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').

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.

August 26, 2019 at 12:10am

#964735

Dolla Bill Yo

A while back, I shared an article on the origins of the semicolon: "Semicolonoscopy"

Today, I want to talk about a far more important symbol.

https://www.bbc.com/worklife/article/20190528-the-curious-origin-of-the-symbol

The curious origins of the dollar symbol
Despite its ubiquity, the origins of the dollar sign remain far from clear, with competing theories touching on Bohemian coins, the Pillars of Hercules and harried merchants.

Fascinating that this is from BBC, yes? Also note the author's name is Hepzibah, which is on my list of all-time most awesome names.

It’s shorthand for the American dream and all the consumerism and commodification that comes with it, signifying at once sunny aspiration, splashy greed and rampant capitalism. It’s been co-opted by pop culture (think Ke$ha when she first started out, or any number of fast-fashion t-shirts)...

I still call Ke$ha "Key-dollar-ha." When I mention her name. Which isn't often.

If you had to find letters lurking in its form, you might spy an ‘S’ overlain with a squeezed, bend-less ‘U’ providing its vertical strokes. In fact, this accounts for one of the most popular misconceptions about the sign’s origins: it stands for United States, right?

That's actually what my mom taught me at an early age. Even then, I think I had my doubts, mostly because most of the dollar signs I saw had the one upright instead of two.

That’s what writer and philosopher and famed libertarian Ayn Rand believed. In a chapter in her 1957 novel, Atlas Shrugged, one character asks another about what the dollar sign stands for.

Yeah... anything Ayn Rand says is bullshit until proven otherwise beyond a reasonable doubt. Still, just because she had a character say it, that doesn't mean she believed it.

The dollar, meanwhile, has a far shorter history. In 1520, the Kingdom of Bohemia began minting coins using silver from a mine in Joachimsthal – which roughly translates from German into English as Joachim’s valley. Logically if unimaginatively, the coin was dubbed the joachimsthaler, which was then shortened to thaler, the word that proceeded to spread around the world. It was the Dutch variation, the daler, that made its way across the Atlantic in the pockets and on the tongues of early immigrants, and today’s American-English pronunciation of the word dollar retains its echoes.

This, actually, I knew. I once spent a lot of time researching the origins of various money words (though not the dollar sign). For instance, the word "money" itself comes from the temple of Juno Moneta. Juno, you know. Moneta was an epithet that, as I understand it, translates to "she who warns," or, more simply and less gender-specific, "warner." The connection is that that particular temple in Rome was used as a vault, one giant cache. From which we get the word "cash."

Yet another version centres on the Pillars of Hercules, a phrase conjured up by the Ancient Greeks to describe the promontories that flank the entrance to the Strait of Gibraltar. The pillars feature in Spain’s national coat of arms and, during the 18th and 19th Centuries, appeared on the Spanish dollar, which was otherwise known as the piece of eight, or peso.

Ever wonder why a quarter dollar is sometimes known as "two bits?" That's why. Two bits to a quarter = eight bits to a dollar, or, as noted, historically a peso. It wasn't that long ago that stock market prices were quoted in eighths of dollars - that only changed in, like, the 1990s I think? When computers started taking over. Point is, that method of quoting stock prices harks back to the pieces of eight. Still no good reason to quote gasoline prices to 9/10ths of a cent, though.

As with everything American at the moment, there’s a partisan dimension to the debate about the dollar sign’s ancestry: for duelling political reasons, one faction favours the idea that it’s homegrown, another that it was imported.

Figures.

And as for the first printed dollar sign, that was made on a Philadelphia printing press in the 1790s and was the work of a staunch American patriot – or at least a vehemently anti-English Scotsman – named Archibald Binny, who’s today remembered as the creator of the Monticello typeface.

I remember no such thing. I only care about who created Comic Sans.

Anyway, I just find this sort of thing fascinating, so here it is.

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