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').

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.

May 12, 2020 at 12:24am

#983365

Free Willy-Nilly

I've ranted at length in here about things that people think are illusions being real, such as time and consciousness. Today, you get treated to the opposite.

PROMPT May 12th

Write about a fork in the road in your life, and how you made the decision to go the direction you did. What would have happened if you chose the other path?

Sure, I hit a fork in the road once. It blew out my tire and I got stuck there.

Humans put a lot of stock in decisions. A popularized interpretation of the Many Worlds Hypothesis in quantum theory is that for every decision we make, the universe splits. This is not only a mischaracterization of the actual hypothesis, but it puts sapients back into a special place in the cosmos. After all, no one seems to believe that a cat's decision splits the universe; at best, the cat's a victim of radioactive decay, alive in one universe and dead in the other, with its actual status a superposition of the two until a (human) observer checks on it.

I won't be convinced of the MWH in general until someone explains to my satisfaction how it's compatible with conservation of something. But that's not really relevant to the larger question, which is: Do we really make decisions? That is, who's the "we" in that question?

It's unlikely that free will exists

in the form that most of us believe it does. I can't pretend to understand all of the science behind it, but based on my understanding, we tend to make decisions even before we're consciously aware of making the decision, and then find ways to justify them afterward. In this sense, we're not rational - we're rationalizing.

There's significant pushback to this idea, because as a whole, we're kind of wedded to the idea that we consciously make decisions. It's the basis for most of our systems of government and justice. But there's growing evidence that, if it were possible to replicate the exact set of circumstances leading to a perceived decision, we could not have made a different one.

I realize this goes against the grain of everything we've experienced. After all, if you're playing a video game, for example, you can always go back and make different choices that lead to different outcomes. But that leaves out the fact that you've already played through the simulation once, and you know what happens.

Arguments for free will rely on the idea that consciousness is a sort of ghost in the machine, guiding our steps. You can choose, the argument goes, whether to put your emptied shopping cart into the corral or leave it in the middle of the parking lot; you can decide whether to return a lost wallet, and with or without its contents. And it certainly seems that we can make these choices. But that turns out not to be the case.

The philosophical pushback to this usually takes some form of "Well, if we can't make the decision, then why even bother? Just leave the shopping cart for someone else to deal with." This misses the point. You know you should put the shopping cart away, and if you don't, you've marked yourself as unfit for society. You know this, so you return the shopping cart, right?

Yes, free will is an illusion. But it's an important one. We need to feel we have some agency, that we're not just automatons or instinctual creatures like beetles. By doing so, we can feel good about our choices. Or, alternatively, feel bad about them and try to do better next time. "But Waltz, you just said you can't act any differently." Yeah, but future actions are determined by an entirely different configuration of matter and energy; the exact set of circumstances cannot be replicated. You're always changing, and so is your environment. As I've said before, not only can you not step into the same river twice, you can't even step into the same river once -- because by the time your foot hits the bottom, the river has changed. Hell, even putting your foot in it changes it. Which is true for any antecedent of "it," including one's mouth.

I should also mention that determinism doesn't imply either predestination or predictability. Plenty of things are deterministic but, to a greater or lesser extent, unpredictable -- the weather past a week or so into the future, for example. Same for our decisions. They spring from configurations of neurons, atoms, electric fields, chemistry, and so on, but that doesn't mean you can know exactly what's going to happen at some point in the future. The best we can do is come up with scenarios and think about what we'd do in them.

The point of all of this digression -- and no, I don't expect people to agree with me -- is that it does me no good to put past (illusory) choices under a microscope. What if I'd overcome my hatred of cold weather and gone to MIT instead of UVA? Or accepted the spot at Annapolis and joined the Navy instead of staying in civilian life? I'd be a different person, or at least the same person with different experiences, and likely done more with my life than designing parking lots for assholes to leave shopping carts in. Going back, knowing what I know now, I probably would have gone to MIT (Virginia has a slightly shorter winter season, but not significantly so), but I don't think I'd have joined the Navy (ships get cold). Nor would I have married my first wife, which would be a moot point if I hadn't been living in this town, where I met her. But I didn't know then what I know now, and those choices were the only ones I could have made at the time.

I'm not saying these things are mistakes, or that I wish I hadn't done them. But what's done is done, and the best I, or anyone else, can do is learn from the past. Such learning affects our illusory decision-making, and, ideally, improves it. Knowledge is part of the equation, and that's one reason I seek it out. Or, maybe, the desire for knowledge is just part of my particular configuration of matter and energy.

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