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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 20 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

about 24 hours ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

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.

3 Like • Comment • View 1 comment

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