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 8 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 10 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.

June 29, 2022 at 12:03am

#1034432

Gut Feeling

You know how people are always saying "trust your gut?" Well...

When Gut Bacteria Change Brain Function
Some researchers believe that the microbiome may play a role in regulating how people think and feel.

Maybe that's what your gut wants you to think.

By now, the idea that gut bacteria affect a person’s health is not revolutionary. Many people know that these microbes influence digestion, allergies, and metabolism.

I mean, the digestion part at least couldn't have been a surprise.

The trend has become almost commonplace: New books appear regularly detailing precisely which diet will lead to optimum bacterial health.

Most of which only contribute to authors' financial health. Not that there's anything inherently wrong with that, but there's a whole lot of nonsense out there when it comes to nutrition.

But these microbes’ reach may extend much further, into the human brains.

*cue evil laugh*

Scientists have found evidence that this assemblage—about a thousand different species of bacteria, trillions of cells that together weigh between one and three pounds—could play a crucial role in autism, anxiety, depression, and other disorders.

*record scratch*

Autism, while technically labeled a "disorder," isn't even in the same ballpark as those other things. I know I'm no expert on the subject, but there's a whole range of behavior labeled as "autism" (which is why it's called a spectrum) and I highly doubt that every diagnosis has the same root cause. Nor is it always a thing to be cured.

Still, a good portion of the article deals with autism, and while I take issue with its description of the condition, it's still some interesting stuff.

For decades, doctors, parents, and researchers have noted that about three-quarters of people with autism also have some gastrointestinal abnormality, like digestive issues, food allergies, or gluten sensitivity.

For a while there, approximately three quarters of the population of the developed world claimed "gluten sensitivity," or at least got the idea stuck in their head that gluten was somehow bad. The sound of the word doesn't help, I'll admit. While that little fad seems to finally be dying out, the result is I automatically don't believe anyone who says they're "sensitive" to gluten (celiac disease, I tend to believe). Similarly, a whole lot of people who claim "allergies" simply don't like certain foods and don't want to seem rude about it. I know I should take it at face value when people mention these things, but it's kind of like when you get those fake service animal tags for your emotional support crocodile: it ruins things for people with legitimate needs.

But okay, for the purposes of this article I'll assume actual diagnoses of these issues.

This recognition led scientists to examine potential connections between gut microbes and autism; several recent studies have found that autistic people’s microbiome differs significantly from control groups.

Yeah, I just hope they got the causality arrow pointing in the right direction there.

Exactly how the microbes interact with the illness—whether as a trigger or as a shield—remains mostly a mystery.

Again, "illness?" And I guess they still don't have that arrow pointing in the right direction after all.

Scientists have also gathered evidence that gut bacteria can influence anxiety and depression. Stephen Collins, a gastroenterology researcher at McMaster University in Hamilton, Ontario, has found that strains of two bacteria, lactobacillus and bifidobacterium, reduce anxiety-like behavior in mice... In one study, he and his colleague collected gut bacteria from a strain of mice prone to anxious behavior, and then transplanted these microbes into another strain inclined to be calm. The result: The tranquil animals appeared to become anxious.

Around this point in reading the article, something occurred to me:

This is mind control.

As I've noted before, science is value-neutral. It can be used for good or nefarious purposes, or both (I brought up nuclear fission as an example recently). If I were inclined to write science fiction, I'd probably put in a story something about using manipulation of intestinal microbiome to make people more pliable for their corporate overlords. Oh, wait, I am inclined to write science fiction.

It's great that they're learning shit about the gut-brain connection, something I've long suspected ever since my mother, who had digestive issues, declined and died of unspecified dementia. That, of course, was a sample size of one; anecdote, not data. And it can absolutely be used for good: calm anxiety, maybe help some more withdrawn people become more social, that sort of thing -- if that's what they want. But I can absolutely see this being used for crowd control.

A paper published in the May 2015 issue of Psychopharmacology by the Oxford University neurobiologist Phil Burnet looked at whether a prebiotic—a group of carbohydrates that provide sustenance for gut bacteria—affected stress levels among a group of 45 healthy volunteers. Some subjects were fed 5.5 grams of a powdered carbohydrate known as galactooligosaccharide, or GOS, while others were given a placebo.

"I never put anything in my body that I can't pronounce." Stop with the willful ignorance. If you'd only seen the word "quinoa" written, not spoken, you would probably pronounce it wrong.

In this case, it's ga-lac-to-ohl-ih-go-saccharide. (We lost something when we quit putting a diaresis, or umlaut, over a second or third vowel to indicate that there's a glottal stop in between, as with naïve). There. Now you can pronounce it. Let's not be afraid of chemicals. Practically everything is a chemical.

Perhaps the most well-known human study was done by Mayer, the UCLA researcher. He recruited 25 subjects, all healthy women; for four weeks, 12 of them ate a cup of commercially available yogurt twice a day, while the rest didn’t.

Thus perpetuating the idea that men don't eat yogurt.

To Mayer’s surprise, the results, which were published in 2013 in the journal Gastroenterology, showed significant differences between the two groups; the yogurt eaters reacted more calmly to the images than the control group. “The contrast was clear,” says Mayer. “This was not what we expected, that eating a yogurt twice a day for a few weeks would do something to your brain.”

And no, it won't make you trans.

It’s not yet clear how the microbiome alters the brain. Most researchers agree that microbes probably influence the brain via multiple mechanisms.

Hence why we do more science on the stuff.

This interconnection of bugs and brain seems credible, too, from an evolutionary perspective. After all, bacteria have lived inside humans for millions of years. Cryan suggests that over time, at least a few microbes have developed ways to shape their hosts’ behavior for their own ends. Modifying mood is a plausible microbial survival strategy, he argues that “happy people tend to be more social. And the more social we are, the more chances the microbes have to exchange and spread.”

Just to be clear: this is speculation. Also, "humans" have probably not existed for "millions of years." Granted, our evolutionary predecessors had a gut biome, but I'm willing to bet it goes back way more than a few million years; the concept of a digestive tract has probably been around for over half a billion years (quick google search), and bacteria since almost the very beginning of life, over four billion years. This is their world; we're just living in their tiny shadows.

As scientists learn more about how the gut-brain microbial network operates, Cryan thinks it could be hacked to treat psychiatric disorders. “These bacteria could eventually be used the way we now use Prozac or Valium,” he says.

See? Mind control. It's not like no one has misused Prozac or Valium to keep people compliant.

Another thing occurred to me while reading this article. Well, actually, it occurred to me a while back, but I got to think about it again because I'm currently bingeing The Umbrella Academy which (not much of a spoiler here) involves time travel.

Time travel may be impossible. But for the sake of argument, say it's not. You know how I know it didn't happen? Doesn't happen. Won't happen. No, not because we'll have to invent new verb tenses. Because every living thing evolves, including bacteria, which includes our microbiome. There's a famous and often-referenced Ray Bradbury short story called A Sound of Thunder, which (it's not a spoiler if the damn story is older than I am) deals with the changes made to the timeline by someone inadvertently killing a random butterfly while... I don't remember, running from a dinosaur or something. Look, it's been a while, okay? Guy goes back to the future and the world's changed. Point is, it wouldn't take a butterfly (did they even coëxist with dinosaurs? Can't be arsed to look it up right now, and it hardly matters because Bradbury was awesome anyway); leaving some of today's gut bacteria lying around in the Jurassic or whatever would leave obvious traces on the evolutionary record.

So, either time travel is impossible, or we wipe ourselves out before inventing it. Or both.

In other words, if you did manage go back in time, you wouldn't have to kill a butterfly to change the present to something unrecognizable.

All you'd have to do is take a shit.

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