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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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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.

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Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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November 17, 2022 at 12:01am

#1040741

Yeast to West

Speaking of food, I continue to see people raving about sourdough, which apparently a lot of people played with when they were stuck at home over the past couple of years. Me, I just continued buying bread from the bakery as nature intended. Every time I've tried to bake bread, bad things happened.

Anyway, this article isn't about pandemic sourdough bakers; it goes back a bit further than that.

San Francisco’s Famous Sourdough Was Once Really Gross
Gold miners made themselves sick on smelly, hard loaves.

Long before it became a viral food trend or social-media sensation, American sourdough was surprisingly gross.

So, more like the other meaning of "viral."

San Franciscans proudly trace their city’s iconic bread back to the Gold Rush of 1849.

That's fair, but it should be noted that before people knew what yeast actually was, almost all bread was "sourdough." I think some was made with repurposed beer yeast, but the origins of sourdough extend way before San Francisco was a thing.

The men who flocked to Northern California in search of gold made bread in their wilderness camps not with store-bought yeast, but with their own supply of sour, fermented dough.

Yeah, I could be wrong, but I don't think there was a lot of store-bought yeast there at that time.

Letters, diaries, and newspaper articles written by and about the 49ers, lumberjacks, and pioneers of the American West are full of complaints about horrible and inedible sourdough. Could bad bread really have inspired San Francisco’s most beloved loaf?

Or it could be gold miners protecting their hoard. "Don't come out here. The weather is shit, people are camping everywhere, and the bread sucks." You know, a bit like San Francisco today.

In 1849, when gold miners began arriving in San Francisco, most Americans didn’t bake or eat sourdough bread. American bakers typically leavened their bread with “barm” (a yeast derived from beer brewing) or one of several relatively new commercial yeast products.

Yeah, see? Look, I comment on these things as I go, and it's nice to turn out to be mostly right.

Hm, I wonder... why, yes, that is the origin of the mostly British word barmy.

These commercial yeasts were easy to work with, didn’t require constant maintenance, and produced reliable results. They also produced bread that appealed to American taste buds.

American taste buds suck. They think Budweiser is beer, pasteurized process cheese food is cheese, and Wonder Bread is bread.

Most 19th-century Americans preferred bread that was sweet rather than sour. According to one 1882 advice book for housekeepers, the “ideal loaf” was “light, spongy, with a crispness and sweet pleasant taste.” Sour bread was a sign of failure. As a result, bread recipes from the period used commercial yeasts along with considerable amounts of sugar or other sweeteners to speed up fermentation and avoid an overly sour flavor.

The ideal bread is a French baguette with a crispy crust. Period.

Sourdough required only flour, water, and fresh air. A sourdough “start” needed care, attention, and regular feedings but offered an inexhaustible, self-perpetuating supply of leavening agent, even in the wilderness.

Some brewers make beer with wild yeast, too. The results are all over the place. Some of them are also described as "sour."

Bread was baked under difficult circumstances—outdoors, over a campfire or hot coals, and sometimes in the same flat pan used for panning for gold—leading to inconsistent and unsanitary results.

But they could have sold it as artisanal sourdough with flaked gold.

Sourdough baked by pioneers wasn’t just gross and unappetizing; it could also make you sick.

Well, duh. Not all the microorganisms floating around are beneficial.

Across the American West, sourdough was considered a food for unmarried men who didn’t know how to cook.

As opposed to married men who didn't know how to cook.

So it’s worth asking: If sourdough bread baked by miners was so terrible, how did it become one of San Francisco’s most beloved foods?

It all came down to the success of the city’s French and Italian bakeries.

Yep. That'll fix it, alright.

By the second half of the 20th century, tourism boards in San Francisco were placing the 49ers at the center of the city’s history, idealizing life on the frontier and playing up links between the Gold Rush and the City by the Bay. San Francisco bakeries joined in, crafting stories about partnerships between bakers and miners and attempting to market the bread nationwide.

And once again, we see that no matter how disgusting something may be, if you market it right, it'll become popular.

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