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 2 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 4 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.

January 31, 2023 at 12:01am

#1043924

In The Beginning Was the Emoji

Mostly, this is just an interesting article from Vice to share. But naturally, I have comments. Some are serious, others, not so much.

A Total Amateur May Have Just Rewritten Human History With Bombshell Discovery
Ben Bacon is "effectively a person off the street," but he and his academic co-authors think they've found the earliest writing in human history.

The idea that an "amateur" might make a discovery isn't all that shocking. People with experience sometimes let that experience get in the way of coming up with fresh ideas, and nothing says "fresh ideas" like a newbie. Hell, Einstein was famously working as a patent clerk when he figured out how most of the Universe worked. This, of course, doesn't mean that an amateur is always going to get it right.

What's more important is the "discovery" itself, and whether it will hold up under scrutiny.

In what may be a major archaeological breakthrough, an independent researcher has suggested that the earliest writing in human history has been hiding in plain sight in prehistoric cave paintings in Europe, a discovery that would push the timeline of written language back by tens of thousands of years, reports a new study.

This, folks, is how you write a lede. And it's even in the first paragraph.

These cave paintings often include non-figurative markings, such as dots and lines, that have evaded explanation for decades.

Samuel Morse went back in time and left messages?

Ben Bacon, a furniture conservator based in London, U.K. who has described himself as “effectively a person off the street,” happened to notice these markings while admiring images of European cave art, and developed a hunch that they could be decipherable.

BEHOLD THE POWER OF BACON

Now, Bacon has unveiled what he believes is “the first known writing in the history of Homo sapiens,” in the form of a prehistoric lunar calendar, according to a study published on Thursday in the Cambridge Archeological Journal.

Technically, if it is writing, then it's not "prehistoric." By definition.

Intrigued by the markings, Bacon launched a meticulous effort to decode them, with a particular focus on lines, dots, and a Y-shaped symbol that show up in hundreds of cave paintings.

This supports my Samuel Morse time-traveling theory, if we also assume he was horny and thinking about the pubic regions of females.

Previous researchers have suggested that these symbols could be some form of numerical notation, perhaps designed to count the number of animals sighted or killed by these prehistoric artists. Bacon made the leap to suggest that they form a calendar system designed to track the life cycles of animals depicted in the paintings.

I was wondering how that relates to a "lunar calendar," but fortunately, the author continues to practice good journalism:

The researchers note that the paintings are never accompanied by more than 13 of these lines and dots, which could mean that they denote lunar months. The lunar calendar they envision would not track time across years, but would be informally rebooted each year during a time in late winter or early spring known as the “bonne saison.”

Hey, that's French. I didn't need years of study to know that this means "good season."

On a more serious note, finding out when the calendar ticked around would be pretty cool. Our Gregorian calendar begins nearly equidistant from the winter (northern hemisphere) solstice and Earth's perihelion (that bit's a coincidence). The original Roman calendar on which it was largely based rolled over at the beginning of spring. That's why the names of the ninth, tenth, eleventh, and twelfth months start with Latin prefixes for seven, eight, nine, and ten, respectively... but I digress.

It's a cycle, so it doesn't really matter what you call the end/beginning, but it might shed some light on the ancients' thought processes.

The “Y” symbol, which is commonly drawn directly on or near animal depictions, could represent birthing because it seems to show two parted legs.

What did I tell you? I told you.

“Assuming we have convinced colleagues of our correct identification, there will no doubt be a lively debate about precisely what this system should be called, and we are certainly open to suggestions,” they continued. “For now, we restrict our terminology to proto-writing in the form of a phenological/meteorological calendar. It implies that a form of writing existed tens of thousands of years before the earliest Sumerian writing system.”

I'm not an expert, as you know (I even had to look up "phenological"), but I feel like calling it "writing" or even "proto-writing" is a stretch. "Counting," maybe, I could see.

As far as I've been able to learn, writing came from earlier pictograms, and those pictograms stood for actual things in the world. The letter A, for example, can be traced to a pictogram for an ox. Basically, all writing starts as emoji, becomes a system for communicating more abstract thoughts, and then, after centuries of scientific, cultural, and technological advancement, we start communicating in emoji again.

But counting? What I don't think a lot of people appreciate is how abstract a number is. There is no "thing" in nature that you can point at and say, "that is the number three." There was a huge leap when someone figured out that three oxen and three stones have something in common; to wit, the number three. So if you only know pictograms, how do you represent three? "3" hadn't been invented yet. You use, maybe, three dots, perhaps representing three stones. It's not a painting of something that exists in nature, like an ochre ox on a cave wall, but a representation of an abstract concept.

This may be a classification problem. Numbers are a kind of language, too. And that ochre ox isn't an ox; it's a painting of one.

The only way the people of the past can communicate to us is through metaphor. Okay, and genetics.

It would be hard to overstate the magnitude of this discovery, assuming it passes muster in the wider archaeological community. It would rewrite the origins of, well, writing, which is one of the most important developments in human history. Moreover, if these tantalizing symbols represent an early calendar, they offer a glimpse of how these hunter-gatherers synchronized their lives with the natural cycles of animals and the Moon.

This bit I'm going to quibble with. I question whether early humans separated themselves and their works from nature, as we do today. But that's kind of irrelevant to the story.

In short, if the new hypothesis is accurate, it shows that our Paleolithic ancestors “were almost certainly as cognitively advanced as we are” and “that they are fully modern humans,” Bacon told Motherboard.

They couldn't have been fully modern humans; they didn't have beer. Jokes aside, though, I wasn't aware that this was in dispute. They didn't have our enormous body of knowledge and experience, but they were just as smart (or dumb) as people are today. Ignorance is not the same thing as lack of cognition.

Ignorance can be fixed. Stupid can't.

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