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

March 19, 2021 at 12:03am

#1006640

I'll Do It Later

Well, here's a source that's new to me. Or at least it was when I found this article. It's been in my queue for a while. No, I haven't been putting it off, but I'll say I have for the joke.

How to Beat Procrastination Like a Stoic Philosopher
Seven tactics from the ancient world that have stood the test of time

Seven tactics? What is this, a numbered list? That's Cracked's schtick. Well, okay, these aren't numbered.

For tens of thousands of years, people have been procrastinating just like you do today: They put things off, delayed, made excuses, and wished their deadlines would disappear. And just as it does with you, this caused them anxiety, made them piss off their colleagues and families, and, worst of all, wasted time.

Yes, but at least it's a source of comedy material.

Fortunately, unlike our ancient counterparts, we have ages of wisdom to help us avoid the mental traps that lead us to procrastinate.

Unfortunately, unlike our ancient counterparts, we have Netflix and social media.

Action by action

Don’t let your imagination be crushed by life as a whole… Stick with the situation at hand, and ask, “Why is this so unbearable? Why can’t I endure it?” You’ll be embarrassed to answer.—Marcus Aurelius, Meditations, 8.36

"Action by action?" That doesn't even make sense. Or, rather, it can make at least three different kinds of sense.

While it can be productive to think about the troubles that might lie ahead — the Stoics used an exercise called premeditatio malorum, or “premeditation of evils,” to prepare for potential adversity — imagining the worst usually just causes us to become paralyzed with fear.

No, imagining the worst means I can only be pleasantly surprised, or right. Either way, I win.

This is why Marcus Aurelius’ advice was to keep in mind that a life is built action by action.

Oh, so that's the sense the heading means.

No author ever writes a book, he would say. Instead, they write one sentence and then another and then another.

That's, like... the worst way to write a book.

Create a routine

In many circumstances, we do not deal with our affairs in accordance with correct assumptions, but rather we follow thoughtless habit.—Musonius Rufus, Lectures and Fragments, 6.7

Thoughtless habit can be a good thing. Not always. But it can be.

Really, I won't be quoting all of it; the article speaks for itself. But that's a taste.

I'll just have one more thing to say:

Well-being is attained by little and little, and nevertheless is no little thing itself.—Zeno

That's rich, coming from the guy who famously said it's impossible to go anywhere.

It's almost as if philosophers have always been self-contradictory.

Anyway, yeah, I'm a professional crastinator, but this advice might help. If I ever get around to reading the rest of the article.

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