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!

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.

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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.

June 29, 2019 at 12:03am

#961723

Authentic Thoughts

https://blogs.scientificamerican.com/beautiful-minds/authenticity-under-fire/

Authenticity is one of the most valued characteristics in our society. As children we are taught to just "be ourselves", and as adults we can choose from a large number of self-help books that will tell us how important it is to get in touch with our "real self". It's taken as a given by everyone that authenticity is a real thing and that it is worth cultivating.

"Everyone" except me, apparently.

One big problem with authenticity is that there is a lack of consensus among both the general public and among psychologists about what it actually means for someone or something to be authentic. Are you being most authentic when you are being congruent with your physiological states, emotions, and beliefs, whatever they may be? Or are you being most authentic when you are congruent with your consciously chosen beliefs, attitudes, and values? How about when you are being congruent across the various situations and social roles of your life? Which form of "being true to yourself" is the real authenticity: was it the time you really gave that waiter a piece of your mind or that time you didn't tell the waiter how you really felt about their dismal performance because you value kindness and were true to your higher values?

My father wasn't exactly a paragon of virtue. Like most people, he embodied those values we call "good" as well as those we call "evil." While I think, on balance, he was more of the former than the latter, there were things about him that I consciously chose not to emulate in my own adulthood.

Somewhere in the 90s, he started showing symptoms of dementia. Fortunately, the worst of those symptoms didn't appear until after my mother died in 1999. Until then, while she languished in a nursing home with her own version of dementia, he'd visit her every day, bringing her fresh food from the garden, or just sitting with her for a time. Though when they both had most of their mental faculties, sometimes he could be cruel, at the end, it was all kindness.

Which one was his true self? A lot of people seem to think that someone who is mostly kind, and then does something evil, well, that means they're showing their "true colors." And a lot of people with Alzheimer's do become hateful - for whatever reason, they lash out, lose their filters, or generally act like assholes.

And my father did, sometimes. But usually, he was concerned about other people. I'd visit him in his nursing home - I couldn't do that every day like he did with my mom, a thing that still eats at me sometimes - and he'd offer to share his meager rations, or give me his coat if I'd come in on a warm day wearing a t-shirt. Yeah... he'd literally offer to give me the shirt on his back, and at that point, he didn't even know who I was. As far as I'm concerned, that was his "true colors."

So, that's how I choose to remember him - that, and his abiding curiosity and love of scientific investigation. A sailor by profession, he went to college later in life, earning a degree in, of all things, chemistry. He taught himself to read Russian so he could understand the science coming out of what was then the USSR. Point is, though, the times when he was unkind were just as much a part of him as those positive traits; I just make the decision to usually focus on the good parts.

I don't always pay myself the same courtesy.

As long as you are working towards growth in the direction of who you truly want to be, that counts as authentic in my book regardless of whether it is who you are at this very moment. The first step to healthy authenticity is shedding your positivity biases and seeing yourself for who you are, in all of your contradictory and complex splendor. Full acceptance doesn't mean you like everything you see, but it does mean that you've taken the most important first step toward actually becoming the whole person you most wish to become.

It's probably a good thing that I try to be aware of when I'm being an asshole. More, I try not to be an asshole; it's just that sometimes, it happens. Because being aware of when you're being hateful is the only way to work on being a better person.

I would argue, then, that it is impossible to not be authentic. Yes, even if you're a character actor. That makes this "authenticity" thing a semantically null concept. But that doesn't mean you can't learn to improve yourself in the direction you want to go. If you don't give much to charities, for example, you can make the decision to do so in the future - and then do it - and you go from "someone who doesn't give to charity" to "someone who gives to charity." It's sometimes that simple. Other things, like learning to control what one does when one is angry, well, maybe that takes more work.

But one is no less "you" than the other.

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