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 12, 2020 at 12:02am

#990565

I Got a Rock

When the answer to the question, "Have you been living under a rock?" is "Yes."

https://www.cracked.com/article_28076_a-brief-history-giant-rock-americas-most-b...

A Brief History of Giant Rock, America's Most Bonkers Landmark

Americans are not known for their creativity in naming landmarks.

It's called Giant Rock, because desert living requires many things, but wild flights of poetic fancy aren't one of them.

Nor are umbrellas.

There's certainly nothing remarkable nearby, except for the time machine.

Way to bury the lede.

Over the years, it's also been a family home, a center of pilgrimage, the scene of a bloody standoff, a potential rival to Las Vegas, and the galactic antenna for the wisdom of interdimensional aliens.

Pretty sure I saw this episode of Star Trek.

The giant rock's modern history starts in the 1930s, when a German guy named Frank Critzer stumbled on it, conveniently located in a part of the Mojave Desert so remote the scorpions had to get two buses and a cab just to murder you.

Now, now, surely that is hyperbole. Scorpions have their own custom means of transportation. Pretty sure it involves teleportation gates in shoes.

Besides, Critzer was reportedly quite fond of waving a shotgun at any distant neighbors who came snooping around. Since ornery old shotgun weirdos are a crucial part of the desert ecosystem, nobody asked too many questions.

This part rings true.

Using only dynamite, pickaxes, and insanity, he managed to excavate a large cave underneath the rock...

You can accomplish quite a bit with those tools alone.

Before long, he was welcoming over a plane a day, and fantasizing about converting the whole area into a Vegas-style winter resort.

Oh, the missed opportunities.

These plans were destroyed by the arrival of WWII, when the other scattered inhabitants of the desert reported seeing mysterious flares in the mountains, as well as missing dynamite.

I have to think that at least one person made the obvious connection, but this is the Mojave.

Public opinion turned against the mysterious German with the massive radio antenna on top of his rock, particularly after the FBI swept up the Silver Shirts, a group of Nazi sympathizers attempting to convert a Hollywood mansion into a self-sustaining headquarters for American fascism.

Eighty years too early and on the wrong coast.

The group went down into the house under the rock, where Critzer exploded. Which probably answered their questions about the dynamite.

"Oh, there it was."

In any case, Critzer was instantly converted into "wallpaper," while all three deputies were badly wounded. One just about managed to drive 40 miles to the nearest telephone to get help.

This should be a movie. This needs to be a movie.

The house under the rock was left abandoned until the arrival of George van Tassel, who worked as an aircraft engineer for Howard Hughes and claimed to have met Critzer back when he first moved to California, although his story of the meeting might be the most implausible thing in this whole article.

That's... that's a high bar to clear.

In 1947, he decided to move his family out there, which involved scrubbing Critzer's dried blood off the walls. Just remember, if your husband ever comes home and says "Honey, we're moving to the desert and make sure to bring your blood sponge!" there's no need to even divorce him, the law says you can just empty the bank accounts and go.

I know better than to take legal advice from Cracked, but this sounds about right.

In 1952, van Tassel announced that his time at Giant Rock had led to him being visited by a race of aliens, who looked like tan white people and spoke in posh English accents.

No mention of a blue police box in the area at the time?

The story goes on to relate how van Tassel next began construction of a time machine called the Integratron, which explains the bit in the first paragraph above.

In 1957, Van Tassel used the rock as a platform to announce his run for president in the 1960 elections.

Thus locking up that all-important scorpion demographic.

But somehow the machine was never finished (in fairness, time machines are complicated, and it would be cynical to bring up the ongoing donation money) and he died in 1978, without ever publishing a complete plan for the machine.

Too bad the original British time travelers took their TARDIS with them.

The internal machinery of the Integratron mysteriously vanished shortly afterward...

Only to reappear suddenly at some point in the future, obviously.

So, yeah, four out of five on TripAdviser, good job George van T!

In some ways, I hope we never cure this kind of insanity. I mean, the human cost aside, stories like this are just too much fun. Well, except for the whole "dynamite wallpaper" bit. That's sad.

Mini-Contest results!

Great answers to my question from yesterday, some serious, one amusing. I thought about giving it to the funny one, in honor of today's amusing article, but the late entry by Words Whirling 'Round is well-written and generally aligns with my own thoughts on the matter. (It's dated August 11, yesterday, so that's not a link to some older item; therefore, I decided it counts.) So the Merit Badge today goes to Words Whirling 'Round. I highly recommend reading the essay; here's another link to it: "Consciousness"

[E].

But as always, I appreciated all the comments and seeing how other people think about the subject of consciousness, and I hope to do this again soon so everyone will have another chance. Probably with something less... mentally weighty.

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