand all of it. But I don't have to.

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