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.

August 1, 2021 at 12:16am

#1014825

If You Choose Not To Decide, You Still Have Made A Choice

Great... it's Hangover Night and the random numbers give me this one. Oh well, I have no choice but to post it. Just don't expect much in the way of commentary.

The clockwork universe: is free will an illusion?
A growing chorus of scientists and philosophers argue that free will does not exist. Could they be right?

Oh hey look, a rare example of the answer to a headline question not being "No."

But because it's philosophy, it's not necessarily a "yes" either.

It isn’t unheard of for philosophers to receive death threats.

Just in case you were thinking, "Oh, I think I'll become a philosopher. Sounds like a nice safe profession."

They argue that our choices are determined by forces beyond our ultimate control – perhaps even predetermined all the way back to the big bang – and that therefore nobody is ever wholly responsible for their actions.

That last bit doesn't necessarily follow from the first bit. Coyne (he gets quoted later in the article) distinguishes between moral responsibility and practical responsibility. But he's a biologist, not a philosopher.

“This sort of free will is ruled out, simply and decisively, by the laws of physics,” says one of the most strident of the free will sceptics, the evolutionary biologist Jerry Coyne. Leading psychologists such as Steven Pinker and Paul Bloom agree, as apparently did the late Stephen Hawking, along with numerous prominent neuroscientists, including VS Ramachandran, who called free will “an inherently flawed and incoherent concept” in his endorsement of Sam Harris’s bestselling 2012 book Free Will, which also makes that argument.

This is a rare case of someone whose writings I follow showing up in a major news publication (yes, I consider The Guardian a major news publication). I've linked to Coyne's blog here before, probably mostly for his (to me) decisive takedowns of panpsychism. He's had quite a bit to say on the subject of free will in there, too.

Arguments against free will go back millennia, but the latest resurgence of scepticism has been driven by advances in neuroscience during the past few decades. Now that it’s possible to observe – thanks to neuroimaging – the physical brain activity associated with our decisions, it’s easier to think of those decisions as just another part of the mechanics of the material universe, in which “free will” plays no role.

And this is why it's more than a philosophical question: there's science backing it up. It's also why scientists get to have an informed opinion on the topic. I should note, however, that it's not necessarily settled science.

Arguably, we would be forced to conclude that it was unreasonable ever to praise or blame anyone for their actions, since they weren’t truly responsible for deciding to do them; or to feel guilt for one’s misdeeds, pride in one’s accomplishments, or gratitude for others’ kindness.

"Arguably" is right, because if there is no free will, we also can't help feeling gratitude, guilt, or pride.

The conviction that nobody ever truly chooses freely to do anything – that we’re the puppets of forces beyond our control – often seems to strike its adherents early in their intellectual careers, in a sudden flash of insight. “I was sitting in a carrel in Wolfson College [in Oxford] in 1975, and I had no idea what I was going to write my DPhil thesis about,” Strawson recalled. “I was reading something about Kant’s views on free will, and I was just electrified. That was it.” The logic, once glimpsed, seems coldly inexorable. Start with what seems like an obvious truth: anything that happens in the world, ever, must have been completely caused by things that happened before it. And those things must have been caused by things that happened before them – and so on, backwards to the dawn of time: cause after cause after cause, all of them following the predictable laws of nature, even if we haven’t figured all of those laws out yet. It’s easy enough to grasp this in the context of the straightforwardly physical world of rocks and rivers and internal combustion engines. But surely “one thing leads to another” in the world of decisions and intentions, too. Our decisions and intentions involve neural activity – and why would a neuron be exempt from the laws of physics any more than a rock?

That's about all I'm going to quote from the article because, like I said, hangover. But that last paragraph pretty much sums up the argument; the rest is commentary. And if you're sitting there going "Well, that's utter bullshit," well, I suggest you go to the link and read the whole thing. I mean, I don't blame you -- you can't help it -- but part of who we are is our experiences and memories thereof, all of which contribute to our decisions, including on what to believe and not. And articles are part of those experiences and memories.

I will say this, though: the opposite of "free will" isn't necessarily "determinism;" the article goes into a brief discussion of quantum effects, which are by nature unpredictable. And then there's chaos theory, which has a specific meaning in physics, having to do with the unpredictability of certain systems, such as the weather, past a certain point. And I'm betting the brain is at least as complex as a weather system.

But the most important corollary, in my estimation, has to do with moral responsibility, and the article delves into that at length.

So there it is. Hell of a way to start a new month, but what can I say? I'm a slave to the random number generator.

Because I choose to be.

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