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 19 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 21 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.

February 8, 2020 at 12:18am

#975077

Regeneration

And now for my monthly (kinda) dose of confirmation bias.

https://www.vox.com/science-and-health/2019/11/12/20950235/ok-boomer-kids-these-...

Why old people will always complain about young people
Prediction: Today’s “OK boomer” Gen Z will complain about the youth one day. Blame human memory.

If all this article had to offer was a rejection of arbitrary and misleading "generations" nonsense, I wouldn't bother. That's kind of my argument in the first place. But it takes a few side trips that I found compelling.

To recap, I think "generations" are bullshit because they're over-generalizing, and because, in extremis, you can get situations like a pair of twins born bracketing midnight on January 1 of the arbitrary cutoff between generations, thus forever pigeonholing them into different cohorts. There will (as this article agrees) always be older people and younger people, and they'll always complain about each other, but this whole Boomer / Gen-X / Millennial / whatever crap is mostly just useful for marketing purposes.

So I'll skip the parts of the article that I already agree with and point out some of the interesting asides.

If you’re just tuning in: “OK boomer” is a clapback...

Normally, at this point, I'd stop reading. I may have a lot of time, but I have zero time for trendy catchphrases like "clapback." But, again, since the article was going in a direction I agreed with, I gave it a chance. Kids these days and their slang... wait.

Protzko became interested in the “kids these days effect” — the tendency for older people to say the youth of today are somehow worse than youth in the past — after surveying a few hundred of the top experts in developmental psychology.

The complaints adults make about kids through the ages are always the same: “They’re suggesting young people are lazy, they’re entitled, and they act in self serving ways,” said Cort Rudolph...

That's funny, because I'm lazy, I'm entitled, and I act in self-serving ways. I guess I'm actually young. Yay!

And this particular bias starts to make some sense in light of memory science.

This is where things start to get interesting, though it's still not anything I didn't already know from other sources.

So what does memory have to do with how adults see young people? In order to make a judgment call about today’s youth, you have to try to remember how kids used to be. But that’s hard! Who has objective information on how kids used to be? Since we don’t have it all, we use information from the present to fill in the gaps.

I'd imagine there are other biases at work. While some people have bad memories of childhood, and sometimes for good reason, I think most people tend to view it as an idyllic time (with maybe a few hiccups along the way).

Other possible causes: “kids these days” is cultural trope that people are taught. Or, it could be linked to the belief that the past, overall, was better than the present.

Note to self: read on before bringing up points like that. Anyway, I've always used "kids these days" humorously, as I do with "get off my lawn!"

If anything, data shows the rising Gen Z, at least, is better in many ways than generations past: They’re doing fewer drugs, drinking less alcohol, and having sex at older ages

Yeah... I'm not sure that's "better."

The differences we see in the generations, he argues, aren’t necessarily unique to those generations. They could just be more of a general reflection of age. Young people are always more self-centered and narcissistic. Older people are always a little more set in their ways.

That's what I've been trying to say.

Without the data, we talk about generations a bit like we talk about astrological signs. “The distinction between ‘my’ zodiac sign or generation and ‘their’ sign or generation gives people an opportunity to identify with a group while differentiating themselves from others,” Rudolph and some colleagues argue in a recent paper.

Yes, because that's what we really need now: more tribalism.

That was sarcasm, by the way.

“Millennials are just like everyone else,” he says. “They’re going to grow into older people, who are just going to keep looking at kids and making the same complaints about them. [The] memory bias ... is going to keep affecting us in 1,000 years.”

Bless your heart. You think there will still be people around in 1,000 years.

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