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').

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.

February 5, 2022 at 12:02am

#1026065

The Yeast You Can Do

Today, in Booze News...

A newly discovered cryosphere-dwelling yeast stays alive by making ethanol
Rhodotorula frigidialcoholis was isolated from 150,000-year-old permafrost in the McMurdo Dry Valleys of Antarctica

I need to take a moment to point out that this is the best binomial I have seen in a long time: Rhodotorula frigidialcoholis.

Most of the Earth’s biosphere is permanently cold and contains environments below 0° C, known as the cryosphere.

Not for very much longer.

Microorganisms like bacteria and fungi call the cryosphere home, despite the seemingly inhospitable conditions. Some can even stick around in the ice for thousands of years.

This is why a lot of people think we could find life in otherwise inhospitable extraterrestrial locations, such as Mars. Not cogitating aliens, but microorganisms.

One example of cryosphere adapted fungi are a genus of single celled, pink pigmented yeast called Rhodotorula, which have been isolated and characterized from a range of cold ecosystems.

Hm, pink, much like all these elephants I'm seeing.

The researchers found it has two novel responses to extreme cold temperatures: it can switch its metabolism from respiration to ethanol fermentation as its main pathway, and can overexpress molecules called small non-coding RNAs (sRNAs) that help regulate which genes are expressed after transcription.

That last bit is above my pay grade, but I understood "ethanol fermentation as its main pathway."

...the metabolic switch from respiration to ethanol fermentation by R. frigidialcoholis may help the novel yeast – and potentially others like it – save energy, slowing down the freezing point in their cells as a long-term survival strategy.

Not mentioned in the article: new ways to make booze. Too bad.

One of my favorite stories concerning drunk history involves how Russian Imperial Stout, one of my favorite beer styles, was created. This is the way I like to tell it (and yeah, I may have mentioned it in here before, but it's been a while):

Long ago, during the reign of, I forget, Catherine or Peter (one of the Greats anyway), the British ambassador to Russia introduced the Russian imperial court to the concept of stout: that dark, roasty beer of which Guinness is probably the best-known example worldwide. The Russians loved the stuff, so the ambassador arranged for several casks to be sent to St. Petersburg.

In the winter.

Naturally, the casks froze and burst during the trip through the frigid waters, disappointing everyone involved.

The British, in a fit of genius not seen since Newton invented gravity, found a way to brew the stout with a higher alcohol content, while balancing the flavors of all the other ingredients. Alcohol has a lower freezing point than water, making it an effective antifreeze, so those casks made it to the Tsar, and Russians being Russians, no one complained about the higher alcohol content. And thus was born Russian Imperial Stout -- not a Russian brew at all, but originally British, much as India Pale Ale was invented by the British for parallel reasons -- but that's a story for another time.

The point being that these yeast presumably survive extreme cold by creating their own antifreeze in the form of ethanol. Obviously other yeast produce ethanol too, which is what makes some of the greatest beverages on Earth possible, but apparently this is a different mechanism? Or something; the article is a bit short on details where that's concerned.

But since it involves ethanol, well, here it is.

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