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 7, 2024 at 10:08am

#1079616

Past and Future Intents

I've lined Aeon articles in here before. This seems to be an affiliated site, Psyche. Apparently, someone loves Ancient Greek.

How to do mental time travel
Feeling overwhelmed by the present moment? Find a connection to the longer view and a wiser perspective on what matters

Bullshit sense... tingling!

But let's give this a chance.

You have a remarkable talent – the ability to step outside the present, and imagine the past and future in your mind’s eye.

Hey, you know what else you can imagine? Things that never happened, places that don't exist, and impossible scenarios. It's a rather important ability for fiction writers. Or city planners.

Some people apparently don't have the capability for visualization, while others can experience it more vividly. But I think it's fair to say most people have some ability to imagine.

Known as ‘mental time-travel’, some psychologists propose it’s a trait that allowed our species to thrive.

Well, at least this is the polar opposite of the "staying in the present moment" crap that's been circulating. I'll give it that.

If I ask you to imagine what you did yesterday, or what you’re planning for tomorrow, you can conjure up rich scenes in the theatre of your mind.

Well. Some of us can. I doubt the author intended to be ableist, but this is a bit like saying, "If I ask you to walk a mile, you can get up and walk a mile," ignoring or forgetting that paraplegics exist.

In the accelerating, information-rich, target-driven culture of the early 21st century, the present often dominates thoughts and priorities instead.

And some people seem to want to make "the present" the only priority.

We need to be present-minded sometimes. However, too much focus on the ‘now’ can also lead to the kind of harmful short-termism that infuses business, politics and media – a near-term perspective that worsens many of the long-term challenges we face this century, such as the climate crisis.

I can't really disagree with that. I've said similar things. Whether we can visualize things or not, we can learn from the past and make plans for the future, and both of those things are important.

But it’s also compounded by a host of unhelpful human habits and biases too, such as our ‘present bias’, whereby we tend to prioritise short-term rewards over long-term benefits (the classic example is the marshmallow test, in which some children can’t resist eating a single treat now, rejecting the chance to chomp two later on).

Yeah... that might not be the best example

to quote.

A longer view provides a deeper, richer awareness of how we fit into the human story – and the planet’s – and reveals just how fortunate you are to be here, right now. The geologist Marcia Bjornerud calls this perspective ‘timefulness’.

I'm not sure I like that name any more than I like its apparent inspiration, mindfulness.

Why a geologist gets to weigh in at all should be obvious: they generally cultivate a real sense of deep time, working as they do with rocks that sometimes predate eukaryotic life.

In this Guide, I’ll share practical tips and exercises that can help you escape the unwanted, short-termist distractions of the present, and discover the upsides of a longer time perspective.

The author proceeds to do just that, and at length. I don't think I need to copy anything else; if you're interested, go to the link (hopefully it won't rot anytime soon). I will note, however, that he does turn back to "mindfulness" at one point in there.

As for my bullshit sense, well, jury's still out for me. I tend to distrust pop psychology (for example, the marshmallow study, above), though that doesn't mean it's all bullshit. But right now, I have plans for the rest of the day because, no, I don't live in the present.

3 Like • Comment • View 2 comments

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