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 3 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.

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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.

April 28, 2021 at 12:15am

#1009252

Competition

This article is a few years old now, but I haven't seen anything to contradict it in the last four years, so it's probably still current science.

New Model of Evolution Finally Reveals How Cooperation Evolves
By treating evolution as a thermodynamic process, theorists have solved one the great problems in biology.

It also addresses a few of the misconceptions about evolution, but there are a few points I'd quibble with.

The first misconception is that "survival of the fittest" necessarily equates with "survival of the strongest." Bunnies, for example, didn't get to where they are by being strong; they exist because they're famously fast and, even more famously, prolific reproducers. And they're very cute.

One of the great unanswered question in biology is why organisms have evolved to cooperate. The long-term benefits of cooperation are clear—look at the extraordinary structures that termites build, for example, or the complex society humans have created.

In other words, we know that intraspecies cooperation (and sometimes interspecies cooperation) exists, but how such cooperation evolved has been an ongoing subject of debate.

But evolution is a random process...

Uh... not really. There's a random element, sure, but chance isn't the biggest factor. I mention this because some people have used "random chance" as a specious argument against evolution. No, we humans didn't get to where we were able to invent computers merely "by chance," and bunnies didn't suddenly and randomly develop the ability to hop.

...based on the short-term advantages that emerge in each generation.

That whole sentence is problematic, but I understand that its purpose is to provide background for later arguments. It's close enough for that purpose, I think.

Of course, individuals can cooperate or act selfishly, and this allows them to accrue benefits or suffer costs, depending on the circumstances. But how this behavior can spread and lead to the long-term emergence of cooperation as the dominant behavior is a conundrum that has stumped evolutionary biologists for decades.

I'll take that as given. All I'll add is that neither cooperation nor competition is the only way for a species to evolve. I'm sure you can think of plenty of examples of both, but in many cases it's some mixture of the two.

Today, that could change thanks to the work of Christoph Adami and Arend Hintze at Michigan State University in East Lansing. They have created a simple mathematical model using well understood physical principles to show how cooperation emerges during evolution.

So, basically, while it's a model with solid grounding, it's not like an experimentally verified result. Still, it's worth noting, because the first thing you have to do to draw a conclusion is ensure that your hypothesis is at least plausible.

Their model suggests that the balance between cooperation and selfish behavior, called defection, can undergo rapid phase transitions, in which individuals match their behavior to their neighbors. What’s more, a crucial factor turns out to be the process of punishment.

Well, good news for BDSM fans.

The article goes on to explain, in what I think are very accessible terms (so don't be scared off by the "mathematical model" phrase above), the way they looked at the problem.

That’s an interesting result. It implies that behavior can be manipulated on a large scale by the introduction of certain costs. It also implies that the result can be modeled using relatively simple physics.

Well, they could have just taken the route of "ask an economist." The only danger there is that if you ask n economists a question, you'll get n+2 different answers.

Anyway, it seems to my completely unschooled opinion to boil down to game theory. But it struck me as an interesting result because I'd been thinking about how competition and cooperation interact. Like in a game. Two players sit down to play chess. At first glance, it's a competition: either one will win and the other will lose, or it will end in an unsatisfying stalemate. But the thing is, both players have agreed, formally or not, on a set of rules governing that competition. Someone who breaks the rules is called a "cheater" and no one likes a cheater. The rules are a form of cooperation, and the social disapprobation heaped upon cheaters serves to discourage such behavior.

Doesn't matter if the game is chess, football, Monopoly, or whatever. You have to cooperate before you can compete.

And even when there's not an actual game involved ("game theory" isn't actually about games), there are rules and consequences for breaking them -- even in the ultimate human competition called war. I know the saying is "all's fair in love and war," but that's simply not the case.

Like I said, the title is a bit misleading, because nothing's been settled (or "solved") here -- but it does provide a fresh way of looking at things, and sometimes that's all it takes to eventually make a breakthrough. Another advantage of cooperation.

2 Like • Comment

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