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 6 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 8 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.

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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.

August 9, 2020 at 12:07am

#990304

Wine, Cheese, and Bread

Today I get to talk about some of my favorite things.

https://www.smithsonianmag.com/science-nature/how-cheese-wheat-and-alcohol-shape...

How Cheese, Wheat and Alcohol Shaped Human Evolution
Over time, diet causes dramatic changes to our anatomy, immune systems and maybe skin color

You aren’t what you eat, exactly. But over many generations, what we eat does shape our evolutionary path. “Diet,” says anthropologist John Hawks, of the University of Wisconsin-Madison, “has been a fundamental story throughout our evolutionary history. Over the last million years there have been changes in human anatomy, teeth and the skull, that we think are probably related to changes in diet.”

Well, that doesn't mean I'm a pizza, no, but at least on a fundamental level, we know that we're made up of atoms taken from the substances we breathe and ingest.

I do have some issues with the way this article so cavalierly uses the word "evolution." I get the feeling an evolutionary biologist would have a cogent argument against it. I'm not one, so I'm just going to go with the flow here, for now.

When mammals are young, they produce an enzyme called lactase to help digest the sugary lactose found in their mothers’ milk. But once most mammals come of age, milk disappears from the menu. That means enzymes to digest it are no longer needed, so adult mammals typically stop producing them.

Thanks to recent evolution, however, some humans defy this trend.

Another interpretation: some of us retain childlike features. As should be apparent if you've been reading my stuff.

Hawks explains why being able to digest milk would have been such a boon in the past...

Speculation without supporting evidence. I mean, if you look at the article, the explanation makes sense, but I shy away from such just-so stories on principle. The facts are what they are; the explanation for the facts is suspect.

The article then goes on to talk about grains.

Since wheat and rye became a staple of human diets, however, we've have had a relatively high frequency of celiac disease. “You look at this and say how did it happen?” asks Hawks. “That's something that natural selection shouldn't have done.”

Translation: "I don't understand how natural selection actually works."

Yet despite the obvious drawbacks of celiac disease, ongoing evolution doesn't seem to be making it less frequent. The genetic variants behind celiac disease seem to be just as common now as they've been since humans began eating wheat.

That's because evolution doesn't care if an individual gets sick or has dietary limitations. Evolution only cares if someone lives long enough to reproduce. Well, I'm speaking metaphorically. Evolution neither cares nor doesn't care. It just is. Point is, unless there's something that kills people at an early age, the genes for it happily pass on. Okay, that's another metaphor. Genes don't feel happiness.

Then the author gets into the discussion of how diet could influence skin color, over time and in populations, which is kind of a minefield these days. Keep in mind it's not promoting any shade of skin as better or worse, in terms of evolution, than another, just describing what could have influenced skin tone changes in humans.

But then you get sentences like:

Mostafavi's genetic research also revealed that some variants that actually shorten human life, like one that prompts smokers to increase their consumption above smoking norms, are still being actively selected against.

Again: evolution is concerned with reproduction. Poor health outcomes from things like smoking or alcoholism don't tend to show up until later in life, after a human has had plenty of opportunity to reproduce (whether such habits influence the portion of the process of evolution called "sexual selection," I leave as an exercise for the reader). Point is, you can have some disease that kills you at, say, 30 or 40, but it doesn't stop you from having reproduced by then, so the genes that affect the expression of said disease aren't selected against.

So if there is indeed research on that subject, I hope it's peer-reviewed, because as it's presented here it certainly doesn't inspire any confidence from me.

Overall, I expected better from Smithsonian Magazine.

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