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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🌖 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.

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Stik to My Own Beat

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 wanderlust-words-in-motion

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.

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tj wanderlust-words-in-motion

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.

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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.

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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!

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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.

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BIG BAD WOLF is Surprised

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

September 12, 2010 at 10:29pm

#705885

On a serious note...

Lots of media attention on attempted Koran/Qu'ran/(you know, the Muslim holy book) burnings, lately. And I get all my information from the internet - no cable or newspapers - so I can only imagine the media carnage on those platforms.

Of course, even on the internet, there's been plenty of hot air (pun intended) blown around about it. And a person's opinion doesn't seem to have much to do with the political or religious leanings of the person - that is, not all Jews believed one way or another; not all atheists were in agreement; not all Christians agreed or disagreed. I'm not exposed to enough Muslims to know, but I figure they might be an exception; I can't imagine a Muslim being okay with it. But I suppose it could happen.

Me, I'm conflicted. And even though - or more likely, because - the time for Qu'ran burnings and other inappropriate 9/11 commemorations has past, I'm going to put my thoughts down here, to try to organize them and give y'all a chance to tell me how wrong I am.

My first response, upon hearing about the concept of "Let's burn a Qu'ran," was, "Book burning is Bad." This is my visceral response to any book burning, and I'm not even sure why. I think it's because the Nazis liked to burn books, and my family's Jewish - and we all know that they didn't stop the burning at books. It's also because I personally think that the greatest atrocity against world knowledge, science, and progress was not the Holocaust (which is arguably the greatest atrocity against humanity), but the burning of the Library at Alexandria. Interesting stories behind that, but basically, you had then what you have here today: an intellectual elite that wasn't very religious, and a subset of Christians who had enough intelligence to know how to make fire, but not enough to keep them from using it to burn books (and torture and kill one of the greatest mathematicians of all time, but that's another issue).

Okay, that was my first reaction before I knew anything other than "this preacher wants to burn a Qu'ran." Then my inner libertarian kicked in. "Is the book his property; i.e., did he obtain it through legal means like buying it? And is the fire going to be on private property? And are there safeguards to protect damage to others' property?" If the answer to these questions is "yes," then I believe that the First Amendment grants us the right to free speech, which includes burning private property on private property, assuming you're not causing direct harm to someone else's property.

Nothing, of course, is ever as simple as libertarians like to pretend it is. In this case, there was the threat of retaliation from some of the people whose holy book it was. Now, if the original 9/11 attacks were the result of angry, radical Muslims, is it really smart to make more angry, radical Muslims?

But then I thought about it further, and I thought, "So what?" Freedom isn't free, as they say. I know they think this means that we should support the military, but what it really means is this: The reason we have things like freedom of speech and freedom of religion enshrined in the Constitution is that, by and large, these things didn't much exist except as philosophical constructs before the late 1700s. The thing is, freedom of speech is a radical idea. Before that - and even after, before we decided that the Constitution overrode local ordinances - there were blasphemy laws in this country. Other countries still have blasphemy laws. The basic idea of a blasphemy law is that you can't say something to offend (whatever religion is in power), or you get punished through the civil system. And people died to defend the radical idea that a person's freedom to speak his or her mind, even if it goes against conventional wisdom, is sacrosanct.

To me, nothing is more offensive than the idea of blasphemy laws. Plenty of worse laws out there, yes, but they don't offend me on a personal level.

Now, I didn't hear anyone saying it should be illegal to burn a Qu'ran (or a Bible, or a Torah, or the Book of Mormon, or whatever), just that it was a Bad Idea. Probably someone did; I don't know. I just didn't hear it. Naturally, if I owned a copy of one of these and wanted to chuck it into my fireplace, no one would be the wiser. But these people tried to do it publicly for the express purpose of pissing people off. Well... Okay. I believe we have the Constitutionally guaranteed right to piss people off. There may be consequences. People may get mad and kill other people over this.

That's because free speech is a radical idea.

And the only way to protect this radical idea is to use it, and face the consequences. Face them boldly, willingly, and with full knowledge that you may die for the idea. That, because we as a society hold freedom of speech to be one of our highest ideals, we need to support this freedom no matter the consequences.

Personally, I think we'd do better as a society if we'd actually bother to read the Qu'ran. And the Bible. I don't see how anyone can read either book from start to finish without realizing that religion - all religion - is made-up bunk.

But your opinion will differ. And that's okay. Because we're all free to have one - and we're all free to have one because we're willing to face the consequences if someone doesn't like our ideas.

Me, I'm working on a community center and mosque with some Muslims in my neighborhood. I've worked on church site plans, too. If they asked me to work on a temple, I'd do that. If the secular humanists wanted a headquarters, I'd give them the same professional consideration. I wouldn't burn any of their holy books (not that secular humanists have a holy book, but you know what I mean), because I believe in finding commonalities, not differences. But that's just me.

Do what thou wilt.

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