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 10 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 12 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.

May 3, 2022 at 12:01am

#1031836

Just One Word: Plastics

Not a lot to add to this one, just some interesting science.

Bugs across globe are evolving to eat plastic, study finds
Surprising discovery shows scale of plastic pollution and reveals enzymes that could boost recycling

Because I did an entry about the Pacific Garbage Patch a few days ago ("Garbage Patch Kids"

), this article (which I only found after doing that entry) caught my eye.

Microbes in oceans and soils across the globe are evolving to eat plastic, according to a study.

Just for the record, that's an excellent lede. Concise, accurate (as opposed to the "bugs" in the headline), and gets right to the point. Of course there's more, which is why there's an article, but I do appreciate a good lede.

The study is the first large-scale global assessment of the plastic-degrading potential of bacteria and found that one in four of the organisms analysed carried a suitable enzyme. The researchers found that the number and type of enzymes they discovered matched the amount and type of plastic pollution in different locations.

This should not be too shocking. It's in line with how we know evolution works. If one "bug" has the enzyme and another doesn't, then the second bug, after ingesting plastic, has a higher chance of dying, so the first bug has a higher chance of reproductive success.

The truly amazing thing is that these microbes had the potential to create the appropriate enzyme in the first place. That's what's unexpected from my point of view. These plastics aren't generally from non-manufactured sources.

Reducing the amount of plastic used is vital, as is the proper collection and treatment of waste.

Not if we have mutant microbes to eat it! We're saved! Use ALL the plastic!

The new research provides many new enzymes to be investigated and adapted for industrial use.

In other words, the really useful thing here is being able to replicate these enzymes, not that we can unleash hordes of plastic-eating archaea.

Importantly, the researchers ruled out potential false positives by comparing the enzymes initially identified with enzymes from the human gut, which is not known to have any plastic-degrading enzymes.

...yet.

Another mutant enzyme was created in 2020 by the company Carbios that breaks down plastic bottles for recycling in hours.

Don't these people read science fiction? Sheesh.

5 Like • Comment • View 5 comments

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