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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

April 1, 2023 at 12:25am

#1047267

Lost in Thought Experiments

Well, April 1 is here. Remember, folks, pranks are only funny if they're not played on me.

Today's article is arguably funny, but it's not a prank; it's from Cracked:

4 Dumb Thought Experiments That Need to Be Eradicated From the Hivemind

Even though the thought experiment in theory is supposed to be a pure philosophical discussion — a conundrum cooked up to change your viewpoint on an issue in a compelling way — it’s lost a little bit of that shine, especially thanks to the internet.

Thought experiments have some limitations. That's why they should be left to professionals. Kind of like comedy.

Of course, a good thought experiment is a wonderful thing to kick around the old noggin. You’ve got your classic trolley problem, which only becomes more prescient now that cars are driving themselves (terribly).

People make fun of the trolley problem (and sometimes, rightly so), but it's not just autonomous vehicles that use it as a model. Pretty much any intervention can be likened to a trolley problem: Some large number x of people will face negative consequences if we do nothing; if we do something, some smaller number y will face negative consequences. Difficulty: y is not necessarily a subset of x.

As for AVs, that's a separate rant that I'll probably update at some point.

In particular, here are five thought experiments that the world doesn’t need and nobody asked for…

Four. Dang editors.

4. Buridan’s Ass

Ba-donka-donk.

The ass in question is a boring old donkey, who is starving, and standing equidistant between two bales of identical hay, each precisely as easy to reach and eat as the other.

I'm just going to pause here so you can get ass puns out of your system.

Ready?

Okay:

The question is, which bale of hay does the donkey choose, or — and this is a real situation people present — would the donkey simply be torn by indecision so deeply that it starves.

No. Come the fuck on.

3. The Life You Can Save

Here’s the gist of it: If you were walking in your expensive work clothes and saw a drowning child, would you jump in to save them, even if it would ruin your fancy clothes? The answer to this, unless you are a sharply dressed sociopath, is obviously yes.

No, because while I learned how to swim as a child, I've forgotten everything about it. I jump in, regardless of what I'm wearing, there'll be two victims instead of one. Including, most importantly, me.

But alas! By answering yes you have fallen into Singer’s great trap! Because by the same belief, how could you freely spend money the way you do while a child starves somewhere else in the world! You are undone! How does that petard feel, dummy! This is the most grandiose false equivalence bullshit I’ve ever heard, and it’s created in service of something everybody already understands anyways: Yes, it’s easy to dehumanize people when you can’t directly see them.

All of these things are Not My Problem.

2. Roko’s Basilisk

This is, beyond the merest shadow of a doubt, the stupidest, inanest (?) excuse for a thought experiment ever concocted. I won't even dignify it with further comments or explanation. If you want to learn about it, there's the link up there, or just Google it.

1. Hamlet Monkeys

As much as I love monkeys, I never need to hear this tired-ass “thought experiment” ever again. It’s one of a whole smorgasbord of hypothetical situations created to constantly try to reinforce the idea that yes, infinity really does mean infinity. It’s a delightful mental picture, but it’s wholly unnecessary. If someone doesn’t understand that infinite time creates infinite possibilities, best of luck to them.

If you love monkeys, you've never been around monkeys. Here's the thing most people get wrong about this one: the original formulation is this:

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare. (Wikipedia)

This has gotten twisted over time into, instead of "infinite," "very large." And there's the problem: "very large" is still infinitely smaller than "infinite." While it's quite droll to observe that the idiocy you find on the internet disproves the infinite monkey theorem, we're only talking about a few billion apes (not monkeys) typing for the last, oh, let's be liberal here, fifty years. Neither of these numbers is anywhere even close to approaching infinity. So we shouldn't feel too bad that we haven't been able to surpass Shakespeare.

Well, some of us haven't. Cracked sometimes comes close.

3 Like • Comment • View 2 comments

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