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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 16 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 18 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.

March 7, 2020 at 12:45am

#977380

For A Reason

Today's entry is #2 in a series of 8

assuming I don't fuck this up
for:

Journalistic Intentions

(18+)
This is for the journal keeping types that come to PLAY! New round starts July 1!
#2213121 by Turkey DrumStik

Just as with my usual link entries, I'm picking these at random. Little did I know how many of my buttons it would push. I probably should have known; I've been friends with Turkey DrumStik

for a long time, and we're often on similar wavelengths.

In case you don't want to play the above video - and I wouldn't blame you one bit - here's my executive summary as I'm watching it for the first time.

1) Chick is living her perfect life.
2) Chick gets diagnosed with nasty cancer.
3) Some kind of aside into Prosperity Gospel: "Nothing is impossible if you believe." Calls it "theology."
4) "Spiritual guarantees." Snort.
5) "It is extremely difficult to avoid falling into the trap of believing that virtue and success go hand in hand."
6) In re #2, "Why is this happening to me?"
7) "It was a mindset that served me well... until it didn't."
8) Questions that boil down to "why do bad things happen to good people," specifically "but I'm a good person, so why did bad shit happen to me?"
9) "Everything happens for a reason."
10) "Near Death Experience Research Foundation"

The Real American Dream

I get the impression that a lot of people, when they hear the words "American Dream," think of the suburban house, two cars, 2.1 kids, dog, that sort of thing - the material stuff. But this speaker touches tangentially on what the American Dream really is, and why it can suddenly turn into the American Nightmare: the idea that, if you work hard enough and follow The Rules, you will prosper. This is so embedded in our culture that it's like when you live in a house where something smells bad - eventually you don't smell it any more, until you go away for a bit, and then you come back and you're like "Peeeew."

If hard work was all it took to get rich, then sharecroppers would be millionaires.

Here's the dark side of it: If working hard leads to prosperity, then if you're not prosperous, well, you must not have worked hard enough. Or you did something else wrong. Or you don't believe the right things. Whatever the case, clearly you're only poor because you deserve to be poor, so why should I do anything about it?

This is, of course, bullshit. People get rich by playing the lottery, guessing right in the stock market, or because of their connections (yes, and sometimes by working hard and smart). Lots of other people stay poor by playing the lottery, guessing wrong in the stock market, or because of their lack of connections, and you never hear the stories of people who worked hard and smart and still ended up with nothing.

Theodicy

The question, "Why do bad things happen to good people?" and its inverse, "Why do good things happen to bad people?" are known in theological philosophy as theodicy.

Essentially, as seen in Wikipedia:

As defined by Alvin Plantinga, theodicy is the "answer to the question of why God permits evil". Theodicy is defined as a theological construct that attempts to vindicate God in response to the evidential problem of evil that seems inconsistent with the existence of an omnipotent and omnibenevolent deity.

A great deal of entropy has been generated by people thinking about these questions, including, apparently, the speaker in this video. Whereas the only reasonable response is to remember that your basic premise is incorrect, and lots of times, random shit just happens. In this worldview, there's no need to appeal to a Driving Force of the Universe. A simple hypothesis that satisfies Occam's Razor, and frees the mind to concentrate on less abstract matters, like, say, how to stop the spread of an epidemic. To use a purely hypothetical example.

Recently, a tornado blew through a town in Tennessee, killing about two dozen people. Someone found an intact Bible in the wreckage, and Franklin Graham crowed something like "This is proof of God's love for us!"

Presumably the same God that just killed two dozen people with fucking wind.

I can't be arsed to look it up, but it was one of the Greek philosophers who noted something along the lines of, either God can do something about evil and chooses not to, in which case he is evil, or he cannot do something about evil, in which case he's not a worthwhile god to worship.

By "evil" above, I'm generalizing to "bad things." When I say "evil," usually I mean shit that people deliberately do, like committing rape, kicking puppies, or failing to tip a restaurant server.

Okay, so what?

I don't want you to get the idea that I lack compassion for this speaker. Cancer sucks. It sucks if you're old, and it sucks if you're young. It sucks if you're alone, and it sucks if you have a spouse and a kid. It sucks, and I don't wish it on anyone (though I come close with Franklin Graham and his ilk). I'm glad she found the strength to get through it.

Listening to her, though, there's something that she's very loudly not saying. Lots of people helped her. Medical professionals, directly. Scientists, indirectly, by coming up with the treatments that seem to be working to save her life. Her fans and well-wishers, who send her letters and emails and whatnot. Whoever set her up as a TED speaker. Her editors and readers. And I only listened through this once, so perhaps I missed it, but I didn't hear even one time when she acknowledged that it was because of all of these humans that she could be standing on stage delivering motivational talks.

Not gods. Humans.

This is what we do. We help. If you're of a religious bent, you can argue that it's some spiritual drive to do so, but I say it doesn't matter - we choose to help (with, of course, a few notable exceptions).

I don't begrudge her what comfort she can draw from her support system, her paycheck, her beliefs. Whatever works. She's probably happier for it, and that's great for her

But the reason she's alive is science.

3 Like • Comment

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