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!

about 2 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 4 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.

December 10, 2022 at 12:01am

#1041640

You Mean Like From The Toilet?

Most marketing is about convincing us to buy stuff we don't need.

The ugly story of how corporate America convinced us to spend so much on water
We’re being packaged and sold a bottle/can/box of lies on water.

Water, however, is one of the few things that we legitimately need.

Every now and again I catch an ad for miracle spring water, which promises to cure everything from laryngitis to debt.

Pretty sure it does the exact opposite of curing debt. Except for the vendor.

It’s fairly obviously a scam seeking to separate people from their hard-earned money. Then again, the same goes for the plastic water bottles people buy at the convenience store every day, or the box of water or can of water that promises to be more environmentally friendly but isn’t especially.

"Fairly," my ass. While one could make a case that water cures some things (notably dehydration), that kind of claim is way over the top.

If you live in the United States, chances are that the water coming from your faucet is perfectly fine to drink (though there are, of course, some exceptions).

It's also often bottled and marketed as "fresh spring water."

“We’ve gotten here, step by step, down a dangerous road of converting a public resource into a private commodity,” said Peter Gleick, a scientist and expert on global water and climate and co-founder of the Pacific Institute, a research institution focused on water. “Water utilities don’t have advertising budgets; private companies do.”

As the author notes, tap water is usually, but not always, just fine. There are some good reasons for bottled water to exist: getting through the occasional municipal supply problem; being on the road and not having access to a tap; living in the American Southwest; that sort of thing. It's not the existence of packaged water I have a problem with; it's that, most places, it costs way too much.

In an economic system where virtually everything can be packaged and sold, of course we’re going to fall for it on water, just like we do everything else.

I don't remember who it was that spoofed the bottled water trend with canned air marketing. I think it was Mel Brooks.

It started to be introduced through imports such as Perrier in the 1970s, explained Gary Hemphill, managing director of research at the Beverage Marketing Corporation, and was facilitated by the proliferation of polyethylene terephthalate (PET), the material plastic water bottles are made of.

As I recall, Perrier at least was sold in glass bottles (I have no idea if it still is), and had the advantage of being slightly effervescent, which you don't usually get with tap water.

I also recall that people laughed about another brand of bottled water, Evian. "It's naïve spelled backwards!" they chuckle. Then, pretty recently, I learned that one of the words for sink in French is évier. Evian, it turns out, is an actual French company. I absolutely wouldn't put it past them to laugh Frenchily at their cleverness in putting an English possessive suffix on it to sell to gullible and monolingual Americans.

Companies have a litany of tactics — and cash — to get people to buy, buy, buy. They position bottled water as a healthier alternative to sodas (which it is) and to tap water (which it often is not). They try to entice people with sleek imagery and promises of purity, positioning the packaging as sporty or sexy or extra-healthy or whatever the brand’s schtick is.

I mean, that's what marketing is. You could sell surströmming by the truckload if your ads featured attractive young people doing outdoor activities and making flirty glances at each other while slurping the disgusting stuff. Moreso if one of them is a celebrity.

There’s a convenience component here, too. If you’re out and about, it’s really easy to pick up a bottle of water from the store around the corner.

I'm all about convenience. It's still way overpriced for that.

Sometimes when I'm traveling I'll find bottled water that costs less than soda. But not everywhere.

What’s more, there are no guarantees that the bottled water you’re drinking is actually safer. Sometimes it’s not, nor is it as tightly regulated as what’s coming out of the tap. There are some water trends that can genuinely make you sick.

I didn't click the hyperlink provided for the last phrase there, but I do remember some bullshit a few years back about how wonderful "raw water" is for you. I think the hype finally died off after some of its drinkers did.

Water has been turned into a highly commercial endeavor, and there are no easy answers on how to roll that back. A step in the right direction is to try to restore people’s faith in the water coming out of their faucet — a faith that’s, rightfully, been eroded over the years. “The first approach that needs to be taken is making it so people know that their water is safe to drink, that’s the first behavioral bias that we have to get across,” Donworth said.

That means water utilities and state and local officials need to get the message out better.

Municipal water systems are generally run by civil engineers. Civil engineers, as a whole, suck at marketing.

It also entails spending on infrastructure.

Which means taxes. People would rather spend $100 for $1 worth of water than pay an additional $1 in taxes.

Anyway, yes, I sometimes drink bottled water. But beer is usually made from the municipal supply, so it's good enough for me.

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