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 8 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 10 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').

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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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

March 18, 2024 at 9:36am

#1066490

Cookie Wars

Here's an article about an important component of life.

The curious case of the disappearing Hydrox cookies
Oreo, the Hydrox knockoff, has been accused of burying its competitor by scoring sweetheart deals with grocers. Does it wield too much power over smaller rivals?

Yes, Oreo was a Hydrox knockoff, not the other way around. But that crap happened around 1910. Maybe stop worrying about who was first?

Audrey Peard is searching for an elusive, chocolatey piece of Americana: a package of Hydrox cookies.

Come to think of it, I haven't seen them in a while, either. Can't go to the grocery store without hearing the siren call of Oreos (which I usually ignore because they produced, then yanked, my favorite ones), but haven't seen Hydrox since before I can remember.

She’s visited multiple grocery stores near her home in the Bronx, followed a Facebook group, and even talked to a manager at a production facility in El Segundo, California, about supply shortages.

El Segundo may sound like an obscure California town, but in reality, it's right next to LAX. Hardly obscure.

Hydrox was the original chocolate sandwich cookie, predating Oreo. With a mildly sweet creme and a crunchier cookie that has a darker chocolate taste than its rival, Hydrox developed a reputation as the dessert of the discerning eater. It was, according to legendary food writer Calvin Trillin, the “far superior” cookie.

Sure, but this is the US. We ignore the superior in favor of the cheap and/or flood-marketed. That's kind of our thing; we even apply it to countries.

Hydrox, meanwhile, was discontinued in 2003. It came back in 2015 thanks to Leaf Brands, a San Diego-based company that specializes in reheated nostalgia.

Well, that might explain why I haven't seen any for a while.

Like almost anything else, you can buy Hydrox on Amazon (in bulk).

Don't fucking tell me that. Dammit!

In March 1912, the first batch of Oreos produced for sale was shipped from Nabisco’s six-story Romanesque-style headquarters in Manhattan to a grocer in Hoboken, New Jersey. It would’ve been a historic moment of American innovation if not for an inconvenient fact: Nabisco totally copied the cookie.

Another US thing. See also: Thomas Edison.

A smaller rival, Sunshine Biscuits (then known as The Loose-Wiles Biscuit Company), was started by Jacob and Joseph Loose in 1902.

Now, this is where my inner linguist starts asking questions. Nabisco was originally National Biscuit Company. Obviously, they both mostly made cookies, not biscuits. Except that in British English (and also in French), what we call cookies, they call biscuits. And what we call biscuits aren't technically biscuits at all ("biscuit" comes from Latin words for something like "twice baked"), though obviously they're delicious in their own way. But these were clearly American companies, as evidenced by their locations in New York and Kansas City. So why "biscuits?" Did we only split off from British usage after the watershed moment when someone invented the first delicious creme-filled sandwich biscuit/cookie?

Apparently not.

And it looks like we can blame the Dutch for, among other things, "cookie."

To complicate matters further, some cookies are called crackers.

All of which is to gloss over the fact that it's too bad they didn't stick with Loose-Wiles.

Anyway, the article goes off on a tangent about corporations and unions in the US, before going on with:

In 1908, Sunshine created one of its most popular products, Hydrox.

If you're thinking that's a portmanteau of hydrogen and oxygen, well...

According to company lore, the name stemmed from the elements of water — hydrogen and oxygen — chosen to symbolize the cookie’s cleanliness at a time when materials like chalk and plaster of Paris routinely appeared in baked goods.

I mean, technically they should have called them Carbohydrox, but whatever. Remember, this was before some other marketing convinced people that chemicals are bad, despite everything being made of chemicals.

Sunshine bragged that chemists tested Hydrox for purity and that workers made them at “thousand window” factories with natural light.

Yeah, right.

Hydrox also stood out against Oreo by promoting its more adult taste and kosher ingredients — in advertisements that bordered on erotic.

The "kosher" thing wasn't just to grab the Jewish market, either. It had the cachet of being more "pure" back then, kind of like "all-natural" is today. Whether it was/is or not is up for debate.

The article goes on with history for a while, and I think lays out a compelling case for why Oreos ended up reigning supreme. There's also quite a bit about the anti-competitive world of grocery stores, which I find fascinating.

Usually, doing articles like these makes me hungry. Not this time. All I can think of is the abomination known as Swedish Fish Oreos,

so... no, thanks.

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