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:

Turkey DrumStik

about 7 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').

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

November 11, 2018 at 12:31am

#945325

Freedom Freeze

https://www.bbc.com/news/world-europe-46137121

A sudden cold snap in the Pyrenees cost a young man his life as he slogged through the snow, smuggling cigarettes for a few €10 notes.

The 20-year-old Algerian was found unconscious in the snow by a French police patrol last week. He died of hypothermia.

Note that the article isn't unclear about who's being assigned the blame for this death:

"Traffickers are the ones who benefit from these weak people who are impoverished, and who are ready to take any risk for a few dozen euros," Perpignan prosecutor Jean-Jacques Fagni told France 3 TV.

But let's be clear on this: the ultimate blame lies with government.

Look, I'm no fan of unregulated free markets. There needs to be some oversight, or unscrupulous vendors could sell whatever they can while misrepresenting the product. Sure, people in government can be corrupt, too, but it's still an additional layer of protection for the consumer. I just wanted to state this up front lest I be mistaken for a pure libertarian.

The only reason I know of to smuggle like this is to get around taxes and/or tariffs. The health risks of smoking are well-documented, and so governments slap taxes on cigarettes ostensibly to discourage their consumption. They do the same thing with alcohol. Other drugs are banned completely, which, as we all know, doesn't stop people from using them. We tried that with alcohol here in the US, and they tried it in other countries as well, with disastrous consequences.

But if the purpose of such a tax is to reduce health problems, then a) it utterly failed in this case, because here's a dead dude; and b) why, then, don't these taxes go directly toward mitigating the health problems caused by the product in question?

I get that secondhand smoke is an issue, but ultimately, the choice of how to treat one's own body is a personal one - whether the product in question is alcohol, nicotine, methamphetamine, opiods, or kale. Sure, it can take a toll on one's friends and family, but Western concepts of personal freedom and choice dictate that we allow people the liberty to make their own decisions, though it's perfectly acceptable to encourage or discourage certain behaviors - and necessary to provide education on consequences.

Had the laws been devised with an eye toward personal freedom, there would be no incentive to smuggle smokes, and this guy might still be alive, if possibly earning less money. But that's the free market for you - when there's a demand, there's going to be a supply; and market efficiency will dictate the remuneration for products and services.

All prohibition does is encourage this sort of illicit behavior - whether said prohibition is all-out banning, or excessive taxes. I've seen it here in Virginia, where someone will buy multiple cartons of smokes at a convenience store, then load them into a truck with, say, New York plates, New York being a state with draconian taxes on cigarettes.

The tax thing raises another question of governmental ethics, as well: governments come to rely on whatever revenue they receive. If they're putting the taxes into the general fund, or using them for anything other than mitigating the health risks involved in whatever they're taxing, then they start to need that income just as surely as a junkie needs that next shot of heroin. Everyone, whether private citizen, corporation, or government, responds to incentives - this is a basic axiom of economics. And with all that sweet, sweet sin tax revenue, governments have less or even no incentive to discourage the use of whatever substance we're talking about. They'd rather make a big show of discouraging it through excessive taxes, while actually promoting it in order to keep the revenue stream alive.

So yeah, shame on the crooked traffickers for paying a pittance to impoverished people who have to make a mountain trek in snowstorms, but mostly shame on the governments involved for setting up a system that results in that situation in the first place.

Solution? Any such taxes should be reasonable, and the proceeds should go exclusively toward healthcare and education.

In other words, that'll happen in the US about the same time as Hell freezes over, trapping cigarette smugglers crossing the borders between its various circles.

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