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 11 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').

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.

May 7, 2022 at 12:21am

#1032067

Dien Bien Phu

Thought I'd experiment with a new, occasional format today. I learn something, maybe you learn something.

On This Day
7-May-1954
End of the Battle of Dien Bien Phu

I have an appreciation for history, sure, but one aspect of it that never really grabbed my attention is detailed war chronicling. War is usually too depressing, even for me, so normally I content myself with having a vague idea of the timeline, major players, and general outcome.

And even there, some military actions are more on my radar than others. The big US conflicts, of course; the Sino-Japanese wars, the Seven Days War in the Middle East, that time the UK took back the Falklands and... ummm... well, apart from the really historical ones like the time Hannibal crossed the Alps on elephants (come on, just picture that), I'm pretty ignorant.

I was almost completely ignorant about the Battle of Dien Bien Phu.

I mean, I had a vague notion that the French got their asses handed to them in Vietnam about a decade to two before we went in and got our asses handed to us, but that's about it. I wonder if this particular battle is where they got their offensive and undeserved reputation as surrender monkeys? Look, everyone's lost a war. Including us. Several times to Canada, for fuck's sake.

I'm not going to rehash most of what's already at that link in the title, just make a few of my own observations, as is my wont:

The United States was officially not a party to the war, but it was secretly involved by providing financial and material aid to the French Union, which included CIA contracted American personnel participating in the battle.

I doubt this was the first time we were secretly egging on a war, nor was it the last. The Cold War wasn't, always.

The Peoples Republic of China and the Soviet Union similarly provided vital support to the Viet Minh, including most of their artillery and ammunition.

Yay, proxy wars!

By 1953, the First Indochina War was not going well for France.

This being Wiki and not Cracked, and having little historical perspective of the overall conflict, I'm not sure if this is comic understatement or not. I'm going with yes.

Most of the page deals with the actual weeks-long battle, and is of value, but my purpose here is to a) snark on things and b) focus on this particular day in history, the day the battle officially ended.

But the "Battle" section is interesting mostly because of the names the French gave their fortifications. French women's names. It's just so... very... French.

So, on May 7, 1954, they record this message:

The last radio transmission from the French headquarters reported that enemy troops were directly outside the headquarters bunker and that all the positions had been overrun. The radio operator in his last words stated: "The enemy has overrun us. We are blowing up everything. Vive la France!"

Bad. Ass. Even in defeat.

Afterwards:

Public opinion in France registered shock that a guerilla army had defeated a major European power.

Yeah. History doesn't repeat itself. It echoes.

One final note, from the page:

The French government in Paris then resigned, and the new Prime Minister, the left-of-centre Pierre Mendès France, supported French withdrawal from Indochina.

Pierre Mendès France. For whatever reason, I'd never heard of him. But that name. It would be like if we elected a President named Rock America. If that President was from a liberal Brazilian Jewish family.

Which, you know... we could do worse.

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