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 9 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 11 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.

February 7, 2022 at 12:02am

#1026190

The Real Thing

Sometimes, I just share things because I find them interesting. This is one of those times.

7 Ways Coca-Cola's Cocaine Connection Goes Deeper Than We Realized

This is from Cracked, so it's a countdown...

Everyone knows that Coca-Cola used to contain cocaine back in the olden days. But then they stopped putting the coke in Coke, and now the only white powders in that glorious sparkling elixir are sugar and caffeine. Right?

Wrong! Turns out Coke still contains coca leaf extracts, and the Coca-Cola company is actually deeply involved with the world’s biggest dealers of legal cocaine!

Now, I didn't fact-check any of this, and I probably don't need to remind you that this is, first and foremost, a comedy site, and I think it follows Waltz's First Rule of Comedy: "Never let the facts get in the way of a good joke. Or a bad one. Especially a bad one."

Still, I don't think it's way off.

7 The First Cocaine Drink: Vin Mariani

Let’s start at the beginning. The story of Coca-Cola is intertwined with the story of cocaine, and the story of cocaine starts thousands of years ago when an early human decided to try chewing on some leaves from a coca bush and discovered their invigorating and pleasurable properties.

We here in the US, and I think in other anglophone countries, there's a polluted river of puritanism to contend with. We have prohibitionist tendencies when it comes to anything that affects the mind, calling them "drugs" and trying to legislate against them. I'm not trying to diminish the problems associated with cocaine, but the properties of the coca leaves from which it's refined are not much different from coffee. And for some reason, coffee is perfectly acceptable in society, even if one is addicted to it.

Just like marijuana is more than just THC, coca leaf is more than just cocaine. Actually, coca leaves don’t even contain cocaine, they contain an alkaloid very close to cocaine called ecgonine, which is converted into cocaine during the extraction process.

Like I said, I didn't fact-check anything, but this tracks with what I already knew.

The most popular application of coca was created in the 1860s by a French chemist named Angelo Mariani, who mixed concentrated coca leaf extract into wine and called it "Vin tonique Mariani à la Coca de Pérou"—better known as Vin Mariani.

This, though, I've never heard of. I assume that the French is easy enough to work out.

Unbeknownst to Mariani, he had created something completely new by blending cocaine and alcohol together. The two drugs have a synergistic effect when mixed—they combine to form a third unique drug called cocaethylene, which produces an even stronger euphoric effect than either substance on its own.

I can only imagine the Puritanical reaction to that.

6 The Origins Of Coca-Cola

In 1884, a Confederate veteran in Georgia named John S. Pemberton created a Vin Mariani knock-off called Pemberton's French Coca Wine. He had found that coca drinks helped him reduce his use of morphine, to which he had become addicted after war injuries.

Some might consider that to be like treating radiation poisoning with more radiation.

In 1885, Atlanta passed some of America’s first liquor prohibition laws, so Pemberton replaced the wine with carbonated water, added some kola nuts as a caffeine source and for flavoring, and renamed his drink Coca-Cola.

So... maybe something good did end up coming out of Prohibition. (Coke is my soft drink of choice. Also, fuck Pepsi.)

In 1899, when Coca-Cola started selling their drink in bottles, it suddenly became accessible to the Black community, which led to a racist backlash against a drink associated with cocaine.

Gosh. That sounds familiar.

This section goes deeper into the history of Coca-Cola, which, again, tracks with what I already knew.

5 Coca-Cola Under Attack, For Caffeine!

The Pure Food and Drug Act had become law in 1906, and the feds had two problems with Coca-Cola. First, did it contain cocaine? If not, then wasn’t putting “coca” in the name false advertising? Second, the new law forbade adulterating food and drink—so was Coca-Cola “adulterated” with caffeine, and if so, was the high level of caffeine in the drink harmful? Was Coca-Cola pushing a dangerous drink on the nation’s kids?

Kind of a paradox, isn't it?

4 Coca-Cola Helps Write Anti-Coca Treaties

When the Single Convention on Narcotic Drugs was ratified by the United Nations in 1961, it mandated the destruction of all wild coca bushes and complete eradication of coca leaf from the planet. Indigenous people have been using coca leaves for millennia, but the new global treaty demanded a forcible end to their coca-related culture and traditions.

Are... are we the bad guys?

Anyway, not a lot of point in pasting more extracts (pun intended, as always). The whole thing reads to me like the usual American corporate/government hegemony shutting out developing economies, similar to the whole Banana Republic thing (the concept, not the clothing store).

None of which is enough to make me stop drinking Crack Zero.

Just one more comment, though:

Coca comes from a leaf, not so different from how coffee comes from a seed. The fact that coffee is celebrated and coca leaf is vilified is an arbitrary accident of history which could have easily gone the other way. Frankly, the world would be a better place if coca leaf wasn’t demonized and forbidden, and if Coca-Cola didn’t have a global monopoly on coca drinks. That way, people everywhere could enjoy a variety of coca-based beverages with all the wonderful alkaloids included, and hopefully more of the profits could be going back into the coca-farmers’ pockets.

I figured out a long time ago that the reason we have illicit trade in cocaine in the first place is largely because it's way easier to refine it and ship the powder than it is to ship much larger bales of the leaves. Crack takes that process to a further extreme. And prohibition of both is inextricably tied up with racism, as this article points out.

I have no dog in the fight -- as you know, my mind-altering drug of choice is mostly-legal alcohol -- but perhaps if we'd drop the whole Puritan act, policies surrounding this sort of thing would become more rational.

But it ain't gonna happen.

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