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:

🌝 HuntersMoon

Sep 20, 2025 at 12:12am
In response to "Numbers End"

Boy, am I behind thee times. I thought you just put this on hiatus. Of course, in March I wasn't a complete devotee of your brilliance yet.

I'm more than happy with Carrion Luggage, however.

1 Like

Elisa, Snowman Stik

May 1, 2025 at 7:09pm
In response to "Shhhhhhh"

Anytime I hear/read about the Quiet Zone, I am always reminded of this video.

TJ says, "keep on keeping on!"

Mar 12, 2025 at 8:16pm
In response to "Penultimatum"

As my Grandmother used to say, "Everything changes except that which stays the same." I never did understand why she said it, but it sounds good.

1 Like

TJ says, "keep on keeping on!"

Mar 12, 2025 at 8:13pm
In response to "Numbers End"

Looking forward to the new book and hoping you don't encounter a meteor.

1 Like

Lazy Writer est 4/24/2008

Mar 12, 2025 at 4:19pm
In response to "Numbers End"

Sorry this one is closing..I will look for the new one tomorrow! Looking forward to it.

2 Like

Aiva Raine

Mar 12, 2025 at 3:45pm
In response to "Numbers End"

Thanks for a great 3000 entries.

Looking forward to the new blog, whatever it might be.

2 Like

Wordsmitty ✍️

Mar 12, 2025 at 1:36pm
In response to "Numbers End"

OK. What have you done with the real Robert Waltz?

Also, for the scientist engineer in me, have you actually tried to make entry 3001 and been rejected. Until there's proof, it's not real.

Thank you for your indulgence and sharing of wisdom.

R.I.P. "Complex Numbers"

3 Like

Anni Pon

Mar 12, 2025 at 11:20am
In response to "Numbers End"

Sorry that this one is coming to an end, but I'm excited to see what you have in store for the new one!

2 Like

Wordsmitty ✍️

Mar 11, 2025 at 1:42pm
In response to "Penultimatum"

It's been an enlightening run. We wish you well in your next adventure.

It will be interesting to see if after the next entry that WDC actually does stop you from adding another.

We're all waiting on pens and pencils for the unveiling of the new blog and to discover what it will be called.

1 Like

BIG BAD WOLF is Merry

Mar 11, 2025 at 10:46am
In response to "Penultimatum"

I'm reminded of this scene from The Princess Bride.

(Vizzini has just cut the rope The Dread Pirate Roberts is climbing up)
Vizzini: HE DIDN'T FALL? INCONCEIVABLE.
Inigo Montoya: You keep using that word. I do not think it means what you think it means.

3 Like

August 24, 2009 at 11:59pm

#665047

Something to Think About

I rarely wax political here, but this article caught my eye.

http://www.ourfuture.org/blog-entry/2009083205/fascist-america-are-we-there-yet

All through the dark years of the Bush Administration, progressives watched in horror as Constitutional protections vanished, nativist rhetoric ratcheted up, hate speech turned into intimidation and violence, and the president of the United States seized for himself powers only demanded by history's worst dictators. With each new outrage, the small handful of us who'd made ourselves experts on right-wing culture and politics would hear once again from worried readers: Is this it? Have we finally become a fascist state? Are we there yet?

There's been a lot of rhetoric fired across the news and internet lately, and what at first glance seems like the same old tired arguments may actually be the front line of a concerted emotional attack on our future. The people involved know that they don't have to win our minds; they only have to win our feelings.

Top of the list for me was one of the latest right-wing attacks on Obama, painting him with a Hitler mustache and calling him a fascist.

Normally I'd be like, "Whatever." It should be self-evident that Obama is not a fascist - at worst, he's a proponent of limited socialism, which, once the Right decided to turn that into a pejorative, I decided isn't a bad thing at all. Basically, these days, whatever the right-wing loonies decide is a Good Thing, I figure is a Bad Thing. Which is too bad, because I tend to agree with the fiscal conservatives - just not the social conservatives.

Well, now the two are conflated, and I can't be in the middle anymore. I have to choose a side. The problem is this:

The right wing is going to win.

When it comes to the war between heart and mind, mind doesn't stand a chance. It never did. There are no right-wing intellectuals anymore; they've all defected. Now they're all Glenn Beck and Rush Limbaugh and Sarah Palin - idiots at first glance, but there's a method to their madness, and that method is to convince us all that black is white, up is down, freedom is slavery, and war is peace.

Consider that, when the war in Iraq was in full excrement under Bush, the left protested. "No unjust war." "Don't kill people for oil." That sort of thing. Rhetoric. I mostly ignored it, while keeping tabs on it. Meanwhile, the smug segment of the right kicked back and counted their cash, secure in the knowledge that they'd be in charge through the New American Century.

Now they're not in charge, and their replacement is trying to institute a public option for health care. I don't know what the details of the plan are right now; I can't find out because they're obscured under a haze of rhetoric, mostly shots fired from the right-wing that scream "socialism" and "fascism" - two forms of government which, anybody but a complete moron knows, are completely incompatible. Okay, yes, the name of the Nazi party was "National Socialists," but that's just an example of a political party using language to convince everyone that they're something other than what they are, like the Red Chinese calling themselves a People's Republic.

So on the left we have: Don't take my taxes and use it to kill people.

On the right we have: Don't take my taxes and use it to heal people.

So that's it. That's the ideological divide that's been nagging at me for a while, and it finally came to me that there's nothing right about the right. Okay, yes, there is, but whatever is good and useful on the Republican side has been obscured under a miasma of town-hall screeching designed to... what? Keep the status quo? If some of these people aren't being paid directly by health insurance companies, I'll munch on my fedora.

Notice that I'm not saying we should or should not have a public option for health care. I don't have an ante at that table, and, like I said, I can't figure out what's actually being proposed. It's the larger questions, and the tactics, that are perturbing me.

What is fascism?
The word has been bandied about by so many people so wrongly for so long that, as Paxton points out, "Everybody is somebody else's fascist." Given that, I always like to start these conversations by revisiting Paxton's essential definition of the term:

"Fascism is a system of political authority and social order intended to reinforce the unity, energy, and purity of communities in which liberal democracy stands accused of producing division and decline."

Given the choice between fascism and socialism, give me socialism. But I don't believe it's a binary choice at all - there's room for a lot of compromise in our system.

The trouble is, I'll be forced to make that choice. And if I come out as a liberal, I could be in severe danger.

I may have to don some protective coloration, soon. I can pass as one of them. It may be the only way to survive.

Fortunately, I already have a future history in place for my science-fiction stories that presupposed a fascist theocracy in place in most of what is now the United States.

Sometimes, I hate being right.

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