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:

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

1 Like

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

October 31, 2024 at 9:44am

#1079221

Phases

Samhain, and the temperature is supposed to get up into the 80s (Freedom Units) today. Not that I'm unhappy about that, but it is somewhat unnerving for the end of October.

This Halloween night coincides closely with a new moon (8:47 Eastern tomorrow), so it's not nearly as cool as when it coincides with a full moon.

So I'm doing... absolutely nothing for the occasion. Except, of course, trying to keep both of the black cats indoors, because some people are idiots.

Speaking of the moon, though, here's a lunar mythology article from Discover:

Myths of the Moon Shaped Ancient Cultures and Modern Cultures of Today
From ancient myths to modern superstitions, all cultures — even modern ones — tell stories about the moon.

The moon is something all people on Earth, no matter where or when they’ve lived, have in common.

Unless, I suppose, you spend your entire life underground.

That we all share the moon does not mean that we imagine it the same way, however. In our myths and stories, the moon plays many different roles.

We know more about the moon than ever before, and we continue to learn more. That dead hunk of rock in the sky still has secrets to give up. Myths and legends don't tell us much about the moon, but they do reveal a lot about us.

The relationship between the moon and the tides, for example, was clear to early peoples who lived near the sea, explains Tok Thompson, an anthropologist at the University of Southern California who specializes in folklore and mythology.

That's probably an example of noticing a correlation without understanding the deeper effects (in this case, gravity) behind the correlation. In other words, the "observation" part of the scientific method.

The moon, with its regular phases, also functioned as a calendar. You could track the days by the sun, but for longer periods, the moon was useful. The English word moon comes from mensis, the Latin word for month, which is also the origin of the word ‘menstruation.’

Much as I enjoy a good etymology, I'm not sure this is right. Wiktionary

traces the word back through Proto-Germanic, all the way to PIE. This suggests to me that, rather than having its origin in Latin, the Latin and the Germanic words share a root.

I'm not an expert, by any means, but this is enough to make me question the "comes from Latin" assertion.

Pretty sure Luna is from Latin, though; that was the name of a moon goddess.

All that aside, let's stop pretending that Gregorian calendar months are related to lunar cycles.

So yes, ancient cultures knew a lot about the moon, and they wanted to share that knowledge to keep a record of it.

I also question the qualifier "a lot" in that sentence.

Thompson’s favorite myth involving the moon is a creation story from the Tlingit people of the northwest coast of North America. In this story, an old man keeps all the light in the world stashed away in a box. Through wiles that vary from telling to telling, the trickster Raven steals the box and releases the Sun, the moon, and all the stars, bringing light to the world.

One could be forgiven for wondering how Raven was able to see the box, considering there was no light at the time, but myths follow their own, dreamlike, rules.

In Chinese mythology, a woman named Chang E drank a magic elixir, whereupon she floated all the way to the moon, and there she lives still — with, in some versions, a rabbit.

Radical feminists: "Hey, gimme some of that elixir!"

We still have some fascinating myths about our favorite satellite. Probably the most persistent is that the moon can drive us mad.

I find that it's necessary to separate the two meanings of "myth." A foundational story, like the ones about Raven or Chang E, is not the same thing as a persistent falsehood. I don't think this article draws the distinction.

Many people — including some healthcare workers and police officers — believe that crime, traffic accidents, psychiatric hospital admissions, and even suicides spike during a full moon.

Or that could be cognitive bias. You notice things more when they fit your pre-existing worldview. Like when you're frustrated by machines and bureaucracy, and think "Mercury must be in retrograde."

These days, we don’t blame the effect on moon goddesses or magic elixirs but on something at least quasi-scientific: the moon’s gravitational effect on the water in our bodies. On its surface, that makes a lot of sense.

No. No, it does not.

After all, the moon creates tides, and roughly half our body weight is water.

Okay, but have they ever observed tides in a mudpuddle?

And indeed, though a few small studies have found some possible effects of the moon on mental health, research over the past decades has not borne this out. For example, a Swiss study published in 2019 looked at almost 18,000 cases of in-patient psychiatric admissions over ten years and found no correlation between the phase of the moon and psychiatric admissions or length of stay.

There's some nuance there, but basically, the "moon causes madness" bit is the second definition of myth.

Pausing to gaze at the moon can create an intense feeling of both awe and peace. Rather than making us mad, the moon might just make us a little bit more sane.

Nice to think, but there's little evidence for that, too.

Still... go look at the moon. Might want to wait a few days, though; it's too close to the sun right now.

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