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.

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.

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.

2 Like

February 27, 2023 at 12:51pm

#1045623

What You Will

Ugh. I would get this weighty topic on a Monday.

Why the Classical Argument Against Free Will Is a Failure
Despite bold philosophical and scientific claims, there’s still no good reason to doubt the existence of free will.

There's no good reason not to, either. But I'll get to that.

My problem with this article is it uses the strawman fallacy. But if you subscribe, as I do, to the notion that we don't have free will, the author couldn't have written otherwise. Not then. Now, they could, once the strawman is pointed out. But they won't; I've found that some people are invested in the idea of free will, and get upset if you present evidence to the contrary.

In the last several years, a number of prominent scientists have claimed that we have good scientific reason to believe that there’s no such thing as free will — that free will is an illusion.

I've talked a lot about illusion in here. In short, let's not confuse "illusion" with emergent or bulk properties. For example, consciousness is an emergent property of our bodies, but it's, by definition (cognito ergo sum) real. But some things that people think are real are, necessarily, illusions; the obvious example of that is, well, look up any number of optical illusions online; until they're explained, they certainly seem real.

If this were true, it would be less than splendid.

Assertion without evidence.

And it would be surprising, too, because it really seems like we have free will. It seems that what we do from moment to moment is determined by conscious decisions that we freely make.

"Seems" is the key word here.

We need to look very closely at the arguments that these scientists are putting forward to determine whether they really give us good reason to abandon our belief in free will. But before we do that, it would behoove us to have a look at a much older argument against free will — an argument that’s been around for centuries.

The older argument against free will is based on the assumption that determinism is true.

And that's the strawman here. I'm sure there are scientists who still accept determinism, but once quantum theory began to take shape about a century ago, all that went right out the window (the article notes this later). I won't copy their version of the strawman here, but it's there at the link if you want to read it.

There’s a big problem with the classical argument against free will. It just assumes that determinism is true.

On this, we can agree. But again, no one's promoting the classical argument anymore. Or, at least, very few are.

The idea behind the argument seems to be that determinism is just a commonsense truism.

Which is another reason why common sense is neither. Another person's "common sense" might lead them to conclude that we do have free will. After all, as the article noted up there, it certainly seems as if we do.

Quantum mechanics has several strange and interesting features, but the one that’s relevant to free will is that this new theory contains laws that are probabilistic rather than deterministic.

True.

We can understand what this means very easily.

False. No one understands quantum physics.

But if this is right, then it means that at least some physical events aren’t deterministically caused by prior events. It means that some physical events just happen.

This is the crux of the issue, though. These "physical events that just happen" include the quantum effects in your brain. It's not hard to understand that the result of this can be some truly random firing of neurons, which in turn affects our thoughts and/or actions. Despite metric shit-tons of ink wasted on "you can control your quantum reality" bullshit pseudoscience books, it's really the other way around.

And so we don’t know whether all physical events are completely caused by prior events. In other words, we don’t know whether determinism or indeterminism is true.

And as regards free will or lack thereof, it doesn't matter. Any quantum effects, inside or outside the brain, that we observe or are otherwise affected by, inform our perceptions, which in turn inform our decisions.

But now notice that if we don’t know whether determinism is true or false, then this completely undermines the classical argument against free will.

Yes, but only the classical argument.

Now, to be fair, the article goes on to present that as an argument, along with a more philosophical argument. I take no issue with they way they're both described. But there is no good reason to devote most of the article to debunking determinism, as that has already been done, and thoroughly.

My own view is that neither of these new-and-improved arguments succeeds in showing that we don’t have free will. But it takes a lot of work to undermine these two arguments. In order to undermine the scientific argument, we need to explain why the relevant psychological and neuroscientific studies don’t in fact show that we don’t have free will. And in order to undermine the philosophical argument, we need to explain how a decision could be the product of someone’s free will — how the outcome of the decision could be under the given person’s control — even if the decision wasn’t caused by anything.

Yes. "It takes a lot of work." Because they're good arguments. I followed the same arguments this author (who is a professor, but let's not add "argument from authority" to our list of sins here; he's a philosophy professor) describes and came to a different conclusion.

Well, I guess in the end, we can agree on one thing: More study is necessary.

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