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 20 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 22 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.

February 5, 2020 at 12:02am

#974890

Maths and Polymaths

Ever feel inadequate?

Well, today's link isn't going to help.

https://www.bbc.com/worklife/article/20191118-what-shapes-a-polymath---and-do-we...

Why some people are impossibly talented
Polymaths excel in multiple fields. But what makes a polymath – and can their cross-discipline expertise help tackle some of society’s most pressing challenges?

If it weren’t for an actress and a pianist, GPS and WiFi might not exist.

In the late 1930s and early 40s, Hedy Lamarr was already the toast of Hollywood, famed for her portrayals of femme fatales. Few of her contemporaries knew that her other great passion was inventing.

Even today, most people don't know how awesome Hedy Lamarr was.

Besides helping to outline the specific traits that allow some people to juggle different fields of expertise so successfully, new research shows that there are many benefits of pursuing multiple interests, including increased life satisfaction, work productivity and creativity.

Despite having once served as an "expert witness," I don't consider myself an expert at anything. I do have a lot of interests, I think, but I can't say I score high in life satisfaction; I don't care about work productivity; and I'm about as creative as a paramecium.

There are, of course, some good reasons why we might be hesitant to pursue multiple interests. One is the fear that we might spread ourselves too thinly if we devote ourselves to more than one avocation. With a divided attention, we would fail to achieve success in any domain – the idea that the “Jack of all trades is the master of none”.

Yep, that's me. Hardly "all" trades, though.

In reality, there is some evidence that developing diverse disciplines can fuel creativity and productivity. So while the pursuit of a second or third interest may seem like a distraction, it can actually boost your success in your primary field.

Now we get to why I bothered to read and link this article. Presumably, the people reading this are writers, and what have we writers always heard? "Write what you know." I've said before that this advice should probably be turned around to "Know what you write," but either way, a lot of writers can benefit greatly from cross-discipline study. Some of the things I learned about personal finance, for example, can be applied to weight loss; both involve budgeting and tracking.

And when we write, we draw from all kinds of different sources. Sometimes it's sufficient to look something up when you need it, but better still if you have some background in the subject in question. While I'd wager none of us have experience in microgravity -- few people do -- some basic knowledge of orbital mechanics and Newtonian physics can help one write more compelling science fiction.

Some people figure they just go to school for some number of years and train in their chosen field. That's one way to get through life. Me? I'll stop learning when my brain turns to mush, through disease, injury, or death. I can't be afraid or unwilling to learn new things, because I never know when some obscure fact will pop up in my writing.

Problem is, the universe of human knowledge is expanding exponentially. I doubt that it's possible for any one person to know everything, as much as some people like to act as if they do. So we all have to be selective. But you can't know ahead of time what will be useful and what won't, so for me, I just assume that all knowledge is potentially useful.

There's no such thing as useless trivia, and yes, all of this will be on the quiz.

2 Like • Comment • View 3 comments

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