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

April 6, 2023 at 7:51am

#1047614

Mind Your Own Mind

Today, we're looking within.

Imagination is a spectrum – and 1% of people can’t mentally visualise things at all

I'm sure this comes as no surprise to at least 1% of your readers.

I saved this one because, fairly recently, it came to my attention that while I see spoken words as text in my head, not everyone does that. It may be a reason I'm pretty good at spelling: lots of practice.

When you hear someone talk, do you see the words in your mind’s eye? Or do you see what they’re saying as a movie? It’s easy to assume that the way you perceive the world is the same for everyone.

I know I've said this before, but when I was a kid, I had a deep, philosophical conversation with another kid, one which amounted to "how do I know that the color I see as 'red' is the same color as what you see as 'red?'" Much later, on the internet, such a question popped up as a profound revelation. Me, I'd spent the intervening years occasionally wondering how one would go about investigating such a thing.

The reason I bring this up is to note that I've been open to the idea that we each perceive things differently for a long time, and yet, it still sometimes surprises me.

We range from those who are “mind blind” and cannot visualise things mentally to those who have brilliant images in their mind. Some people see shapes in their mind when they hear music, or imagine colours when they see a number (a phenomenon called synaesthesia).

Yes, the article uses British spellings. Those don't usually appear in my mind when I hear words, but I can see them when someone speaks with a British accent.

There’s even a type of synaesthesia in which people’s minds run a written text on a mental ticker tape. Even though ticker tape (or subtitle) synaesthesia (TTS) was first studied in 1883 by Charles Darwin’s cousin Francis Galton, little was known about it until lately.

Yeah, I don't think it's quite like that for me. I do remember some instances of synesthesia (American spelling), like when I was a kid, each day name (Monday, Tuesday, etc.) had a different color associated with it. What those colors were, I can't describe. But ticker tape was already obsolete by the time I started reading. I saw it in action, once, when I was very young, and never since. I'd expect that a person's inner experience would be based on stuff they're familiar with; today, one might see words in their mind as text on a small screen.

Any reason to mention Darwin here other than name-dropping? Your science should stand on its own, regardless of the accomplishments of your more famous cousin. (Darwin, incidentally, had a lot of cousins. Hell. He married one.)

A study published recently, one of the first to explore this condition in more depth, found that of the 26 participants with TTS, most had additional types of synaesthesia, most commonly space-time or number-space, where they experience time or numbers as a location.

26 is hardly a compelling sample size, but considering the apparent rarity of TTS, it's understandable. I just wouldn't make any far-reaching leaps based on the study.

While many adults can imagine written words when listening to speech if asked to do so, people with TTS are different because of the ease with which it happens. In fact, some cannot stop, even when it makes it difficult to follow conversations when lots of people are talking at once. But the ability to process information from different senses at the same time is often helpful. There is an evolutionary explanation for this.

Of course there is, but it's speculative and possibly wrong.

FFS, I have to explain this again: Yes, evolution is fact. But asserting that such-and-such is the case in our minds "because our ancestors needed it on the savanna" or whatever is, at best, an interesting thought experiment, and, at worst, utter nonsense. Unless they can back it up with actual evidence.

In this case, a moment's thought might suffice to convince you that our savanna-dwelling ancestors didn't see written words when someone spoke, because writing hadn't been invented yet.

The more general informational-processing argument, okay, sure. But that would also apply to all the other animals that share, to one degree or another, our sensory array: sight, hearing, etc. And thus be a holdover from way before apes.

Okay, enough of my perennial ragging on evo-psych. Skipping that part of the article.

Except for this gem:

So if we also saw, or even smelled an animal in the scrub behind us, we could more easily determine if it was a dangerous predator we needed to escape from, or a fluffy little rabbit. As they evolved, our brains became experts in tying information from different senses together.

Which of course made my mind conjure an image of the Killer Rabbit from Holy Grail.

Researchers recently suggested fewer people are born towards the end of the low-visual-imagery spectrum. Extreme forms of aphantasia, people who do not have visual imagery at all, are rare. Less than 1% of people have this form.

While I've known about synesthesia for years, this is probably the first time I've encountered aphantasia, and I love the word, even though it describes a condition I hope I never experience.

Some research suggests we are not born with the ability to imagine. Instead, visual images emerge and develop during early childhood. This is followed by a decline in visual imagination in adulthood.

Okay, this strikes me as being like saying "we are not born with the ability to speak." Obviously, in reality, no child has ever popped out and immediately said, "Hello, Mother. Nice tits." No, we learn that later, but our capacity for speech (or imagination) is inborn. For most people.

Despite my issues with some of the claims here, I don't think there's anything inherently wrong with the basic idea: that we each have a different interior world. And, as the article notes at the end, there's a lot more to discover. We just have to begin by imagining...

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