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

June 30, 2023 at 12:14am

#1051896

Oopsies

I can't for any amount of money recall why I saved this particular Cracked article, but I'm going to link it anyway.

5 Times in History Somebody Seriously Snoozed and Losed

History is full of success stories. These are not those.

The idea that you need to “strike while the iron is hot” has morphed from a straightforward explanation of blacksmithing into an endlessly repeated cliche. It, or some more unwieldy modern rejiggering, is plastered over everything from horoscopes to aviator stock photos on hustle culture Instagram accounts.

Every cliché was once profound wisdom.

Plenty of people throughout history have filled their metaphorical blade with microfractures by waiting too long, while others have snapped the sword in half entirely because of ill-advised hesitation — losing a couple million dollars or an entire city in the process.

It's not enough to invent something; one must also have the imagination to see the implications in one's invention.

5. Greeks Almost Inventing The Steam Engine

Eh. This one wouldn't have gotten very far, anyway. The math wouldn't be ready, also the fault of the Greeks, who insisted that irrational numbers were of the devil.

4. Not Signing The Beatles

This is fairly well-known, I think, and usually held up as an example of big mistakes.

Now, Decca Records weathered the blunder and has still been plenty successful in their own right.

I think maybe I had in mind, when I saved this article to my queue, something like "just because a publisher rejected your manuscript doesn't mean it sucks." However, it still might. It's just not a sure thing.

3. Passing on the Patent for the Telephone

Orton and Western Union were offered the patent to a newfangled communication device invented by Alexander Graham Bell, one we know as “the telephone.” Sure, the asking price was high at $100,000, but you’d like to think the head of a communications company might see the value in the fucking telephone. Instead, he balked, apparently not seeing a future in the device that would, you know, literally define the future.

This is where that lack of vision comes in. Western Union was used to doing business a certain way, and the telephone would have changed that. See also: Sears, who could have seen this whole Internet coming and positioned itself to take advantage of it, but no, Amazon won that race.

2. The Byzantine Empire Not Buying A Huge Cannon

In general (pun absolutely intended), if you're at war, don't turn down enemy-destroying weapons.

1. Not Letting Hitler Drown

By contrast to some earlier entries, this one's not a lack of vision. I'd like to think most of us would save a drowning kid, if we had the opportunity and didn't know for sure he'd grow up to be a genocidal autocrat.

In 1894 in Germany, a man named Johann Kuehberger saw a drowning boy in a river and, without a second thought, strode in to save his life. Incredible, awe-inspiring, medals all around, right? At the time, sure.

Time-travelers, take note: this is your moment.

Except killing baby Hitler is the biggest cliché in time-travel fiction, and we don't know if the result would have been worse. "But what could be worse than 11 million dead people?" Well, someone just as evil and actually competent might have risen to power, someone who wouldn't have done the 20th century equivalent of turning down a war-ending cannon by losing all their rocket scientists and nuclear engineers.

4 Like • Comment • View 1 comment

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