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.

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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.

March 23, 2024 at 10:24am

#1066768

Montserrat

Got a long one for you today. It's from Truly*Adventure, a source I've never used before, and it's a few years old.

Every Living Creature

As the title is kinda vague, and the subtitle is kinda long, I'll sum it up as quick as I can: A volcano erupted, people fled, and other people saved the animals left behind.

What makes this interesting to me is that I've visited the volcano. While it was active.

The Soufriere Hills Volcano had been threatening the small Caribbean island of Montserrat for two years.

I mean, technically, the island of Montserrat is the Soufriere Hills volcano. It was dormant for so long they thought it was extinct. Turns out it wasn't.

The article is, as I said, long. So this is more to share it, and recount my own experience, than to make comments on it.

There exists a photo of me—rare, I know— standing next to the volcano exclusion zone sign on Montserrat. Really, I should have put at least one foot over the line to demonstrate just how rebellious I am.

But it's probably wrong for me to joke (though that's never stopped me before). What happened to Montserrat is a real tragedy. Thousands displaced. City destroyed. Fortunately, they're a Commonwealth country, so England took a bunch of them. But how do you go from living on a tropical island paradise to cold and windy Old Blighty?

Not that Montserrat isn't windy. There's a distinct difference between the windward and leeward sides of the island. One's mostly barren, with sporadic trees that grow diagonally because of the winds off the Atlantic. The other's a tropical rainforest. Or, well, was, before it turned to ash.

And a few holdouts remained on the island. To avoid the exclusion zone, they migrated to the inhospitable (barren and windy) northern lobe. I haven't been there for 20 years, but I recall makeshift dwellings and such. Hopefully, things have improved by now.

At the time, I considered myself lucky to get a tour of the island, at least those parts open to travel. Unless you live in a volcano zone, it's not every day you get to observe an active one, and even rarer that you can do so in relative safety. Standing on a windy hillside, I watched the mountain blow plumes of smoke while the ground vibrated beneath my feet. Clouds hovered around the summit, and, despite the wind, never cleared.

There's a metaphor in there somewhere, but it's too early in the morning for me to tease it out.

6 Like • Comment • View 1 comment

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