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:

Write_Mikey_Write!

about 23 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

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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.

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.

December 24, 2019 at 12:02am

#971952

Dreams on a Christmas Eve

Despite occasionally falling prey to sleep paralysis, sleeping is one of my primary favorite activities. So yeah, occasionally I post articles about it, like this one.

https://aeon.co/essays/the-cure-for-insomnia-is-to-fall-in-love-with-sleep-again...

Falling for sleep

When wakefulness is seen as the main event, no wonder so many have trouble sleeping. Can we rekindle the joy of slumber?

Sleep has been transformed from a deeply personal experience to a physiological process; from the mythical to the medical; and from the romantic to the marketable. Our misconstrued sense of sleep and consequent obsession with managing it are the most critical overlooked factors in the contemporary epidemic of sleep loss.

Okay, look, science is a good thing overall. The more we learn about ourselves, the world, and the universe, the better. That doesn't mean we have to stop being poetic about things.

People with insomnia also suffer immensely by night. Beyond obvious frustrations around their thwarted efforts, and growing anxieties about the consequences of ongoing poor sleep, many also experience loneliness, shame and hopelessness.

Causation, correlation, or reverse causation?

We can no longer deny the striking parallel between global warming and chronic inflammation. Ultimately, our excessive consumption of energy overheats the planet and its people, both of which need to cool at night to sleep. Chronic inflammation can literally raise our core body temperature to a near-feverish point, overheating our brains and bodies and disrupting our sleep.

This is what is known as "a stretch."

Thomas Edison led a popular charge to dominate and even eliminate sleep.

Thomas Edison was a dick.

Medicalisation reduces sleep to just another health regimen, so we attempt to manage it the way we do exercise, stress and diet. We compulsively tweak our sleep with an endless stream of expert tips from countless books, articles and blogs. When these patchwork fixes fail, which they inevitably do, we are vulnerable to the seduction of direct-to-consumer advertising for sleeping pills.

Who's this "we?" I've had trouble sleeping in the past, but always avoided sleeping pills. Though I will admit to occasionally popping a muscle relaxer (prescribed for other reasons) to help me get to sleep.

Great philosophers have taught that most of us mistake the limits of our own perception for the limits of the universe. Nowhere is this conundrum more relevant than in our contemporary take on sleep. We are mired in a pre-Copernican-like, wake-centric era regarding consciousness. We presume waking to be the centre of the universe of consciousness, and we relegate sleeping and dreaming to secondary, subservient positions.

This, now... I like this analogy.

Is it possible that our more modern sense of the divine encourages hyperarousal? After all, the Judeo-Christian God rarely descends from his abode in the heavens and is known to have worked relentlessly for days on end before taking a day of rest. In sharp contrast, although capable of flight, the gods of antiquity were grounded. They lived on earth, atop a mountain, or like Nyx and Hypnos, underground. Although Nyx ascended nightly in devotion to her duty, she faithfully returned to her subterranean abode for daily rest.

And then another stretch. Though I think there might be a causal influence from organized religion, it's, again, the Puritanical drive to always be productive - and one cannot be productive while sleeping.

As the body settles into bed, our challenge is to let go of our ordinary mind, our waking sense of self. This part of us, the part of us we usually call I, is simply incapable of sleeping. It can walk us to the shoreline of the sea of sleep, but it can’t swim.

Ever been frustrated that you can't point to a single instant and go: This is where I fell asleep? I have.

Anyway, the article provides an interesting perspective, I think. While the author sometimes trails off into breathy numinous fogs of speculation, I liked the basic theme, and it's worth reading especially if you have a troubled relationship with sleep.

I've seen a few answers to the question, "why do we sleep?" Some of them are purely scientific: it's when memories get fixed in our minds; it cleans the brain of excess proteins; etc. But I also accept personal and psychological reasons. For me, it's a reset button. Like sometimes you have to reboot your computer because it's doing weird shit? That's sleep, for me.

Which doesn't explain why I do weird shit all the time, not just when I'm tired.

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