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.

October 14, 2019 at 12:05am

#967806

Time is Money

https://www.cnbc.com/2019/08/08/self-made-millionaire-this-is-the-greatest-parad...

Self-made millionaire: This is the greatest paradox of wealth—and most people fail to recognize it

I think the author and I have different ideas of the meaning of the word "paradox."

*gratuitous photograph of Warren Buffett*

It's remarkable to me how many internet articles about wealth will use stock photos of Warren Buffett. Buffett is undeniably rich - he was once the richest man in the world, and I believe he's still in the top five - but he got that way through diversified investments, not traditional focused wealth-building. And, more importantly, he has nothing to do with the points made in this article; he's known to live a relatively modest lifestyle.

"Oh, I just got back from the grocery store," he said.

After a quick pause, I asked, "Have you ever considered having someone else do your grocery shopping?"

He looked at me like I was crazy. Pay to have someone do his grocery shopping? What kind of elitist would do that?

*raises hand* Me. This kind of elitist right here. The benefits of having groceries delivered, for me, are manifold. Here are a few:

Driving to the grocery store, finding a parking spot, dodging around people, having to come face to face with other people, and trying to make small talk with the cashier are all annoyances to me.

Schlepping the groceries the thirty feet from my car to my door is an even bigger annoyance, especially when I have to make more than one trip.

Done right, it's not much more expensive than doing my own shopping. Instacart Express for the win.

A grocery store is a den of temptation and iniquity. The last time I went into one, because I had to, there was an Oreos display. Resistance was futile. Now, I'm not averse to occasionally indulging, but if I'm shopping every week, well, do the math.

If I go to the store, I will buy beer and wine (VA doesn't do private liquor sales, just fermented beverages). Again, I indulge occasionally, but I have a fine selection of beer and wine (and scotch and gin and tequila and...) here at home, and I don't need to buy any more.

With these last two points, I'm actually saving money in the long run. I buy exactly what I want to buy, no extraneous psychological temptations, and since I'm choosing to eat healthy foods these days, that's a big plus.

Your experience may vary, of course. I understand some people actually enjoy (shudder) going out and mingling with people with all their germs and whatnot, but not this introvert. And some people are better at resisting temptation.

Here was an adult earning $750,000 a year, but behaving as if he still earns $50,000.

That's generally how you save money. It doesn't matter how much money you make; if you spend it all, you're boned.

This is a huge paradox of earning more money: Many people claim they value time over money, but if you look at their calendars, you'll find that the opposite is true.

Again, not a "paradox." Cognitive dissonance, maybe.

I used to scoff at people who flew first-class and think, Why would anyone spend an insane amount of money something so pointless? We're all getting to the same destination.

Because I actually fit in the seats, and because there's nothing more satisfying than sipping on your first complementary gin and tonic whilst watching all the harried passengers struggling past to get to steerage.

If you're making more money than ever, aim to save at least one hour per week. Think about all the responsibilities that you hate (e.g., doing the laundry, grocery shopping, managing your finances) where there are great solutions available to outsource or systematize the work.

Despite my snark above, I think the author makes good points. For myself, I'd rather spend a lot of time up front automating a task - or contracting someone else to do it, like I do with my lawn care - than do the task myself, especially if the task is repetitive. That's how I got good at spreadsheets.

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