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.

2 Like

July 25, 2023 at 9:15am

#1053028

Á La Mode

It's unlikely that any individual is "an average human." To claim so would be mean.

What does the average human do every day?
The Human Chronome Project finds that the average human sleeps for 9 hours but only works for 2.6 hours.

Pun intended, of course, if a bit forced.

While each of us has a decent conception of how we spend our own time, the actions of our fellow humans — from our next-door neighbors to people living in faraway countries — can seem quite mysterious. Do they watch as much TV? Work as many hours? Fiddle with their smartphones as frequently? Cook as often? Spend as much time watching their kids?

Even some of those questions reveal a developed-world bias.

First and foremost, the average human spends about 9.1 hours sleeping or resting.

As the article notes, "human" includes everyone, including children. I usually manage something like that, but certainly did not when I was working full-time.

While awake, the average person spends close to one-third of their day on passive, interactive, and social activities. These roughly 4.6 hours include reading, watching TV, making art, playing games, going to gatherings with friends or family, or simply doing nothing at all.

The idea of "simply doing nothing at all" seems incomprehensible to most Americans.

Eating and food preparation accounts for another 2.5 hours.

Yes, fast food isn't universally available.

Hygiene, including grooming, washing, and dressing, takes about 1.1 hours.

Oh, that's what I'm missing.

Cleaning and maintaining the spaces we inhabit costs us 0.8 hours of the day.

And that.

Though employment can be hugely time-consuming for working individuals, when looked at through the lens of the global human day, it appears as a mere sliver of time, just 2.6 hours.

Coincidentally, that's about how much actual productive time most office drones achieve.

Education also isn’t very demanding, only 1.1 hours.

That's because a whole lot of 0s skew the average.

The researchers were also curious about how average human time use changes with wealth.

One big advantage of being rich is grasping more leisure time by paying others to have more working time. For instance, I (not really rich for an American, but close enough) pay people to mow my lawn. To be clear, though, it used to take me about 4 hours to do it, while they knock it out in 15 minutes.

Additionally, people in the richest countries spend an average of just five minutes a day growing and harvesting food, while people in the poorest countries spend well over an hour doing so.

As this illustrates, you do, too, if you go shopping at a grocery store instead of growing your own food and butchering your own hogs.

By assembling the human chronome, the researchers say that we can compare ourselves to civilizations from the past.

Right, because we know exactly how they used all of their days. Also, civilization itself changes its individuals' time utilization.

More importantly, we can see from a high-level, empirical perspective what our species is doing on our planet and make more informed decisions about reallocating our collective time to change the world and society for the better.

Like that's going to happen. Also, the worst excesses are perpetrated by a very small, average-skewing group with greater resources (aka "the Rich").

So, while this research may be useful (no such thing as useless knowledge), I don't think it has the lofty practical potential that they claim. Not to mention that just looking at an "average" (however that's computed) tells you enough about the range. It's just another way of manipulating with statistics. For example, "the average human" possesses fewer than 2 legs. Think about it, and you'll realize I'm right. But saying that can be misleading, as the vast majority of people have exactly 2 legs.

Like me! I always knew I was above average at something.

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