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 21 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day 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🎁

Dec 23, 2024 at 12:56pm
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

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
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

August 20, 2024 at 10:33am

#1075503

Some Clouds Are Just Clouds

This one's from Forbes, and to me is an exercise in suppressing my confirmation bias, because it's saying the same shit I've been saying, only by an actual mental health professional.

A Psychologist Explains ‘Oppressive Optimism’—And The Toxic Side Of Self-Care

Being a professional in a field doesn't mean you're always right; thinking that's the case would be another fallacy. Still, I think the article is worth contemplating. Whether my comments are is an open question.

Self-care has become a buzzword in wellness culture and for good reason—it’s essential for our mental, emotional and physical health.

Assertion without evidence can also be a problem.

These practices can work wonders, but when chasing positivity turns into an obsession—it can lead to what researchers call “toxic positivity.”

Which I've ragged on in here before, repeatedly... though not so much as to have exhausted my patience with the topic.

In essence, toxic positivity invalidates negative experiences by promoting the idea that one should always maintain a sunny outlook, regardless of circumstances.

I believe I've used examples of that before like "My dog died." "That's great! Now you can get a new puppy!"

Here are three specific examples of how self-care can slip into toxic positivity and end up doing more harm than good.

As this is not Cracked, the list actually counts up.

1. The Social Media Mantra—“Good Vibes Only”

While surrounding oneself with positivity can be uplifting, the insistence on “good vibes only” creates an environment where negative emotions are not just unwelcome but are seen as personal failings.

I'm not sure this is so much an insistence on toxic positivity as a desire to not get your own vibes harshed. Like, you're at a party, and everyone's having a great time, and one of your friends staggers in late and bleeding. "Got in a wreck," she moans. Normally, you'd probably be there for your friend, call an ambulance for her, maybe even wait in the ER with her. But you're at a party. How dare someone turn it into an emergency!

2. Oppressive Optimism—“Everything Happens For A Reason”

I've railed against that duck-billed platitude before, and I'll do it again.

Optimism is crucial when coping with negativity...

Another assertion without evidence.

...but the belief that everything happens for a reason can invalidate genuine feelings of fear and uncertainty.

This is related to "it's all part of God's plan," though that one's more specific to certain religious people.

3. Affirmation Overload—“I Am Strong, I Can Handle Anything”

Picture someone going through a tough divorce, constantly telling themselves, “I am strong” and “I can handle anything” to keep up their confidence and stay positive. While this might seem helpful, it might stop them from sharing their pain and struggles with loved ones, out of fear that admitting vulnerability will make them look weak or uncertain about their decision.

And maybe that no one wants to be around them if they're not going to give off good vibes.

There's not much more to the article, really; the conclusion boils down to "seek help," which is fine advice, but what I'd need help with would be finding a shrink. Too much work. I'll just stay over here and be grumpy by myself.

3 Like • Comment

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