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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10 Most Recent Blog Comments:

Write_Mikey_Write!

a day 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

a day 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').

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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.

June 1, 2019 at 12:49am

#959965

On the Bias

A few days ago, there was a prompt about bias.

https://dariusforoux.com/thinking-errors/

I keep getting links from this guy. A lot of what he says is like baby formula: pre-processed, imbalanced, and not as good as the real thing. Why do I keep getting links to him? Because I've clicked on a few. The powers that be don't care if I like something or not; if I click on it, they assume I want more like it. This is why I keep getting links to New York Times articles: I click on them, discover that I can't read them without paying, and dismiss them.

We need a name for that. Analytics bias? Trending bias? I'm open to suggestions.

Regardless, I can't find much to rant about in this one, except maybe that it's not very comprehensive - but he links to more thorough resources.

If you've read the link, you might be asking, based on some other things I've posted here, "Waltz, how can you agree with an article that includes the sentence, 'What it comes down to is this: Avoid making decisions based on beliefs, obvious logic, and even science?'"

Well, I don't necessarily agree with how it's phrased, but the key takeaway for me is this: like it or not, we're emotional beings. Any decision that fails to take emotion into account is going to be risky. But that doesn't mean we should be slaves to our emotions; we can still use science, logic, etc. to inform them.

I'll use as an example my efforts at weight loss. Yeah, I know I talk about it a lot, but that's because my intention is focused on it. That, I've found, is how I get results. If it's an afterthought, it doesn't work. Anyway, the example: this may come as a shock to you, but I really like beer (and other distilled and fermented beverages). But beer isn't conducive to weight loss. Overindulgence also has its drawbacks, in terms of short-term impairment (though sometimes that's the whole point) and long-term damage. So, logically and scientifically, not drinking may sound like a perfectly good solution: you don't risk making an ass out of yourself while drunk, and maybe you live longer, and you don't have to worry about it sabotaging any efforts at losing weight.

That works for a lot of people. Great. But that's not a tenable solution for me, emotionally. There's no point, to me, in living a life without the things that make life worth living.

Consider this: What is something that brings you great joy? You don't have to tell me; just think about it. Your kids? Your car? Playing pranks on your boss? A particular friend? Reading? Whatever it is, imagine it. Now imagine being told - having solid scientific as well as anecdotal evidence - that doing without this one thing might extend your life and/or make you healthier.

Could you give up, say, your kids, never speaking to them again, never seeing them, and maybe only hearing second-hand about what they're up to, for the rest of your life? Even if you could be 99.999% certain that they'd do just fine without you?

I'm guessing the answer is no. In which case, by the rules of the situation I've set up here, you're deliberately choosing to die earlier and/or be less healthy in order to experience something that gives you great joy.

Well, that's the case with me and beer. (And scotch, tequila, wine, gin, etc.) and some other things that science says is bad for me. So I've made my decision, informed by science, but ultimately fueled by emotion, to live an alcohol-positive lifestyle.

So yeah, I didn't mind this link so much. But the next one, I'll likely have something to rant about.

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