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 5 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.

1 Like

🐕GeminiGem🎁

about 9 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

about 12 hours ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

about 12 hours 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

November 11, 2024 at 9:17am

#1079805

What Is the Meaning of This?

Still not quite sure what to make of this Quartz article. It's from 2018, but new to me, though I don't know if the book it's promoting is still available or not.

The secret to a meaningful life is simpler than you think

I don't know; a while back, I arrived at a pretty simple meaning of life: it has no meaning except for what we impose upon it. Impose a complicated meaning, and of course your life is going to be complicated.

Some people seem to spend their whole lives dissatisfied, in search of a purpose.

Or at least in search of that One Perfect Product that will finally live up to its advertising and fix everything wrong with your life.

But philosopher Iddo Landau suggests that all of us have everything we need for a meaningful existence.

The first thing you need for a meaningful existence is an existence.

Philosophers’ answers to this question are numerous and varied, and practical to different degrees. The 19th-century philosopher Friedrich Nietzsche, for example, said the question itself was meaningless because in the midst of living, we’re in no position to discern whether our lives matter, and stepping outside of the process of existence to answer is impossible.

I'm not sure Nietzsche is the best example to feature, but okay, whatever.

Those who do think meaning can be discerned, however, fall into four groups, according to Thaddeus Metz, writing in the Stanford Dictionary of Philosophy. Some are god-centered and believe only a deity can provide purpose. Others ascribe to a soul-centered view, thinking something of us must continue beyond our lives, an essence after physical existence, which gives life meaning. Then there are two camps of “naturalists” seeking meaning in a purely physical world as known by science, who fall into “subjectivist” and “objectivist” categories.

I'm not well-read enough to pick sides there, but the only context in which I know the word "objectivist" has to do with philosophers like Ayn Rand, who can go fuck off.

Landau argues that meaning is essentially a sense of worth which we may all derive in a different way—from relationships, creativity, accomplishment in a given field, or generosity, among other possibilities.

It seems self-evident that we each value different things, so what's meaningful to one person won't do much for another. For instance, lots of people seem to get a sense of accomplishment from working. I, on the other hand, feel accomplished if I can avoid work.

For those who feel purposeless, Landau suggests a reframing is in order. He writes, “A meaningful life is one in which there is a sufficient number of aspects of sufficient value, and a meaningless life is one in which there is not a sufficient number of aspects of sufficient value.”

I don't claim to be an expert on logic or philosophy, but that statement strikes me as both circular and weaselly.

Landau argues that anyone who believes life can be meaningless also assumes the importance of value. In other words, if you think life can be meaningless, then you believe that there is such a thing as value.

I'm not sure I agree with that. If you believe life is meaningless and that's okay, I think you're acknowledging the existence of value, but not necessarily its importance.

In Philosophy Now, Tim Bale, a professor of politics at Queen Mary University of London in the UK, provides an extremely simple answer: “The meaning of life is not being dead.”

See? This sort of thing is why I picked engineering over philosophy.

There are a few other examples of simple answers to the question of life's meaning, and I'm really quite proud of myself for making it this far without quoting "42" as the answer, which, honestly, I'm only doing to forestall the inevitable comment about it.

I'm not sure I buy any of the arguments, though. Perhaps I find meaning in skepticism.

4 Like • Comment

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