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

December 21, 2024 at 11:36am

#1081375

Neutral Zone

A neutron walks into a bar. "What'll it be?" asks the bartender. "How much for a beer?" "For you, no charge!"

Stop being positive and just cultivate neutrality for existential cool

While this Quartz article is from the faraway year of 2017, I found enough to snark on to make it worth sharing.

You’ve likely been asked how you see the proverbial glass: half full or half empty? Your answer allegedly reflects your attitude about life—if you see it half full, you’re optimistic, and if you see it half empty, you’re pessimistic.

I'm an engineer. All I see is an overdesigned glass. Or, depending on my mood, an inefficient use of available storage space.

I'm also a pessimist, but at least I'm a pragmatic one.

Implied in this axiom is the superiority of optimism.

Also, I don't know if even the most dedicated pessimist, not knowing and deliberately following this particular cliché, would seriously consider "half-empty" to be a thing. Almost everything is related to full. Like, if your fuel gauge is in the middle, you say "we have half a tank of gas," not "the tank's half-empty."

Thus, the good answer is to see the glass half full. Otherwise, you risk revealing a bad attitude.

Shut the fuck up about my attitude.

Actually, the glass isn’t half full or half empty. It’s both, or neither.

Come on now. No. It's not in a state of quantum indeterminacy. Or, at least, no more than any other object in view.

Things aren’t mutually exclusive, awesome or awful. Mostly they’re both, and if we poke around our thoughts and feelings, we can see multiple angles.

On that bit, though, I'm fully on board. I really hate it when people put things into two boxes: "awesome" and "suck." The moment Netflix went to shit was the moment it switched from star ratings to thumbs up or down. Of course, I'm fully aware that by hating it, I'm putting the idea of the awesome/suck binary into the "suck" box. Everyone is a hypocrite, including me.

Neutrality sets us free. It helps us see something more like the truth, what’s happening, instead of experiencing circumstances in relation to expectations and desires.

Ehhhh... nah. Pessimism, and only pessimism, sets us free. An optimist is doubly disappointed when their imaginings fail to materialize: from the positive outcome having not worked out, as well as the ego hit from being wrong. A neutral person risks never experiencing the joy of anticipation. It is only the pessimist who, if their prediction falls flat, still takes a win: either something good happens, which is good; or they turn out to be right, which is a pleasant feeling.

The article goes on to relate the quality of neutrality to Buddhism, I suppose in an effort to give neutrality some ancient gravitas, but instead, it only makes Buddhism seem even less appealing to me.

But hey, it's not about me. On the subject of whether this applies to anyone else or not, well, I'm neutral.

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