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:

Write_Mikey_Write!

about 15 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 17 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

April 11, 2020 at 12:02am

#980742

All In A Dream

Many people dislike nightmares. I'm not a big fan myself, but they do sometimes give me story ideas.

https://www.theatlantic.com/magazine/archive/2019/04/dreams-what-do-they-mean/58...

Bad Dreams Are Good
Your night life prepares you for what’s to come.

I've come to be wary of anyone who states, conclusively, that they've finally figured out what purpose dreams have. But speculation? That's fine.

A handful of theories predominate. Sigmund Freud famously contended that they reveal hidden truths and wishes.

I'm pretty sure every single thing Freud said has been debunked, which is remarkable, given that even Donald Trump has been caught telling the truth at least once. You'd think Uncle Sig would have been right once or twice by accident, but no, I don't think so.

More recent research suggests that they may help us process intense emotions,

"process?"

or perhaps sort through and consolidate memories,

init defrag

or make sense of random neuron activity,

But what's doing the sense-making? Other neurons.

or rehearse responses to threatening situations.

Great, if I'm ever in a situation where I can fly away from a threat, I'll know exactly what I need to do.

Others argue that dreams have no evolutionary function, but simply dramatize personal concerns.

I'd be even more wary about "evolutionary function" arguments, pro or con.

Eight percent of dreams are about sex, a rate that holds for both women and men—though women are twice as likely as men to have sexual dreams about a public figure, while men are twice as likely to dream about multiple partners.

Sex? These days, some of us are wistfully dreaming about even casually touching another human being.

Anxiety is also rife: A study of Canadian university students found the most common dream topics, apart from sex, to be school, falling, being chased, and arriving too late for something.

So, Canadians are hyper-worried about being late. Interesting.

Which reminds me of a recurring dream I have. I suddenly remember, most of the way through the semester, that I signed up for a course I never showed up for, and now I have to take the exam.

I still have those dreams even though I haven't been in school for centuries.

A 1958 study determined that compared with Japanese people, Americans dreamed more about being locked up, losing a loved one, finding money, being inappropriately dressed or nude, or encountering an insane person. Japanese people were more likely to dream about school, trying repeatedly to do something, being paralyzed with fear, or “wild, violent beasts.”

Kaiju!

Speaking of trying repeatedly to do something in a dream, does anyone else have the thing where you keep trying to enter a number into your cell phone (or wherever), but you keep messing it up and have to start over? Or is that just me? It's like, in any dream, I find it impossible to ever type in the right numbers or spell anything correctly.

So the next time you dream about an education-related sexual experience in which you are both falling and being chased, don’t worry: It’s probably totally meaningless. Then again, your brain might be practicing so you’ll be ready if such an event ever comes to pass.

And like I said, if I'm ever in a situation where I can escape danger by zipping into the sky like Superman, or running away on all fours, my dreams will have prepared me well.

Here's a bit more on dream theory: https://www.sleep.org/articles/dream/

I probably would have gotten more out of the Atlantic article here if I'd bothered to follow the reference links. As it is, I remain unconvinced.

Perhaps I'll sleep on it.

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