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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Show Recent Comments

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 26, 2022 at 12:01am

#1042151

Symbiosis

Today's article is mostly informational; I don't have the background to really comment on it.

What Is the Human Microbiome, Exactly?

But then, I also don't have a background in making movies, but I comment on them anyway. Review below.

Anyway, the article, which is from 2019; because it engages with current science, something might have changed in the last few years:

Are you an ecosystem? Your mouth, skin, and gut are home to whole communities of microscopic organisms, whose influence on your body ranges from digesting your food to training your immune system and, possibly, impacting your mood and behavior.

I've discussed the gut microbiome in here before, but apparently our prokaryotic fellow-travelers aren't limited to the lower digestive tract.

What are these tiny tenants, and how do they change the way we think about human health, disease, and even identity?

The health questions I leave entirely to the experts. As for identity, well, that overlaps with philosophy.

The human body is made up of trillions of cells—well, trillions of human cells. Around the beginning of the 21st century, scientists learned that in fact the human body contains many trillions more microbial cells—possibly three times as many.

The history of microbiome studies goes back significantly further than this, though it's my understanding that the last 20 years or so have seen quite an expansion of knowledge.

I will also point out that some older estimates were "maybe 10 times as many," but that's outdated.

One might wonder "How can there be more microbes than human cells? Where would they fit?" Well, again, I'm no expert, but I've heard that prokaryotic cells are an entire order of magnitude smaller, on average, than eukaryotic cells (those with nuclei and mitochondria). I'm unclear on whether that's volume, area, or a more linear measurement, and I can't be arsed right now to figure it out. The only point is that they do fit. An analogy would be a jar filled with pebbles and grains of sand; there would be significantly more grains of sand by number.

It is an especially curious discovery—it has been with us, evolving, interacting, and helping to determine our fate as organisms, since before the emergence of the human species itself.

The implication there is that other larger organisms have microbiomes, too. This tracks with other things I've heard; namely, that some of the studies are done in mice. It wouldn't surprise me if most or all complex organisms have one.

We now know that the microbiome contributes a substantial amount to human growth, development, and function. Perhaps the most popular is the gut microbiome, which impacts human digestive health (this is the science behind your daily probiotic yogurt).

I remain unconvinced that probiotics work as advertised—mostly because I don't think the food makers' claims have to pass any real standards. Like labeling something "natural," it might very well be meaningless.

Aside from digestive health, some scientists are studying the relationship between the composition of the microbiome and the development of the central nervous system, and some psychologists want to take this a step further to investigate the relationship between the microbiome and phenomena like emotion, learning, and social behavior.

You might recall I recently linked an article that claimed a correlation between digestive disorders and autism. Here it is if you don't recall: "Colony"

I'll emphasize again that all of this is still preliminary.

Two biologists, Nicolae Morar and Brendan Bohannan, of the University of Oregon, recently surveyed the metaphors scientists use to talk about the microbiome (an “organ” or a “part of the immune system”) and the human-microbiome complex (a “superorganism,” a “holobiont,” or an “ecosystem”). These metaphors influence scientific understanding and can shape medical treatment.

This is really what led me to link this. Metaphors aren't limited to writing.

For example, some physicians support fecal microbiota transplantation (FMT); that is, swallowing a pill full of someone else’s poo to treat malfunction of the gut microbiome.

Ew.

You'd think they could deliver that as an enema or something. For some reason that seems less icky to me. Just goes to show that gut microbes are our friends, and with friends like those, who needs enemas?

I am by no means apologetic for making that pun.

This view is attractive because the human body and its microbiota work to do things that neither could do alone. For example, humans could not get nearly as much energy from digestion without the help of their microbiota, and the microbiota could not survive without a host.

Again, I'm not a biologist, but that sounds to me a lot like the definition of symbiosis.

In general, the problem with each metaphor is that it only captures a part of what the microbiome is and does, and so the researchers conclude that there is no best metaphor, and we need all of them to truly appreciate and understand the complexity of the microbiome and its role in our bodies.

That sort of thing also sounds familiar.

The fact that there still isn’t one “best” metaphor for grasping the microbiome might tell us something much deeper about the world—that perhaps the most promising approach to understanding it is to play with a variety of perspectives, prizing none over the others.

The thing about metaphors is that they have their limitations. Still, when it comes to science communication, it seems to me that we're stuck with them. Just remember the picture of a thing isn't the same thing as the thing.

Speaking of metaphors, I saw my traditional Christmas Day movie yesterday, as I mentioned I would.

One-Sentence Movie Review: The Whale:

This movie is an adaptation of a stage play, and it shows; there's basically no action and it mostly takes place within a single set, but that works with the subject matter, which is highly metaphorical and finally made me take Brendan Fraser seriously as an actor.

Rating: 4/5

5 Like • Comment

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