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.

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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.

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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.

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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.

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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.

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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.

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July 15, 2023 at 6:46am

#1052625

The Proverbial Trap

This guy looms large around here. Sometimes literally, what with all the statues.

Thomas Jefferson’s 10 Rules for Life

Like many of the Founding Fathers, Thomas Jefferson was a dad.

I mean, it's right there in the group's name.

Article is from Art of Manliness, which is biased toward fatherhood as a prerequisite for "manliness," whatever that is. I'm not defending the site here, or Jefferson's personal life.

And like a lot of dads, he often took the opportunity to dispense unsolicited dad advice to his children.

I like to think he made numerous, cringeworthy puns as well. They're called "dad jokes" now, but I'm living proof that you don't have to be a dad to make them.

In an 1825 letter to John Spear Smith, Jefferson laid out his refined list of adages that he called his “Decalogue of Canons for observation in practical life.”

You might recognize "decalogue" as the Greek name for what we call the Ten Commandments. Jefferson, as should be widely known, was not above editing the Bible.

I'm not going to get much into the explanations of the rules; you can go to the link for that. No, as a fellow Charlottesvillian and graduate of the University he founded, I'm going to note my own version of the Rules, updated for life nearly 200 years later.

1. Never put off till tomorrow what you can do today.

Waltz: Never put off until tomorrow what you can put off until the day after tomorrow. Life's too short to focus on productivity all the time.

2. Never trouble another for what you can do yourself.

Waltz: Never do yourself what you can pay someone else to do for you. (And I can't resist pointing out that this Rule is pretty fucking ironic coming from a slaveholder.)

3. Never spend your money before you have it.

Waltz: Own, don't rent.

4. Never buy what you do not want, because it is cheap; it will be dear to you.

Waltz: Never buy what you do not want. (This Rule took me a while, I guess because language changed in two centuries. I think that last phrase can be translated as "it will be more expensive in the long run.")

5. Pride costs us more than hunger, thirst, and cold.

Waltz: We really need to come up with a different word; there's good pride and bad pride. (Did Jefferson, landed gentry, know hunger or thirst? Cold, I have no doubt of.)

6. We never repent of having eaten too little.

Waltz: Don't go hungry if you can at all avoid it. It's distracting.

7. Nothing is troublesome that we do willingly.

Waltz. Doing nothing is always an option. Unless you're hungry.

8. How much pain have cost us the evils which have never happened!

Waltz: How much pain have cost us the embarrassing things we did that we only remember at 3 a.m.

9. Take things always by their smooth handle.

Waltz: Laziness is productive.

10. When angry, count ten before you speak; if very angry, a hundred.

Waltz: Actions speak louder than words.

Okay, I stole that last one from my mom, who was fond of repeating proverbs (which are things entirely different from pronouns, again illustrating how freakin' weird English is). And obviously, I don't actually believe it, because here I am typing words.

Anyway, I'm sure you'll have your own opinions on these things. Let's not fall into the trap of thinking that because a famous person from history said it, it's any more profound than if a nobody from today says it.

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