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:

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

October 29, 2023 at 11:11am

#1058242

Revisited: "Man"

Today's callback is a difficult one, and I'm not sure I have the mental energy for it this morning. The original entry was from about two years ago: "Man"

The article I referenced there was pretty new when I wrote that, and it's still up. If you don't want to read the earlier entry (understandable, though I promise the entry, if not the article, is short), here it is

again.

My intro then: "I can admit when I just don't understand something. This is one of those times."

And I can't say I've developed a full, nuanced understanding now, two years later. The comments and later blog entry on the topic from Turkey DrumStik

did help me to comprehend some things, but it's just not a subject I'm attuned to.

For one thing, I thought, for the vast majority of my life, that being a man simply meant that I was in possession of a todger. Presumably, when I was born, the doctor or nurse checked me out, saw a tallywhacker, and said something like "It's a boy!" I learned of this practice at a fairly young age, and also saw it used on farm and domestic animals, so my mind went "If you see a wangdoodle, that means it's male. Therefore, 'man' is simply defined as an adult human with a pecker."

All that other superfluity, such as "boys have shorter hair" (I didn't always, and don't now) or "girls wear dresses" (they don't always, and historically, boys wore dresses too) just seemed like a decision we collectively made as a society, one which could be reversed. Now I see that this stuff is what people mean by "social construct." And it still feels optional to me.

After all, my cats don't have a gender identity. Their gender is simply their biological sex. The queens don't wear makeup or hair bows, and the toms don't deliberately grow bushy beards or smoke pipes. Even being surgically altered doesn't change what pronouns you use for the animal.

More recently, I realized that this kind of thinking gets you labeled a transphobe, and no amount of denial of that on your part can ever change that label. It's far easier to change your gender than to get people to stop calling you a transphobe. It's kind of like if you say "I'm not an alcoholic," everyone will conclude that you are, in fact, an alcoholic. Especially if you're drunk and slur the words.

Anyway, that kind of thinking is why I always considered those outward markings of "manliness" to be superficial and optional.

Then, later in life, I realized that I was wrong. I can admit that, too; I'm man enough.

So I won't bother offering any support for my assertion that I in no way fear (or hate) trans people. Think what you will of me; I don't care.

And that brings me to the actual crux of the matter, at least for me, and for now: people seem to care a great deal about what others think of them, even, in some cases, when they claim not to. In another grand cosmic coincidence (see yesterday's entry for the last time this happened), one of my favorite webcomics touched on this very subject today. {xlink:https://www.smbc-comics.com/comic/escape-2}This link should be as permanent as anything is on the internet.

In the comic, a man (you can tell from his clothes, hairstyle, and facial features) goes to visit the archetypal Hermit on the Mountain (you can tell from the mountain, his robe/dress, and his beard). His question begins: "Wise Master, how do I escape? I'm a social ape. I'm obsessed with status. All my actions, even private ones, can be perfectly explained if you assume I'm seeking the esteem of other apes."

I'll tell you what, though: once I found a way to live that does not require the approval, or escaping the disapproval, of other social apes, I found a freedom that most people can't even conceive of. Once I stopped trying to peacock or scrabble for this mysterious "status" crap, life became a whole lot easier.

In short, I'm becoming the Hermit on the Mountain, and I'm okay with that.

One other unrelated thing about my earlier entry: Yes, I used to do Merit Badge Mini-Contests. I stopped because readership dwindled (or at least the number of commenters; being not obsessed with status, I don't often check my readership statistics) and because I quit doing entries at midnight in favor of being free to drink in the evenings. The schedule is much more flexible, these days, so it's harder to do deadlines.

But I'm also inclined to be generous, so who knows? I may start that up again in some form.

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