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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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August 22, 2024 at 11:52am

#1075583

Got Nothing

Here's SciAm with an article that's all for nought:

The Elusive Origin of Zero
Who decided that nothing should be something?

I'm sure lots of people wanted to make something out of nothing.

Sūnya, nulla, ṣifr, zevero, zip and zilch are among the many names of the mathematical concept of nothingness.

As well as, of course, nought, which is why the proper term for the first decade of this century is the noughties.

Historians, journalists and others have variously identified the symbol’s birthplace as the Andes mountains of South America, the flood plains of the Tigris and Euphrates Rivers, the surface of a calculating board in the Tang dynasty of China, a cast iron column and temple inscriptions in India, and most recently, a stone epigraphic inscription found in Cambodia.

I'd never heard the Andes one. It's also such a useful concept that I wouldn't be surprised if more than one culture came up with it independently.

For a country to be able to claim the number’s origin would provide a sense of ownership and determine a source of great nationalistic pride.

Kind of like how Russia and Poland claim to be the inventor of vodka. In this case, though, it's bragging rights to say "We're Number Zero!"

Throughout the 20th century, this ownership rested in India.

All the symbols we call Arabic numerals? Yeah, turns out they're of Indian origin. But we're still not completely sure about 0. I've read an entire book on the subject and came away with nothing. Pun intended.

However, a series of stones in what is now Sumatra, casts India’s ownership of nothingness in doubt, and several investigators agree that the first reference of zero was likely on a set of stones found on the island.

I think part of the problem here is that records set in stone or clay tend to last longer than those on papyrus or paper. Any art or math done on a more ephemeral medium is long gone, and the invention may have preceded its carving into stone.

Researchers at the Center for Civilizational Dialogue at the University of Malaya in Kuala Lumpur have been investigating the history of early numeration systems of Southeast Asia. Their findings further strengthened Sumatra’s claim, to which we, the authors, agree.

The article goes into some detail about who found what and when. I should mention, though, that the concept of zero shouldn't be confused with our symbol for zero. Earlier efforts might have rendered it as a dot or dash or some other symbol; the important thing is that some symbol was used to designate the lack of something. Like in our number 107, the 0 indicates the lack of a tens place.

While the issue requires more deliberation and historical examination, this discovery of a possible nothingness symbol is intriguing. Could zero have been conceptually conceived of and utilized in an ancient and barely known Southeast Asian society?

And why not? Just because it's barely known now doesn't mean it wasn't a major power in its own time. And if there's anything major powers need, it's to keep careful records; zero helps with that.

Did the use of zero spread from this region westward into India and finally into Europe?

Seems to me I've talked about Europe's reluctance to adopt the zero in here before, but damned if I can find it. We have Roman numerals because zero wasn't a thing to them. Nor does one appear in ancient Hebrew, which obviously inspired later Roman and European culture in general. There might have been religious reasons to resist the concept, in addition to the same sort of cultural inertia that keeps the US from adopting SI: "this is the way we've always done things; why change it now?"

Is the credibility of the term “Hindu-Arabic” numerals under serious threat?

Eh, maybe. Maybe not. I'll leave that for the experts, of which I am not one. But, again, don't confuse numeral with number. It's entirely possible that India didn't invent the concept of zero, but maybe was the first to make the symbol for it look like an empty circle.

It's important to note that one of the authors of this article is Malay; he may be biased due to the whole "bragging rights" thing. I mean, come on, don't they have a footy team? That seems to be other countries' main source of national pride, for reasons still unclear to me.

Me, I have no dog in the fight and, in the end, it's not like someone can, you know, patent the concept or issue l'appelation controllée like with champagne. The matter is, to me, purely of historic value, though it's usually satisfying to know the truth about something.

And sometimes, nothing is something.

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