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

May 2, 2009 at 11:00pm

#647927

Quack!

The more I read, the more I think people is mostly dumb.

Case in point:

http://observers.france24.com/en/content/20090413-plight-africa-albinos-cameroon...

"I left my country to escape the widespread prejudice targeting albinos such as me. People are starting to talk more and more about the plight of albinos in Tanzania, but things are much the same in Cameroon. Different parts of our bodies are sought for their supposed miraculous powers: to become rich, win elections or concoct love potions.

In south-west Cameroon, where I studied, there is an old belief that when the local volcano erupts, only albino blood can placate the god of the mountain..."

Seriously, wtf? I've heard that there's a widespread rumor in some African countries that the way to cure AIDS is to have sex with a virgin, and since you never can tell who's a virgin they look for younger and younger "cures" - with the predictable (to us) result that AIDS becomes more widespread. To be fair, I have no idea if this is true or just rumormongering, but the fact that it's believable to me means that either a) I'm terribly prejudiced or b) Africa is so frakked up that it makes Japan look normal. I'm inclined to b) because of things like the albino story above.

"Oh, so what we need is more education!" I hear that a lot. "Education will end superstition!"

Well, actually, no. Education doesn't seem to make a dent in deep-rooted superstition. It wasn't that long ago that folks of Western European descent were burning witches, and we still turn to quackery when things get tough, despite scientific evidence. Here:

http://www.newscientist.com//article/dn17064-quack-remedies-spread-by-virtue-of-...

Eating a vulture won't clear a bad case of syphilis nor will a drink made of rotting snakes treat leprosy, but these and other bogus medical treatments spread precisely because they don't work. That's the counterintuitive finding of a mathematical model of medical quackery.

Ineffective treatments don't cure an illness, so sufferers demonstrate them to more people than those who recovery quickly after taking real medicines...

Okay, so it's not nearly as bad as cutting parts off of albinos to bring good luck (to the cutter, presumably, not the albino), but it's still quackery.

Nor am I immune. When my back was in persistent, unrelenting agony last year, I tried everything I could, including so-called "holistic" remedies. And hell, for all I know, they worked, because when the pain finally stopped, I couldn't point to any one cause for it to stop - though I'd tried standard medical treatments along with physical therapy, massage, herbal pills, and electrostimulation.

(If I had to guess, I'd venture that it was the electrostimulation - but I have no idea. At the very least, it didn't seem to cause harm)

Point is, I'm an educated guy, especially by world standards, and I was still willing to try, well, just about anything to make the pain go away (though I like to thing I'd stop short of harvesting albino parts, it's probably good that I don't have ready access to albinos). I can only imagine what people who have life-threatening or terminal illnesses are willing to do, with or without the benefit of science and education.

But why? Well, I'm not sold on the effectiveness of "scientific" treatments in many cases. From the second article linked above:

And in some cases, one peer-reviewed study may conclude that a drug works, while another shows it doesn't. "Even where there is a bit of clinical research, we don't really know yet whether at lot of medicines are effective..."

And what about prescriptions that turn out to be worse than the disease they're supposed to cure? How are we supposed to know? Hell, even putting aside medications, we can't even get it straight as a culture whether certain foods are good for you, bad for you, or neutral. Chicken eggs, for instance: during my lifetime, they've been nutritious, deadly, okay, harmful, helpful and now I think they're back to being nutritious, but I stopped giving a shit long ago and just enjoyed the occasional omelet.

Or how about aspartame, that artificial sweetener that's in bloody everything these days? No one seems to argue that it's beneficial in and of itself, but some say it's good in that it allows people to satisfy a craving for sweets without the attendant calories that can lead to obesity and complications thereof. Others assert that it's harmful in the long run because it doesn't really satisfy the craving for sweets, encouraging people to eat more to compensate. Me, I don't know; I just know I hate "high fructose corn syrup," which is what they put in Coke these days instead of sugar, so I drink Coke Zero instead. For whatever it's worth, the FDA considers aspartame safe.

But there was an email circulating a while back that blamed it for cancer, AIDS, spontaneous human combustion, and the takeover of the Republican party by religious fanatics (okay, maybe not anything quite as bad as the last item, but you get the idea):

http://urbanlegends.about.com/library/blasp.htm

...I explained that I was there to lecture on exactly that subject. When the temperature of Aspartame exceeds 86 degrees F, the wood alcohol in ASPARTAME coverts to formaldehyde and then to formic acid, which in turn causes metabolic acidosis. (Formic acid is the poison found in the sting of fire ants). The methanol toxicity mimics multiple sclerosis; thus people were being diagnosed with having multiple sclerosis in error. The multiple sclerosis is not a death sentence, where methanol toxicity is... (this text is quoted from the circulating email, which is widely accepted as false).

From page 2: Martini's answer, as I mentioned above, is pat and comes in the form of a conspiracy theory: the NutraSweet Company has used its deep pockets to buy all but a very few critics off. I.e., the only sources you can trust, including doctors and scientists, are those approved by Martini.

You know, if you have an assertion to make, you're not helping your credibility to proclaim that everyone who disagrees with you is part of a vast, overarching conspiracy. Then again, maybe the author has simply consumed too much Splenda. Point being, if we don't have pre-existing superstitions about something, we seem to have to make them up.

So who can we trust, when it comes to deciding what to put into our bodies or not?

No one, of course.

But science is still better than superstition. Unless, of course, science is actually a giant conspiracy that will not rest until we are destroyed.

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