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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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').

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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

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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

September 19, 2021 at 12:02am

#1017695

The Thinkers

Not all philosophers had big bushy beards.

Wise women: 6 ancient female philosophers you should know about

I'm still bingeing Lucifer on Netflix, so this is a short one.

When we conjure up ancient philosophers the image that springs to mind might be a bald Socrates discoursing with beautiful young men in the sun, or a scholarly Aristotle lecturing among cool columns.

Well, there are definitely togas (or whatever they called them in Ancient Greece) involved. And beards.

More than two millennia later, intelligent, verbal women still struggle to have their own voices heard. So here are six ancient female philosophers you should know about.

Some of these I'd heard of. Others I hadn't. Hence my saving the link for the blog.

1. Aspasia of Miletus

She was not only remembered for her captivating beauty, but also for her captivating mind. Socrates himself called Aspasia his teacher and relates he learned from her how to construct persuasive speeches. After all, he tells us, she wrote them for Pericles.

For some reason, male philosophers are never remembered for their hotness.

2. Clea

Clea (most active around 100 CE) was a priestess at Delphi — a highly esteemed political and intellectual role in the ancient world. The religious practitioners at the shrine received frequent requests from world leaders for divine advice about political matters. Clea was part of this political-religious system, but she believed in the primary importance of philosophy.

You know that scene in 300 with the Oracle at Delphi? I have reason to believe Snyder took a few liberties with it.

3. Thecla

When she first appears on the scene in the Acts of Paul and Thecla, Thecla (most active around 1st century CE) is leading a normal middle class life, sequestered at home and about to make an advantageous marriage. But leaning out of her balcony, she hears the dynamic preaching of Paul and decides on a radically different path.

The Acts of Paul and Thecla is part of New Testament apocrypha. I'm guessing it wouldn't have done to have a story about a woman (apart from the obvious) in what would become the Bible, so the Council of Nicaea threw it out.

4. Sosipatra

She was surrounded by male experts, one of whom was her husband Eustathius. But according to Eunapius’ biography in his Lives of the Philosophers, her fame was greater than any of theirs, and students far preferred her inspiring teaching.

I'll have to remind myself to look into this one further, as Neoplatonic philosophy is cool.

5. Macrina the Younger

Macrina (circa 330-379 CE) was the oldest of ten in an expansive, influential well-educated Christian family in Cappadocia.

Count this as one I should have known about, but didn't.

6. Hypatia of Alexandria

Most famous for her dramatic death at the hands of a Christian mob, Hypatia (circa 355–415 CE) was a Neoplatonic teacher admired for her mathematical and astronomical works.

This is the one I knew most about, which unfortunately still isn't saying much. Also, there's an asteroid named after her (the trivia floating around in my head comes out at weird times).

In any case, the article is rather light on information, but contains helpful links. Which I'll follow. When I'm done bingeing Lucifer.

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