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 28, 2023 at 11:47am

#1054756

Law of Rules

Sorry, starry-eyed anarchists, there are always rules, and they're made to be followed.

Some of them, admittedly, don't make a lot of sense, like those old laws about having to get a license to wear a beard or whatever.

According to Cracked, however, some rules are just unjust. (Paradox!)

5 Logical, But Wildly Unjust Rules

Laws exist to maintain order in a chaotic society.

Ideally, yes. In practice, laws exist to protect the interests of the rich and/or powerful.

Sometimes, laws are more in place to keep everything moving smoothly and prevent headaches more than to punish Great Evil.

Because Great Evil makes the laws.

Here are five laws that make logical sense, but are deeply unjust…

As usual, it's a countdown.

5. No Dogs in Antarctica

Up until fairly recently, they were a central part of any activity on the continent, but in 1993 they were banned because of worries about them transmitting diseases like the thoroughly old-timey sounding “canine distemper” to native wildlife, like seals.

Sounds like Paradise to me. Well, except for the whole "desert" and "bitterly cold" and "dark six months out of the year" parts.

4. No Bread in Space

If you’re off to space as an astronaut, though, despite heading into a high-stress environment and situation, you won’t be able to rely on the airy, stretchy internal hug of a tasty sandwich or pastry to calm you.

Look, I like bread. Bread is probably my favorite food, when you consider all of its myriad forms, including pizza crust; as well as its cousin, beer (which I suspect is also a no-no in space). But it's hardly the worst thing you have to give up in space. Gravity comes to mind. So does the ability to use a toilet.

3. Getting Your Taxes Wrong

The American process of paying taxes is basically a massive system of mathematical gaslighting that seems designed to make April a forever horrible month.

Which might explain T.S. Eliot's famous opening line of The Waste Land, and why he abandoned the US in favor of England at the ripe old age of 25. Surely their bureaucracy is less impersonal and stress-inducing than our own.

If you get those calculations wrong, because percentages were probably taught to you by a depressed and underpaid English teacher during a staffing shortage, congratulations on committing tax fraud.

As the article notes, they do have to prove that it was intentional. But it still sucks to make a mistake.

2. Jaywalking

I do this on purpose in NYC just so I can be labeled a scofflaw. I also once spit on the sidewalk there. I'm such a rebel.

This all seems a little ridiculous, and, to be honest, unnecessary. Jaywalking already has its own, independent legal system of punishment, known as “physics.”

Except that the vast majority of jaywalkers never receive that punishment, and when they do, it can be traumatic for the vehicle driver as well.

1. No Putting Your Head Under the Soda Machine and Guzzling

Sadly, this one also applies to beer taps.

Maybe it's better in Antarctica, after all.

3 Like • Comment

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