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!

about 16 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 18 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.

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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.

March 13, 2020 at 12:01am

#977952

To Ape

One thing that we share with all the other animals is the need to eat. But humans tend to take things a bit farther than everyone else.

https://www.bbc.com/news/business-48859333

How do people learn to cook a poisonous plant safely?

In 1860, Robert Burke and William Wills famously led the first European expedition across the largely unknown interior of Australia.

It did not go well. Due to a combination of poor leadership, bad planning and misfortune, Burke, Wills and their companion John King ran out of food on the return journey.

The Master Race, ladies and gentlemen.

The local Yandruwandha people seemed to thrive despite the conditions that were proving so tough for Wills's party.

The Yandruwandha gave the explorers cakes made from the crushed seed pods of a clover-like fern called nardoo.

Burke then fell out with them and, unwisely, drove them away by firing his pistol.

I'm sure it seemed like a good idea at the time.

Within a week, Wills and Burke were dead. It turns out that safely preparing nardoo is a complex process.

Maybe not in hindsight.

Toxic plants are everywhere. Sometimes simple cooking processes are enough to make them edible. But how does anyone learn the elaborate preparation needed for cassava or nardoo?

No single person does, according to Joseph Henrich, an evolutionary biologist.

There's been pushback lately against "processed" food (the verb being ill-defined), but processing makes some materials more digestible, or even edible at all.

That said, the article proceeds to venture into evolutionary speculation, which I have railed on in here before.

He argues this knowledge is cultural. Our cultures evolve though a process of trial and error analogous to evolution in biological species. Like biological evolution, cultural evolution can - given enough time - produce impressively sophisticated results.

Somebody stumbles on one step that seems to make cassava less risky; that spreads and another step is discovered. Over time, complex rituals can evolve, each slightly more effective than the last.

In South America, where humans have eaten cassava for thousands of years, tribes have learned the many steps needed to detoxify it completely: scrape, grate, wash, boil the liquid, leave the solid to stand for two days, then bake.

I've often wondered about the development of bread. It's not like you can look at a stalk of grain and go, "You know, I think I'll grind this up really fine, add some water and other stuff, knead it, and bake it. I bet that will be delicious!" At least, not if you don't have any idea what bread is. It's a complicated process, and the learning curve has been lost to antiquity.

I've wondered the same thing about beer. They're very similar in some respects. But at least grain is, by itself, not usually poisonous, just inedible. Some of these other foodstuffs barely register as food at all, until they're processed, and the techniques of processing are, as the article explains, very specific.

Point is, the thing about humans is we have language and society, so we can pass these things down from one generation to another. But that doesn't answer the question about how you get the techniques in the first place. The article speculates, but that's about it.

Further studies show that the verb to ape, meaning to copy, is ironically misplaced: the only ape with the instinct to imitate is us.

Tests reveal two-and-a-half year-old chimps and humans have similar mental capacities - unless the challenge is to learn by copying someone. The toddlers are vastly better at copying than the chimps.

Now that's more concrete. And it speaks to something I've been tossing around in my head for a while (and even made it a central theme in a novel) -- the idea that while individual humans may not be very bright, we're pretty good at imitating things that smarter people have done. It took the likes of Newton and Liebniz to invent calculus, but anyone of moderate intelligence, with the will to learn about it, can achieve a working knowledge of its principles -- even if we're not smart enough to have come up with it on our own.

If Henrich is right, human civilisation is based less on raw intelligence than on a highly-developed ability to learn from each other.

That's a big "if," but it tracks with other things I've been thinking about. Every time someone brings up the possibility of intelligent alien life, some wag says something along the lines of "but there's no intelligent life on Earth either." The wags are wrong; there is intelligent life here; it's just rarefied, and the rest of us are good at copying.

Case in point: I didn't invent any of the letters or words I'm typing here. I'm copying language I learned from others.

Now, by claiming this, I don't mean to diminish the contributions of the copy-apes. Society takes all kinds of people, and each of us has things we're good at. I'm just saying we can't all be Newton.

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