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

August 1, 2023 at 9:35am

#1053424

Pretty Petty

A couple of weeks ago, I posted a Cracked piece about ideas that started as jokes: "The Step-Parent of Invention"

Today's is similar, only instead of jokes, they started as a giant middle finger.

Four Important Things That Were Invented Out of Sheer Pettiness

Unless you’re an actual deity (in which case, greetings, Ms. Knowles), you’ve almost certainly pulled some petty shit in your day.

Actual deities invented petty. Or, well, as they don't exist, we made up stories about them that projected our own penchant for pettiness upon them.

Slowed to a crawl in front of a tailgater, helpfully transferred your neighbor’s dog’s poop from your yard to theirs, squeezed all your roommate’s fruit after the third day of seeing their dishes in the sink — nothing life-ruining, just a risk-free outlet for your anger.

I'd like to say I've never done anything like that. But I can't.

But have you ever been so annoyed, you changed the world?

One person cannot change the world. Not even Beyoncé.

4. The Myers-Briggs Personality Type Indicator

I've ragged on this before: astrology for psychology nerds.

It was also a fairly useless exercise, as the MBTI has been largely dismissed as pseudoscience, probably because it was invented by a stay-at-home mom with zero psychological qualifications to throw shade on her son-in-law.

Hence the "petty" part.

She took over entire sections of the library as she embarked on a lifelong study of exactly what factors made her son-in-law suck so much, which her daughter finished after her death, which might have stung for poor Mr. Myers if he had any capacity for emotion.

Both of my former mothers-in-law had their issues with me, but not enough to ruin the lives of everyone who ever applied for a job.

3. The Automated Telephone Exchange

Let's be real: this was going to get invented eventually anyway.

Back in the early days of the telephone, you had to call the operator, who would then put your call through to your mom or your side piece or whoever.

And then, if they were bored, they'd listen in. You think internet surveillance is invasive? Ha.

They knew who was calling who at all times, and they might even decide for you who you were calling, which must have been frustrating for people who asked for their side piece and got their mom. That was a problem Almon Brown Strowger kept running into in 1878. He was an undertaker living in Kansas City, and he just couldn’t figure out why his business was, uh, not as lively as it could be.

You want to piss off the undertaker, it's your funeral.

It turned out one of the local telephone operators was the wife of another undertaker, and every time someone called for Strowger, she patched them through to her husband instead.

As evil plans go, that one's rather amusing. Of course, when the operators were replaced by machines, few of them likely found that amusing.

There's also a nice plot twist, but I gotta save something for the article.

2. The Lamborghini

This one, I knew about. What can I say? I don't talk about cars much, but I retain some of Teen Me's fascination with them.

Not going to quote this one; the article is funnier than I can be.

1. The Dishwasher

I've said before (including in that entry I linked above) that while necessity may be the mother of invention, laziness is the milkman. A lot of labor-saving devices came into being not out of pure necessity, but a burning desire to do less work, which of course is a worthy goal that unfortunately only means you end up filling all the time you saved with other busy work, instead of writing poetry or doing philosophy or whatever the labor-saving advocates imagined our current utopia to consist of.

Today, it’s a mark of being Kardashian-level spoiled to not even know how to use a dishwasher, but it was invented by a socialite essentially as an insult to the help.

But this one, if a joke site can be believed, wasn't even to save the inventor time, but to diss the servants.

The socialite in question was Josephine Cochrane, the wife of a wealthy businessman living in Illinois in the mid-1800s.

Must run in the family; their distant descendant, Zephram Cochrane, will invent the warp drive, the ultimate time-saving device.

In summary, the dishwasher was invented because servants tended to chip the china. Which is ironic, because rare is the dishwasher machine today that doesn't chip whatever is banging around in there.

I don't know about you, but if I were a servant, I'd chip the china sometimes just to be petty.

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