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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Turkey DrumStik

about 5 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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').

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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

December 1, 2018 at 1:13am

#946630

Dead Presidents

I like money.

By this, I don't mean that I want to fill my Olympic-sized swimming pool with bills of various denominations and go swimming in it - though, to be fair, I do - but I think the whole concept of money is fascinating.

So I was intrigued by this article:

https://www.bbc.com/news/business-46343965

New £50 note scientist nominations released

The Bank of England has released a list of scientists who have been nominated to feature on the new £50 note.

On the list are computing pioneers Alan Turing and Ada Lovelace, telephone inventor Alexander Graham Bell and astronomer Patrick Moore.

See, this is one area where I think the UK is right and the US is not as right. Oh, sure, some of their bills have the Queen on them, and that's fine, but they also take the opportunity to honor people who actually contribute to society.

In the early years of the US, we couldn't exactly put the King on our currency, because as a new country we had to flaunt our independence and all that rot. But, being a new country, we didn't have a lot of other role models. Historically, coins and, later, bills featured kings, emperors, and other figures we deliberately did away with. So they used, primarily, personifications of the ideals upon which our country was (supposedly) founded; notably, Lady Liberty.

Later, we apparently decided that leaders were worthy of having their portraits on currency after all, so we started putting dead people on them (this was a break from the common practice of putting the current king, emperor, or whatever on the money). Huh... I wonder if that's the origin of the word currency, but probably not and I can't be arsed to look it up. More likely it's from the kind of current you get in a river.

Anyway, so here in 2018 we have the following on our bills:

$1 George Washington, first President, traitor to the Crown, crappy general, slaveholder, rich bastard.
$2 Thomas Jefferson, third President, slaveboner, hypocrite (did design a pretty nice-looking university though).
$5 Abraham Lincoln, some number President, can't think of anything bad to say about him.
$10 Alexander Hamilton, not even a President, couldn't win a duel to save his life, inspiration for the most hipster play in history.
$20 Andrew Jackson, some number President, should have been named Andrew Jackass, racist dickwad.
$50 Ulysses S. Grant, has distinction of having the most badass name of any President ever, otherwise unremarkable.
$100 Benjamin Franklin, not a President. Inventor, philosopher, philanthropist, economist, statesman.

Of those, the only one of those even remotely qualifying as a scientist was Ben Franklin. Objectively also the most awesome of the bunch. I mean, come on, dude was chill as fuck and flew kites before kites were cool, so I guess technically he was a hipster, too. Best role model of all the Founders, and yes, I say this knowing full well what he got up to in France. If every American aspired to be Ben Franklin instead of that douchebucket Andrew Jackson, we'd all be better off as a country.

So yeah, I think it's time we put our own damn scientists on the money instead of dead presidents. We've had a few good ones, like... um... okay, now I'm racking my brains trying to come up with American scientists who were actually born in and lived their lives in the US. Um... Richard Feynman? George Washington Carver? I could google it, I suppose, but I'm lazy. Einstein comes to mind, but I'm thinking it'd have to be someone born in the US, like with presidents. That leaves out Tesla, too. Dammit, most of our scientists were immigrants. I suppose you could argue that they all were, unless you can think of a Native American scientist.

Okay, England, you win the scientist war. We still have better musi- DAMMIT.

Maybe we should just acknowledge reality and put capitalists on money. Rockefeller, Carnegie, that lot. That's one category of people we're good at producing, and besides, their business was money.

Just let's keep Ben Franklin, because he was awesome.

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