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 12, 2022 at 12:01am

#1041703

Colony

I don't have a lot to say about today's article, but it ties in with some other entries I've made; specifically, about the question of mind-body duality—or lack thereof.

The Gut Microbiome Helps Social Skills Develop in the Brain
New research in fish suggests that gut microbes can have a crucial early influence on the brain’s social development.

There are some things that are a bit problematic, but I'll get to that.

Two recent papers have shown that during a critical early period of brain development, the gut’s microbiome — the assortment of bacteria that grow within in[sic] it — helps to mold a brain system that’s important for social skills later in life.

One thing I've come to appreciate about Quanta: they don't usually bury the lede. The occasional typo, I can understand. Happens to all of us.

Scientists found this influence in fish, but molecular and neurological evidence plausibly suggests that some form of it could also occur in mammals, including humans.

So take it as a preliminary result, not settled science. If you corner someone at a holiday party and go, "You know, your gut microbiome has an effect on your social skills," a) you'd be jumping the gun and b) you'd better get your microbiome checked out, because your social skills suck.

In a paper published in early November in PLOS Biology, researchers found that zebra fish who grew up lacking a gut microbiome were far less social than their peers with colonized colons...

Snort.

...and the structure of their brains reflected the difference.

I mean, that's a big deal from what I can tell. I've suspected a connection between gut health and mental health for a long time (my mom had problems with both). Again, maybe or maybe not applicable to humans, but seeing a physical difference in the brain because of something in the intestines—something that's even a different genome—I can't help but think is an important finding.

In recent decades, scientists have come to understand that the gut and the brain have powerful mutual influences. Certain types of intestinal ulcers, for example, have been linked to worsening symptoms in people with Parkinson’s disease. And clinicians have long known that gastrointestinal disorders are more common in people who also have neurodevelopmental disorders, such as ADHD and autism spectrum disorder.

This is the problematic part. Autism isn't necessarily a "disorder," no matter what the DSM might have to say about it. But whatever the label, the important thing is the potential link between microbiome and behavior.

How these anatomically separate organs exert their effects, however, is far less clear.

Finding a correlation is one thing. Figuring out the mechanism is quite another, and it sounds like they haven't done much yet in that regard. There's some discussion of possible future studies later in the text.

Again, though, mostly I just saved this to link because it hints at some deeper truth about the physical and the mental, especially in terms of those microbes that are just as much a part of us as our own cells.

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