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
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.

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October 25, 2020 at 12:01am

#996685

Networking

I've noted this article's thesis in a previous blog entry. Good to know it's not just my drunk ass that picks up on these things.

The Strange Similarity of Neuron and Galaxy Networks
Your life’s memories could, in principle, be stored in the universe’s structure.

I'm not sure that the subheading there is strictly accurate. But the physical similarity between the mid-scale structure of the universe and the neural mapping of the brain is too similar to be ignored. Not that there's anything supernatural at play here; patterns often repeat themselves in nature for perfectly ordinary reasons, such as the way we find Fibonacci's number in all manner of structures... but that's not important right now.

We have predicted that the void-filament boundary is one of the most complex volumes of the universe, as measured by the number of bits of information it takes to describe it.

This got us to thinking: Is it more complex than the brain?

I suspect it has to do with exactly how complexity is defined. The speed of light puts a hard limit on how information can be transmitted through such megastructures, whereas the speed of information in the brain, while much slower than the speed of light, is fast enough for our purposes.

The first results from our comparison are truly surprising: Not only are the complexities of the brain and cosmic web actually similar, but so are their structures. The universe may be self-similar across scales that differ in size by a factor of a billion billion billion.

Physical similarity, however, doesn't automatically translate to similarity of utility.

The article, while fascinating, doesn't really lend itself to a lot more quoting, especially since my own gray matter is perhaps just a bit disordered right now, thanks to a generous infusion of simple molecules involving carbon, hydrogen and oxygen. But it's worth reading, if only as an example of what can happen when you get two vastly different scientific specialties together to compare notes.

It's out of such things that creativity arises, I think -- that finding of similarities in things that are, at first glance, disparate.

But I will quote one more bit, because it relates to a phenomenon I've mentioned in here fairly recently.

Is the apparent similarity just the human tendency to perceive meaningful patterns in random data (apophenia)? Remarkably enough, the answer seems to be no: Statistical analysis shows these systems do indeed present quantitative similarities.

Now, I should probably emphasize that no, this doesn't mean that there's some mystical or metaphysical conclusion to be reached here. But, materialist that I am, I'm also a writer (of sorts) and this kind of thing is one reason why science fiction exists. Could the universe itself have a kind of consciousness? My instinct is to scoff at such an idea, but really, no one really knows what consciousness is, even though one must possess it in order to even ask the question.

But soon it will be time for me to experience its opposite for a few hours. I just thought I'd leave this here for your neural networks to contemplate.

Edit: After completing this entry, as I usually do after one of these entries, I checked the website Astronomy Picture of the Day, which updates around midnight EST. In another cosmic coincidence, today's picture is of a computer simulation that closely resembles the structure of the universe on these cosmic scales, and so I present it

as another illustration of the superficial similarity between the universe and a neural net.

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