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 23 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

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.

December 23, 2019 at 12:05am

#971906

Nothing Exists in a Vacuum

Sometimes in here I like to address the Big Questions: Who are we? Why are we here? Why does my cat knock things off the table?

This is not one of those times.

https://thehustle.co/the-worst-sales-promotion-in-history/

The worst sales promotion in history
27 years ago, Hoover offered two free international flights with any £100 purchase. Today, it’s remembered as the worst sales promotion in history.

There have been a few corporate flubs over the years. The 1986 Coke Crisis comes to mind. I barely survived that dark time.

But this... this amuses me on several levels.

In 1908, a department store janitor named James Murray Spangler was suffering from a mean case of dust allergies.

So, he did what any entrepreneurial asthmatic would do: he mounted a motorized fan motor on a carpet sweeper and filed a patent for the world’s first household vacuum cleaner.

Spangler soon sold the patent to his cousin’s husband, William Hoover, who launched The Hoover Company and began selling the devices all over the US and Europe.

It is often fun and enlightening to speculate how things might be if something had gone just a bit differently in the past. Hell, whole books and movies have been founded upon such alternate timelines. But it's difficult to imagine an alternate universe in which "Spangler" became a household verb. "Yeah, I'm gonna spangler that right up." "It was dusty until I spanglered the hell out of the floor." I suppose an argument could be made that you could back-form the verb "spangle" from it, but that verb already has a meaning, as displayed in the US National Anthem.

For decades, the Hoover brand enjoyed a near-monopoly on vacuum cleaner sales. The machines were so ubiquitous in England that ‘hoover’ became a generic noun (like Kleenex or Band-Aid), used as a synonym for ‘vacuum.’ Its provenance earned it a reputation as one of the world’s most trusted brands.

Some companies consider this to be the pinnacle of marketing - to be able to get people to use their name as a generic noun. Coca-Cola succeeded in parts of the US. But the latest example is Uber. "I Ubered over here," I've heard. Which is unfortunate, because Uber is kind of a shitty company (their drivers tend to be nice, though).

The UK was entering the throes of a recession, and Hoover, a US-based company with a large presence in the UK, faced stiff competition from sexy newcomers like Dyson. In an effort to compete, they rolled out ill-fated products like a “talking vacuum” that warned users when to change the dustbin.

Yeah, come on. Who wants a household appliance that talks to them? "Alexa, who wants a household appliance that talks to them?"

Hoover knew that if everyone who bought a product applied, it’d be in trouble — so it made the process of obtaining these flights as annoyingly time consuming as possible:

1 A customer buys a Hoover product for £100+ and mails in a receipt + application within 14 days of purchase.
2 Hoover sends a registration form; the customer has 14 days to send it back.
3 Hoover sends a travel voucher; the customer has 30 days to select 3 departure airport, date, and destination combinations.
4 Hoover has the right to reject the customer’s choices; the customer can select 3 alternatives.
5 Hoover also has the right to reject these alternatives and select 3 combinations of its own choosing; if they don’t work, the customer is out of luck.

You could get away with that sort of thing before the internet. Now, companies have to be really creative in discouraging people from taking them up on their offers. Like airlines who offer free miles, but only if you fly between June 22 and 23 on flights that have a minimum of four legs, and checked baggage fees are $250.

On November 1, 1992, Hoover expanded its free ticket offer to include flights to America.

Dear Brits: We don't destroy your economy on purpose. Pinky swear.

Initially, things went according to plan. Department stores all over the UK became an “uncivilized scene” as thousands of people clammored[sic] to buy the cheapest Hoover products they could find.

In the US, an "uncivilized scene" in a department store means someone has shit all over the merchandise, someone else's kid is puking on the checkout counter, two women are clawing each other's faces off over the last tube of black eyeliner, and there's a gunfight in Housewares. I suspect that an "uncivilized scene" in a British department store involves a few raised voices and cops going "Wot's all this then?"

So, Hoover began to do everything it could to fleece customers out of the free flights.

It claimed thousands of customers had failed to correctly fill out the forms. It wrote back offering flights that departed from airports hundreds of miles away from customers’ homes. It sent out request forms on Christmas Eve, hoping mail closures would cause people to miss the 14-day deadline to send them back in.

And thus it becomes a textbook case of How To Piss Off Your Employees, Customers, and Shareholders Simultaneously.

Customers who’d followed all the rules were told their letters had “gone missing,” or that they’d failed to spot some arbitrary deadline buried in fine print.

Honestly, I figured every company with generous-looking promotion offers did this.

One of them, Harry Cichy, formed Hoover Holiday Pressure Group, a coalition to hold Hoover accountable for what they’d promised. It swelled to more than 4k members — doctors, lawyers, pig farmers, and electricians.

So, three honest professions, and lawyers.

In June of ‘93, 42-year-old Dave Dixon took a more dramatic measure: Angered that he hadn’t yet received his free flights, he decided to hold a Hoover delivery van hostage in his driveway in Workington, England. It remained blockaded by his horse truck for 13 days until a high court finally ordered its release.

This is the most British thing since drinking tea and eating crumpets whilst watching Doctor Who.

Incidentally, it would be difficult for me to post pics here so you'll just have to click the link and scroll down. Near the bottom is a montage of headlines that presumably ran in UK papers. I am disappointed that only a few of them contain "vacuum" or "flight" puns. I expect better from British tabloids.

And Hoover vacuums, once the star of every living room in the UK, sit in closets gathering dust.

Boy, that sucks.

(Come on, you knew I had to make that pun somewhere.)

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