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 25, 2021 at 12:02am

#1023640

Evolution, My Ass

As I pointed out in "Ain't Talkin' 'bout Love"

, sometimes I get a random link that makes sense with whatever day it is.

Today is not one of those times.

Which came first, the butt or the mouth? New research gives an answer
It's a chicken-and-the-egg question, but "which came first?" might not be the right way to think about it

Given enough time, I'd probably find a way to relate this article to today being Christmas, probably involving some pun on "ass," as in "And Mary rode Joseph's ass all the way to Bethlehem." But, as I like to say, I can't be arsed. Besides, Christmas is just another day for me, and it's always relevant to talk about assholes.

For instance, a couple of months ago, I posted another article about buttholes. It's here: "The End"

Today's article is similar, but shorter and covers a few different aspects of the digestive tract.

First, though, let me address the "chicken-and-the-egg question" mentioned in the subheading above: I don't know why this is even still in debate; eggs existed long before chickens did. I'm sure you've seen the fossilized dinosaur embryo they found recently. If not, here.

Like many people, you may have spent hours pondering the question of which came first, the anus or the mouth?

I assure you, it keeps me up at night.

Most mammals have one major tract for solids going "in," and one for "out"; One for eating, a mouth, and another for defecation, typically considered the anus.

One of these days, maybe I'll try to find out why they start with "most." But I'm really not all that interested in searching for counterexamples.

Some scientists believe that, in evolutionary history, the mouth developed first, based on how embryos develop in the womb. Others dispute this theory, arguing that the anus developed first based on embryo development in different animals. Recently, a paper published in Nature reviewed these theories to get to the bottom of it.

These are legitimate questions because it's not like soft tissue is often preserved in the fossil record.

There's a concept in evolutionary history called Haeckel's Biogenetic Law, developed in 1866 by Ernst Haeckel. Haeckel believed that an embryo's developmental stages provide information about the adult stages of that organism's ancestors. If an embryo looks different at different stages of development, those differences correspond to how the ancestors of the species looked at adult stages.

This "law" is often expressed as "ontogeny recapitulates phylogeny," but it's not really a law and has been shown to not be the case.

Here's the important part, though:

Researchers in the Nature study attempted to review all plausible theories by examining the blastopore, tissues, nerves and bands, and other pieces that make up the development of early human embryos. The three major theories they examined, for the evolution of mouths and anuses in bilaterians (which includes humans), were as follows:

1. The blastopore becomes the mouth, and the anus develops secondarily.
2. The blastopore becomes the anus, and the mouth develops secondarily.
3. The blastopore divides into the mouth and the anus through the fusion and separation of a tubular gut.

They found that the evidence supports the third scenario, in which the blastopore elongates and closes laterally at both ends, giving rise to the mouth and the anus simultaneously (or as simultaneous as embryonic development can get).

And maybe that explains why, with some people, it's hard to tell whether they're talking out of their mouth or out of their ass.

Anal openings first appeared around 550 million years ago, around the time of the first worm-like creatures.

That sounds like a long time, but consider: Life on Earth is probably about four billion years old, eight times longer than the time since the first asshole appeared. Granted, until about 600 million years ago, there was no multicellular life, and single-celled organisms have different means of ingestion, digestion, and excretion.

That still leaves about 50 million years between the first multicellular organism and the time when the first shit was taken. And 50 million years is certainly a long time in human terms The dinosaurs disappeared about 60 million years ago, for instance, and recognizable humans have been around for less than a million years.

So here we are, half a billion years later, surrounded by assholes.

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