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🎁

a day 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 7, 2024 at 10:51am

#1080897

Story of the Century

Speaking of time, here's a Guardian article about people who had more of it than usual.

Never take health tips from world’s oldest people, say scientists
Scientists still trying to work out why some people live beyond 100, but agree it is best to avoid taking advice from centenarians themselves

No, we should definitely take health tips from people who die young, instead.

The death of the world’s oldest person, Maria Branyas Morera, at the age of 117 might cause many to ponder the secrets of an exceptionally long life, but scientists say it could be best to avoid taking advice on longevity from centenarians themselves.

Far as I can tell, the secrets to an exceptionally long life include such gems as "stay alive" and "don't die."

According to the Guinness World Records website, Branyas believed her longevity stemmed from “order, tranquility, good connection with family and friends, contact with nature, emotional stability, no worries, no regrets, lots of positivity and staying away from toxic people”.

Also, unicorns and fairies. I mean, those are probably more real than her litany.

However, Richard Faragher, a professor of biogerontology at the University of Brighton, said that in reality scientists were still trying to work out why some people lived beyond the age of 100.

Because they didn't die.

Or, in some cases, because they assumed the identity of their deceased parent so they could go on collecting the... whatever benefits.

Faragher said there were two main theories and they were not mutually exclusive.

The first, he said, was that some individuals were essentially just lucky.

At some point, though, lucky stops being "you didn't die" and starts being "you died."

The second theory, he said, was that centenarians had specific genetic features that equipped them to live a longer life

So, a different kind of luck. But still luck.

Faragher said both theories, however, resulted in the same warning: “Never, ever take health and lifestyle tips from a centenarian.”

Certainly, if I'm unlucky enough to live that long, I'd troll the hell out of anyone asking me about health tips. "See, now, the key is to kick a puppy every day. Doesn't have to be your puppy. Doesn't even have to be the same puppy. But it's gotta be a puppy, not a dog. Or a kitten or a kid. Puppy."

He added: “What you see with most centenarians most of the time – and these are generalisations – is that they don’t take much exercise. Quite often, their diets are rather unhealthy,” noting that some centenarians were also smokers.

What would amuse the hell out of me would be if no exercise, diets considered unhealthy, and smoking really were the keys to long life.

“The fact that [centenarians] do many of these unhealthy things and still just coast through [life] says they’re either lucky or typically very well endowed [genetically],” he said.

Again, both of these things are luck.

Faragher added that many of the mooted possibilities for why centenarians live longer could actually be examples of reverse causation. For example, the idea that having a positive mental outlook can help you live for a very long time might, at least in part, be rooted in people being more sanguine because they have better health.

Glad they acknowledged this. I mean, they could do studies to gain insight into it, but apparently it's more important to promote "positive mental outlooks," especially in a world falling apart all around us.

“From about 100 years ago, what we started seeing was huge advances in life expectancy driven by improvements in reducing the likelihood that children die,” said David Sinclair, the chief executive of the International Longevity Centre, noting that was largely down to the introduction of vaccinations and clean water.

Well, that lasted about a century.

“What we’ve had over the last 20 years, and we’re going to see over the next 20 years, is a similar focus in terms of old age,” Sinclair said, adding that included improvements in vaccines for flu and shingles, statins, and other medications that would help increase life expectancy among older people.

Sure, because, obviously, life expectancy is a more important metric than life quality.

But he said governments also needed to take action to help individuals to make healthier choices – choices that would ultimately help them live longer – adding that many people lived in environments where it was difficult to exercise, eat well or avoid pollution.

Ah yes. "Want to live longer? It's your fault if you don't, even though you're economically stranded in a polluted area with little access to fresh food, thanks to our policies. So instead of fixing economic disparity, despair and environmental degradation, we'll make it illegal for you to eat cheeseburgers. There, we fixed it!"

As Sinclair said, while news stories about centenarians tended to be upbeat, it often emerged that such individuals faced challenges, such as living alone for many years.

Challenges? Hell, that's probably the real reason they lived longer: not having to deal with other peoples' bullshit all the damn time.

8 Like • Comment • View 2 comments

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