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:

BIG BAD WOLF is Merry

about 2 hours ago
In response to "Stress Marks"

There's always something that one can do.

tj wanderlust-words-in-motion

about 20 hours ago
In response to "Life As We Don't Know It"

There's probably a sign at the edge of the Milky-Way that reads, "Caution, enter at your own risk!"

1 Like

Annette

a day ago
In response to "Revisited: "Real Men of Genius""

My second son was born at almost the exact same time as one of my uncles died. When second son was a baby, we looked out for signs whether there was some reincarnation going on.

If there was, then he is a great example of nurture versus nature. Where my uncle was creative, sensitive, and mentally destroyed by the time he was a teen, my son's creative side is thriving, he has turned his sensitive nature into a passion for anthropology, and he knows he is loved unconditionally.

Maybe reincarnation is simply one generation deciding to put a stopper on past generations' bad habits.

1 Like

🌖 HuntersMoon

Mar 2, 2025 at 12:39pm
In response to "Revisited: "Real Men of Genius""

Elisa, Stik of Clubs - It's what makes you unique and so darn loveable!

The same can be said for Robert Waltz if "irascible" is considered weird.

1 Like

Elisa, Stik of Clubs

Mar 2, 2025 at 10:51am
In response to "Revisited: "Real Men of Genius""

I'm pretty sure some of your beliefs, habits, or activities, and mine, would get pegged as weirdness if we were famous.

I am not famous, but plenty of my habits and activities are pegged as weird, anyway. Thanks, autism!

2 Like

Amethyst Angel 🍀

Mar 2, 2025 at 5:34am
In response to "Opti-Miss"

I struggle with feelings of nihilism, tbh. It's very easy for me to become numb and paralyzed, so I have the desire to cling to optimistic worldviews even though my dualistic tendencies tell me there's no reason to believe in them... Sigh. This is a depressing subject.

1 Like

BIG BAD WOLF is Merry

Mar 1, 2025 at 8:11pm
In response to "Opti-Miss"

One might wonder.

Robert Waltz

Mar 1, 2025 at 8:07pm
In response to "Opti-Miss"

Elisa, Stik of Clubs

- my comment about etymology was just me kicking myself for never seeing the linguistic connection between optimal and optimist. Seems obvious in hindsight. I've railed many times against arguments from the dictionary, and noted that dictionaries are descriptive, not prescriptive.

And I agree; I was too binary in the optimism/pessimism divide. As with almost everything else, there is middle ground.

tj wanderlust-words-in-motion

Mar 1, 2025 at 4:08pm
In response to "Opti-Miss"

difference between an optimist and a pessimist:

1 Like

Elisa, Stik of Clubs

Mar 1, 2025 at 4:00pm
In response to "Opti-Miss"

I have thoughts. Let's get going in breaking them down.

Optimism is, after all, by its nature delusional; ‘realism’ or outright pessimism might seem more justifiable given the troubles of the present and the uncertainties of the future.

"Seem?" No. It absolutely is.

That's a pretty bold claim being made there. In my opinion, bold claims require evidence to back them up. It might make sense to those of us with the right combo of priors (as mentioned in a previous entry) to reach this conclusion about the nature of optimism. For those who do not have those priors, we gotta consider their perspective when backing up claims like this.

or someone with such strong opinions about optimism, and such a keen interest in etymology...Leibniz’s view, put forward in his book Theodicy (1710), did not win instant acclaim.

Etymology can be fun, and knowing the origins/definitions of words can certainly help guide arguments. At what point though, do we (including Leibniz) fall into the trap of the dictionary definition fallacy?

The problem is, lots of people seem to accept it as scientific fact—which it is not.

This is news to me. I've always seen it more as a fun creative exercise among those I've seen discuss it.

In other words: ask yourself, "Can God be surprised?" An all-knowing entity couldn't possibly be surprised, any more than it could create a rock so heavy even that entity couldn't lift it.

I think the difference can be split here. I have heard that there are two ways a story can go for a reader/viewer: that it's a complete surprise how the story ends or the story's ending is predictable but how that ending is reached is surprising. I feel like god could not be surprised by the end result but could be surprised by the methods used to achieve that result.

I think things can change for the better, but, absent well-meaning and well-thought-out human intervention, probably will not.

Here's probably where I have the biggest bone to pick. How often are people willing to work to at least try some sort of intervention? This is where I think a lot of frustration on the optimism/pessimism debate actually lies. It's less a matter of where on the spectrum you fall and more a matter of how actively you behave in that capacity. There are certain combinations that I have issues with: pessimists who don't at least due their due diligence to solve a problem before giving up, optimists who are super passive, optimists who build their optimism in ways that lack empathy and take actions that border on antisocial. Likewise, I'm more willing to give a pass to pessimists who make multiple attempts to set things right throughout their lives. They continuously try to do the work and don't just get all their proactive behaviors out of the way in their youth. In this particular framing (framing optimism/pessimism through a combo of mindset and action rather than only mindset) there is a fair amount of gray area, and oddly enough I think accepting this notion would actually make people feel more comfortable. It's no longer just a binary choice. A bit of flexibility may actually give people more incentive to start that intervention.

[Visit the entry for more...]

1 Like

March 4, 2025 at 9:50am

#1084770

Stress Marks

First, an Exclaimer! ▼
EXCLAIMER
(explanation/disclaimer)

The theme for this round of Journalistic Intentions is "Banned Instagram hashtags! All of these will get posts shadow banned on Instagram." So there are a few things my readers should know:

#1- I do not have, have never had, will never have, an Instagram account.

#2- Consequently, I have no idea what a shadow ban is or what it entails, only that it has something to do with suppression of posts with the indicated tag.

#3- I do not acknowledge the pronunciation of the octothorpe (#) as "hashtag." That symbol is either "number" or "pound," and that's more than enough definitions for one character. Anytime I see the # with letters following, I pronounce it pound, which made #MeToo way more hilarious than it should have been.

However, I hold the belief that I can write about absolutely anything, regardless of prior knowledge. This contest round tests that conceit. All entries, including this one, were selected at random from the list of 16 prompts provided.

Journalistic Intentions (18+)
This is for the journal keeping types that come to PLAY! New round starts February 20!
#2213121 by Elisa, Stik of Clubs

#stressed

There is a myriad of ways to denote stress in online writing. One can bold the desired portion of text, or use italics—though the latter runs the risk of being confused with certain styles of quotation, such as the one I use in this blog.

Another option is to underline the stressed text. Classically, an underline denotes a book title, but I think that style has fallen out of favor. Besides, if you do use it for that, the words generally follow an established capitalization pattern; this and other context clues should be sufficient to distinguish text underlined for stress from text underlined for a book title.

There are also numerous colors that can be used to make words and/or phrases stand out. And let's not forget the potential of increasing a word or phrase's size, though it must be remembered that this can cause problems with formatting and flow.

Don't overlook the impact (pun intended) of font type, either: a word in Impact will definitely stand out amidst the standard font here; Times can give your stressed bits an old-time serif look; and Comic, while sneered at by font snobs, has its uses, especially as the long-sought sarcasm font.

Let's not forget the trick of making certain portions of text *stand out* using only ~keyboard~ strokes, or even

emoticons/emoji

. Just remember, though: the use of quotes like 'single' or "double" quotation marks should be reserved for dialogue or quoted text, and never for emphasis or stress.

Nor should Capital Letters be used for stress. I use them sometimes to indicate Important Ideas, but I'm probably wrong to do so; it's a kind of stylistic signature for me. Also, the technique of typing in ALL CAPS is frowned upon except in rare circumstances, and definitely not to be used exclusively because then it makes you look like you're writing your manifesto.

To *really* make your desired text stand out, you can use almost any combination of these techniques. The possibilities are... well, not limitless, but numerous, and a better mathematician than I am could probably enumerate them more precisely. But as I said up there at the beginning, "there is a myriad of ways to denote stress;" the word 'myriad' comes from a Greek word meaning ten thousand, and ten thousand is a pretty large number, more combinations than you probably truly need. It's a good number, so we'll go with that for now.

The stress on 'myriad,' incidentally, falls on the first syllable.

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