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!

a day 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

a day 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.

August 4, 2019 at 12:25am

#963685

Headline Questions Are Usually Answered "No."

I should probably type the words "trigger warning" here, because I know some of my readers have mental health issues. But I also know some of you are more triggered by trigger warnings. Oops, too late. Sorry.

http://nautil.us/issue/45/power/does-depression-have-an-evolutionary-purpose

I want to call bullshit on this. I really do. But it's not my field - far from it. My only experience with psychology was a) one class in biological psychology in college and b) depression diagnosis and unsuccessful treatment thereof.

So I'm throwing this out there.

Now, usually, I mine quotes from the articles I link, but few of them in this one are really worth isolating; I think one needs to read the article to get the idea.

I have problems with how evolutionary "explanations" for this and that and the other thing are generally portrayed. I know I've said this before. My biggest problem with it is somewhat addressed toward the end of the article, but only after the author throws out a lot of speculation about how depression might be advantageous from an evolutionary standpoint.

One problem is this: When it comes to things like, I don't know, skull shape, we have an intermittent fossil record thereof. We can see, based on bones dug up here and there, what human, protohuman, prehuman, and other ape skulls look like and how their general forms changed over time. It's easy, then, to see a general trend toward a larger cranium, and how that could accommodate our relatively large brains. We can also see changes in hand / finger bones, and can make a good inference as to why our hands were evolutionarily advantageous.

When it comes to behavior, though, the fossil record is necessarily mute.

With evolution, not every trait is a survival trait. Some are vestigial or effectively so. Others are incidental. One way evolution works is that incidental traits sometimes end up aiding survival and/or reproduction, so those traits can get passed on. Vestigial traits like - I want to say the appendix, but I've heard that might actually be part of the immune system - whether or not you can wiggle your ears are generally neutral to survival, but might have had some benefit in a distant ancestor.

So, in my uneducated view, depression could just as easily be a byproduct of cognition, a neutral one. How can it be neutral when you take suicide into account? Well, as the article pointed out, most suicide attempts fail. Hence, it doesn't necessarily take a person out of the gene pool. Also I should add that while I've been depressed, I've never been suicidal; I don't know what the stats on that are, but I think the article conflates "depression" and "suicidal ideation," and that pisses me right off.

Not caring if you live or die isn't the same thing as actively seeking to end one's life.

So anyway, smells like bullshit to me. To be fair, whenever someone combines "psychology" with "evolution," my BS detector swings into the red, but I'm sure some studies in that area are legit.

I'm more concerned with this holding out false hope to people, or, conversely, possibly making someone think "Oh, well, it's okay if I off myself, then, because evolution wired me to do that." As the article itself points out, "Even if depression evolved as a useful tool over the eons, that doesn’t make it useful today." You know, like the appendix or ear-wiggling ability.

See, we also evolved to have conscious thoughts, to use science, and to otherwise work to overcome atavistic impulses. I, for one, have had my appendix removed; whatever benefit it might have been providing was suddenly outweighed by its cost to my health. We can make these choices.

Still, the article makes some interesting points and I'm curious to see if this line of speculation goes anywhere.

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