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:

tj-Merry Mischief Maker

about 21 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

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Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

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Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

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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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