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').

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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.

August 4, 2021 at 12:05am

#1015106

Fun Guy

More science today. Not too much to say about it, because my energy's low, but I wanted to put this out there.

Yeast could soon make psilocybin cheaper than their magic mushroom cousins can
New work could even lead to psychedelic intermediates not previously available in large quantities

I'm neutral on recreational magic mushrooms. Never tried them myself, but they don't seem to deserve the bad reputation that, say, opiods get, and of course they have different effects. But the research on the use of their components in mental health treatment seems compelling.

In recent years, many studies have highlighted the potential benefits of the drug for the treatment of obsessive-compulsive disorder and migraine, in particular the active component of magic mushrooms, psilocybin.

Migraines aren't technically a mental health issue, I suppose, but from what I've seen from people who deal with them, making them go away might keep people from going nuts.

The article goes into why it's hard to synthesize psilocybin -- it's primarily a cost thing -- and it's worth the read, but obviously I'm not going to copy everything here.

Now, researchers at the Technical University of Denmark have found a new, better vehicle to make psilocybin: baker’s yeast, Saccharomyces cerevisiae.

If you've been paying attention, you know what that is. It's the same little magic beasties that make beer. (Technically, ales but not lagers.) Hence the "cerevisiae" part of the binomial.

Whenever I read something like this, the first thing that comes to my mind is "Why not just use the appropriate mushrooms?" Well, the article explains that it's hard to control the dosage that way, and when you're doing clinical trials (or actual treatments), it's very important to have a handle on exactly how much of whatever substance you're metabolizing.

But I can't help but think that the issue here isn't so much dosage control as Puritanism; our culture has a deep-rooted bias against anything psychoactive. It might give people Ideas, including the idea that maybe it's okay to alter one's perceptions for a limited time. Hence opiods having been (partially) legalized, despite their negative effects, while it took a very long time to legalize cannabis, which has few negative effects (and don't give me "but driving..." because the labels on opiods warn against driving, too).

Anyway, I just find the whole thing interesting, especially the part about finding yet another use for brewer's yeast. Beer: is there anything it can't do?

8 Like • Comment • View 4 comments

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