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 8 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 10 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.

July 8, 2020 at 12:11am

#987524

All On That Day

Pretty sure the last saber-toothed tiger died the day I was born.

PROMPT July 8th

What historical events, besides your own birth, occurred on your birthday in the year you were born?

Okay, no, not really.

Still, a whole lot of people are going to have to do some gymnastics - or outright lying - if they're trying to obscure their actual date of birth. Me, I haven't made a big secret of it, so I went looking for events that happened on February 18, 1966.

What I found disgusted me.

Not only did nothing of any global import apparently occur that day, but yeeesh, some of these "what happened the day you were born" websites are crapola. "You're an Aquarius!" Yep, knew that, didn't care. The defining characteristic of Aquarians is that we don't believe in astrology. One day later and I'd have been a Pisces. Hell, they don't even report my moon sign (also Aquarius; I was, auspiciously, born during a New Moon).

Speaking of astrology, according to these sites, it was Year of the Horse, and I'm Gen-X. Still don't care. Apparently apathy is the defining characteristic of Gen-X. Probably of horses too. I mean, how many times have you seen a horse rescue someone trapped in a well?

Let's see... celebrity births? No one I've ever heard of.

Famous people deaths? Ditto. Although one of the early cosmonauts "committed suicide" by train on that day. Since he was Russian, I put the quotes around "committed suicide."

I did find one interesting little bit of trivia: On November 21 of this year, if I've survived all the shit 2020 is determined to throw at all of us, I will have been alive for exactly 20,000 days. Considering I only remember, like, three of them, that's quite a lot of days. Doesn't mean much, of course, but I put it on my calendar so I'll have an excuse to drink, which will reduce my number of remembered days to two.

One other random bit of useless trivia I remember from when I was young (and is hence very suspect as to its veracity): As we all know, Star Trek debuted on TV in September of 1966, after having gone through a difficult birthing process. A network ordered a pilot, it was rejected, and it ordered another pilot. The original pilot eventually ended up getting recut for a two-part episode. Point is, it was a long, involved process (that never would have happened if it weren't for Lucille Ball, but that's another story). The thing I remember reading was that the network finally made a series order for the show in February of 1966. I choose to believe that this momentous event happened on my birthday. I can't find what day it actually was, and I have about a 1 in 20 chance of being right (assuming Hollywood took weekends off), so I'm going with it.

I did find one interesting thing, though, something that I'd never heard of.

https://www.onthisday.com/articles/buried-at-sea:-the-casket-that-carried-jfk

Buried at Sea: The Casket That Carried JFK

February 18, 1966 — The casket used to carry the body of assassinated President John F. Kennedy from Dallas to Washington was, on this day, parachuted into oblivion.

Not JFK himself, of course. He's supposedly at Arlington Cemetery; I've seen the gravesite. My father's buried close by.

The story of the coffin is remarkable. It was ordered from Dallas undertaker Vernon O’Neal by Secret Service agent Clint Hill when futile attempts at Parkland Hospital to save the slain President were finally abandoned. Hill is the man who leapt onto the back of Kennedy’s limousine after the fatal shots were fired.

The story is short, and worth the read; I won't risk violating whatever copyright laws by reproducing the whole thing here. Basically, the Kennedy family didn't want the casket to become some sort of spectacle, so they sent it to Davy Jones' Locker.

Like I said, no real historical import, but an interesting footnote to one of the defining events of the 20th century. And this is the first I've heard of it.

Could be bullshit, of course; I can't be arsed to fact-check something that obscure.

9 Like • Comment • View 14 comments

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