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 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.

December 1, 2022 at 12:02am

#1041189

Never Mind

I don't have a lot to say about today's article from The Guardian; I went to the movie theater yesterday (review below) and I'm worn out from all the sitting, drinking, and popcorning.

Is the body key to understanding consciousness?

Most headline questions are answered "no," but this one's not so simple. Besides, the brain is part of the body.

In 2018, billionaire Silicon Valley entrepreneur Sam Altman paid a startup called Nectome $10,000 to preserve his brain after he dies and, when the technology to do so becomes available, to upload his memories and consciousness to the cloud.

Cheaper than most funerals.

This prospect, which was recently popularised in Amazon Prime’s sci-fi comedy series Upload, has long been entertained by transhumanists.

People who actually read science fiction, like me, have been exposed to the concept for far longer than transhumanists, or even Amazon, have been around.

Although theoretically possible, it is rooted in the flawed idea that the brain is separate from the body, and can function without it.

How, exactly, is it "theoretically possible?" I've never heard a compelling reason why it should be.

The idea that the mind and brain are separate from each other is usually attributed to the 17th-century mathematician and philosopher René Descartes, who believed that the body is made of matter, and the mind of some other, non-physical substance.

Remember a couple entries ago I said Descartes was "right about a lot of things, but...still subject to human biases and observational inconsistencies?" This is an example of those things.

Besides, mind/body duality goes back way before Descartes, though as I understand it he codified it somewhat.

Modern brain research rejects the distinction between the physical and the mental. Most neuroscientists agree that what we call “the mind” is made of matter. The mind is hard to define, but the consensus now is that it emerges from the complex networks of cells in the brain.

I don't know about "made of matter," but there's a lot I don't know about this.

But most people still view the mind and brain as being distinct from the body.

It's a useful metaphor to think of mind and body as separate, but that's all it is: a metaphor. Like thinking of atoms as if they were little solar systems.

Being conscious does not just mean having awareness of the outside world. It means being aware of one’s self within one’s surroundings. The way we experience our body is central to how we perceive our self.

If you want to get really philosophical about it, you can go off on how the "outside world" is filtered through our senses and thus an inaccurate representation of the actual "outside world;" it's all really inner experience. As interesting as that may be to contemplate, it's irrelevant.

Phantom limbs are a striking demonstration of the importance of the body for self-consciousness.

The article then goes into how that phenomenon relates to consciousness, and also gets into body dysmorphias, but what I found most relevant to writers is the bit about how people didn't believe in phantom limbs at first, and it took a fictional story to bring it into public awareness. I mean, behold the power of fiction, right?

So I'm skipping over a lot, but in the end:

If self-consciousness is based in bodily awareness, then it is unlikely that a lab-grown “mini-brain” could ever become conscious, as some ethicists have claimed.

I always wondered about that aspect of consciousness "uploads." Of course, you have people thinking we're all uploads, and that the universe is a big computer simulation, but that raises way more questions than it answers.

By the same token, transhumanists’ claim that we will one day gain immortality by uploading our brains to supercomputers will probably always be science fiction.

I'd be very cautious about saying something will "always" be science fiction. We can be quite clever in figuring stuff out and inventing things. But I would bet that it's a very, very long way away.

One-Sentence Movie Review: Black Panther: Wakanda Forever:

I suppose it would have been asking too much for this sequel to be as good as or better than the first one; it certainly didn't suck (and was very respectful to the memory of Boseman), but it's a little incoherent in parts and significantly short on action for a MCU movie.

Rating: 3.5/5

5 Like • Comment • View 1 comment

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