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🎶

about 22 hours 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

November 1, 2024 at 11:01am

#1079301

Cave In

Did my early voting today, so I can spend Election Day doing more important things, like drinking.

Unrelated to that or, well, anything else, really, is this Atlas Obscura article featuring an obscure city in an obscure state:

The Strange Heat Island Lurking Beneath Minneapolis
An urban explorer ventured deep below downtown in search of Schieks Cave. What he found changed science.

Greg Brick knew it was there, lurking beneath his city, hidden within the Minneapolis water and sewer system: an enticing geologic anomaly called Schieks Cave.

That sort of thing fascinates me, but I can't be arsed to do all the work or break the laws necessary to explore for myself.

When Brick arrived at the cave, he shined a flashlight into the thick darkness of the city’s underbelly. He found not just the natural void but also concrete walls that previous generations of civil engineers had built to support the natural structure.

If those civil engineers were anything like me, they wrote something like "put a concrete wall here" on a map, and some minimum-wage temps did the actual work.

The water was about 20 degrees hotter than it was supposed to be. It was more like the groundwater of Mississippi than that of Minneapolis. Something was warming the water beneath his city.

Well, clearly, that's because Minneapolis is actually a gateway to Hell.

[The cave's] discovery in 1904 by a city sewer engineer was initially kept secret lest the public fear Minneapolis had been built on unstable ground.

Apparently, it was, else generations of civil engineers wouldn't have specced out concrete support walls in the cave.

He finally had a way to access the cave—but there was another problem. Along the route, the raw sewage poured from shafts overhead, shooting bacteria into the air as it splashed down and creating what’s politely known as “coliform aerosols.” Unsurprisingly, Brick got sick.

Another reason for us civil engineers to stay at our desks.

In 2008, a separate team from the University of Minnesota had predicted that heat from Minneapolis’s urban surface was conducting itself deep underground, heating the groundwater there like a metropolitan microwave.

As the article notes, this turned out to be the correct explanation, not the "gateway to Hell" one. Much to my disappointment.

But it’s not all bad news: Canadian and European researchers recently suggested recycling underground heat and using it as a low-carbon way to heat homes, while also cooling the groundwater back down to normal temps.

I hope their "suggestion" included exactly how to do that.

Now, from the headline, I was expecting some discovery of dark matter or a violation of the Second Law of Thermodynamics or whatever. But no. Still, at least it's not boring.

3 Like • Comment • View 5 comments

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