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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 6 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 8 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').

1 Like

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 18, 2020 at 12:02am

#991071

As the Turn Worlds (or whatever)

Maybe you have an image in your head, as I did, of a primitive caveman using stone tools to create a stone wheel.

Turns (pun intended) out that the wheel was invented much, much later.

And it's about time for another Merit Badge Mini-Contest! See below.

https://www.wired.com/story/who-invented-wheel-how-did-they-do-it/

Who Invented the Wheel? And How Did They Do It?
The wagon—and the wagon wheel—could not have been put together in stages. Either it works, or it doesn’t. And it enabled humans to spread rapidly into huge parts of the world.

It's not like massive leaps of invention aren't known in modern times, so it shouldn't be surprising that someone made the leap from not-wheel to wheel. What is surprising, at least to me, is how recently (relatively speaking) it happened.

In July 1880, the archaeologist Désiré Charnay discovered the first pre‑Columbian wheel set in the Americas. It was on a small coyote figure mounted on four wheels, and Charnay found it in the tomb of an Aztec child buried south of Mexico City.

It's been widely noted that the Maya, a civilization in Central America, didn't have the wheel. Their other advancements were quite sophisticated -- the calendar, notably. And yet, the civilizations they would have had contact with, such as the Aztecs, did have that invention. I had the opportunity to speak with an actual Mayan once, and his assertion was that the wheel would have made things too easy. My personal theory is that the wheel might have been too sacred to use for mundane purposes; their fabled calendar was represented as a wheel of sorts. Whatever the actual reason, it's lost to time, and I'm not even 100% sure that they never used wheels.

I mention this because, as the article notes, it seems to have been invented on both major continents, independently - not the only time such a thing has happened. The rest of the article focuses on the Indo-European version, which ties in to the entry I made a while back on the origins of most of the languages in Asia and Europe.

The full‑size wagon first appeared approximately 5,400 years ago, and it may be one of the the first inventions in history to go viral. Archaeologists have discovered full‑size carts from southern Iraq to Germany within a few hundred years of each other at a time when cultural barriers were particularly impermeable. The wagon, it seems, was irresistibly useful.

"If you build it, they will come." This is more of a case of "if you build it, you can go to them."

The article is fairly long, so I won't go into it much further. It's there if you want to read it. But I found it interesting because, as I said, thanks to cartoons and whatnot I always thought the wheel was a much older invention. It's one of those things we take for granted now, but as noted at the link, there's really not anything in nature that could have inspired it, not like, say, you can watch birds and go "Wow, I wish I could do that," and then eventually invent flying machines.

Since this article presents the wheel, as used in wagons, as one of the most important inventions of human civilization, I thought I'd make today's mini-contest about inventions. So...

Merit Badge Mini-Contest!

Other than the wheel, what's the most important invention, or innovation, that we've made, and why?

The response I like best gets its author a Merit Badge that I think is appropriate. As usual, you have until midnight WDC time (Today is August 18.) Just comment here below. I have a few in mind, but I'd rather hear from others. You can be serious or funny; I can certainly go either way. And there's no limit on how far back you want to go; prehistoric or modern will do.

3 Like • Comment • View 8 comments

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