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 9 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 11 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.

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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.

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.

February 8, 2024 at 11:05am

#1063762

Freeze Frame

I'm not going to have too many comments on the article I'm sharing today. It's more for informational purposes; these days, everyone is a photographer, and while phone cameras do a hell of a lot of the work for you (more than most people can appreciate), there are still things anyone can do to improve their picture-taking ability.

How to take better photos
Anyone can learn the principles that are essential to capturing quality images. Follow these tips and see the difference

Before the era of social networking sites, creating and sharing photographs was more deliberate and less instantaneous, which made picture-taking feel special. Families cherished the physical print – a link to the past with sentimental value – and kept their precious memories in albums, rarely showing them outside the family. Today, people commonly take photos not only to serve as memory cues but to tell others who they are, what they care about, and how they feel.

One might be tempted to think because I learned photography and even made a bit of money from it before digital cameras were a thing, that I'd have things to say about how kids these days have it too easy, that photography used to mean something, dammit, and that I'd yearn for the good old days of film and the techniques involved in developing it.

While I'm glad I got that experience, I'm also something of a technophile. Digital cameras can be amazing, and they've democratized the act of recording images for posterity.

In this Guide, I will provide some photographic principles and specific tips for casual, amateur and aspiring photographers on how to take better photos. By ‘better photos’, I mean appealing images that evoke emotions in the viewer and capture the essence of a subject or scene (and that resonate with how our brains make sense of the world).

I find that no matter how good you are, or think you are, at something, learning more about it, even some of the so-called basics, can help. Even us engineers were expected to keep learning throughout our careers.

So. Like I said. I have no real comments on the article; I think it's full of good information, whether you're using your smartphone as a casual snapshot-taking device, or have invested inspent money on a fancy DSLR (it's only an "investment" if you expect to make money from it).

Everyone's a photographer now, but that doesn't mean they're all good at it.

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