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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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.

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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.

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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.

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

August 14, 2020 at 12:01am

#990710

Go Home, Evolution. You're Drunk.

Science. Drinking. A drinking metaphor for science.

https://theconversation.com/evolution-that-famous-march-of-progress-image-is-jus...

Evolution: that famous ‘march of progress’ image is just wrong

Tough to reproduce images here; you'll just have to click on the link to see the one -- but it's enshrined in world consciousness, so you probably already know it: the illustration of how an ancient ape gradually acquired more humanlike characteristics until humanity is attained.

Evolution explains how all living beings, including us, came to be.

This should be uncontroversial. Yet, somehow, it is not. What is controversial is how evolution works at the subcellular level.

It would be easy to assume evolution works by continuously adding features to organisms, constantly increasing their complexity.

That's the popular misconception, sure. What's missing in this picture is that there are living things with a more complex genome than our own, and that every organism on the planet has been subject to evolutionary pressures for exactly as long as we humans have.

Yet this is one of the most predominant and frustrating misconceptions about evolution.

The article links to another article with a list of five common misconceptions, including the idea that we're descended from monkeys. It's worth at least a glance.

Our results show that the origins of major groups of animals, such as the one comprising humans, are linked not to the addition of new genes but to massive gene losses.

The obvious thing one might ask is: why, then, do we exhibit more complex behavior than, say, a caterpillar? Now, I'm no expert, and the article doesn't cover this, but I think of it like this: more genes doesn't automatically translate to greater cranial capacity.

Another fairly obvious question would be: If we evolved through loss of genes, was the first organism incredibly complex? This article does handle this, sort of.

The evolutionary biologist Stephen Jay Gould was one of the strongest opponents of “the march of progress”, the idea that evolution always results in increased complexity.

That's the controversy I spoke of before: whether complexity necessarily increases.

As an aside, one ignorant (either willfully or naïvely) objection to the idea of evolution that I've heard is: "it would violate the second law of thermodynamics." This demonstrates a misunderstanding of both thermodynamics and evolution. It is true that, in a closed system, everything proceeds in the direction of disorder (entropy). But it is also true that the Earth isn't a closed system; there's a giant fusion reactor close enough to provide it with a great deal of energy.

The second law is not violated. The system, Earth and accursed daystar, produces more entropy than it would if it weren't for life. In my darker moments, I sometimes think that this is the primary purpose of life: to increase universal entropy so the Universe experiences its heat death that much sooner.

But then I sober up and remember that there is no purpose of life; it just is.

And so we come to the drinking part:

In his book Full House (1996), Gould uses the model of the drunkard walk. A drunkard leaves a bar in a train station and clumsily walks back and forth over the platform, swinging between the bar and the train tracks. Given enough time, the drunkard will fall in the tracks and will get stuck there.

What's the point of the analogy? Read on.

The platform represents a scale of complexity, the pub being the lowest complexity and the tracks the maximum. Life emerged by coming out of the pub, with the minimum complexity possible. Sometimes it randomly stumbles towards the tracks (evolving in a way that increases complexity) and other times towards the pub (reducing complexity).

Been there.

No option is better than the other. Staying simple or reducing complexity may be better for survival than evolving with increased complexity, depending on the environment.

And that's what evolution is: survival. Not necessarily "of the fittest," but finding an environmental niche to exploit. Drunkenly, without purpose.

But in some cases, groups of animals evolve complex features that are intrinsic to the way their bodies work, and can no longer lose those genes to become simpler - they become stuck in the train tracks. (There are no trains to worry about in this metaphor.)

The lack of trains is a big relief for us drunks, but I'm not sure it's entirely true. Metaphorically speaking. Just ask the woolly mammoth -- oh wait, you can't because it got hit by a train.

Together with Peter Holland from the University of Oxford, we looked into how genetic complexity has evolved in animals.

The article goes on to explain the science involved, and how some extant species evolved through losing or gaining genetic complexity. No need to reproduce it here; besides, there's another graphic.

Our results confirm the picture given by Stephen Jay Gould by showing that, at the gene level, animal life emerged by leaving the pub and making a large leap in complexity. But after the initial enthusiasm, some lineages stumbled closer to the pub by losing genes, while other lineages drifted towards the track by gaining genes. We consider this the perfect summary of evolution, a booze-induced random choice between the bar and the train track. Or, as the internet meme says, “go home evolution, you are drunk”.

So that's worth including a bonus link

, because WTF, Evolution? is never not funny to me.

Anyway, it's not often I get to combine two of my favorite subjects, so I just wanted to share this one.

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