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 12 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 14 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').

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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.

May 15, 2020 at 12:22am

#983603

Liver Let Live

Sometimes I think there are selfish motivations for everything, even if the only selfish motivation is feeling good about doing something for someone else.

PROMPT May 15th

Write about a time when you or someone you know acted selflessly. What was the motivation and what were the results?

That point of view probably says more about me than about humanity in general. It's not that I don't think there are people out there who do good things; it's just that I think we always get something out of it. You give blood, for example -- as I used to do before I couldn't any more -- and you get free juice and cookies and the satisfaction of knowing that you've probably helped someone, and maybe even saved a life.

There is nothing wrong with this, in my view. We're social animals and we gain benefits from helping each other out. Some people might not see the benefits, and are all take no give, but I think most of us would do simple things to help others that might be seen as selfless. This could mean giving blood, as I mentioned, or returning a lost wallet, or giving food to a homeless person. More rare, and less simple, are things like jumping into a river to save a drowning person.

I think the most helpful act -- whether it's selfless or not is up for debate, but there's certainly an argument for it -- that I've ever personally witnessed was when the wife of a former co-worker of mine once donated part of her liver to someone who needed it. This isn't a procedure without risk; any surgery confers the risk of death to some degree. So this lady risked herself and put constraints on her own life to save another.

Well, to be clear about it, I didn't actually "personally witness" it; this was told to me by the co-worker, but I saw the donor after the surgery and she was recovering nicely. Last I heard, she was doing just fine.

Could I do it? Honestly... no. I mean, leaving aside for the moment that anyone who reads this blog should know by now that no one would even want part of my liver, or that I probably couldn't donate for the same reasons I can no longer give blood, I just wouldn't want to do it. I have this recurring fear that someone from my biological family would track me down somehow and turn out to need a kidney or something. Then I'd be in the awkward position of being the dickwad who says "no" to something like that.

Blood transfusion? Sure. Spleen? Nope. Hell, I'm not even letting them take my organs after death, lest they be used to extend the life of someone I don't like. I'm a complete asshole like that, and I'm okay with it. Why I'm okay with giving blood but not being an organ donor is a question I still haven't answered for myself. Maybe it's the free cookies. Before you jump on me for it, I've already decided that I won't accept a donor organ; I'm not that big a hypocrite.

Now, why did I say the question of whether this liver donation was "selfless" or not is up for debate? Well, I'd argue that the donor got something out of it, so it wasn't purely selfless by my definition. First, she got the knowledge that she saved someone's life -- and that's a deep satisfaction indeed, especially as the recipient was someone she knew personally. Second, she's a religious person, and in her mind it got her Jesus points, or whatever it is that Mennonites believe get you into heaven. Third, a lot of people (myself included) respected her for it. And finally, though probably least importantly, she got a free total health examination, which in this country is like finding a unicorn and the pot of gold at the end of the rainbow and winning the lottery all on the same day.

Still, by most definitions, it was a selfless act.

Incidentally, I reserve the right to change my mind about the organ donation thing, just like with everything else. And I wouldn't want anything in return. I wouldn't turn down some cookies, though.

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