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

October 26, 2008 at 6:37pm

#614893

Freedom

Someone asked me the other day what I thought of "socialized" health care.

My off-the-cuff, edited for typos and clarity, answer:

"Socialized" is a loaded term.

Under our current system, those who have the right type of job contribute to a for-profit company for various types of health coverage. In some cases (like my company) the insurance is 100% employer-paid; in others, some lower percentage. Those who need more care don't usually pay more than those who need less, meaning that healthy people contribute to the health care of those who need more care AND to the insurance company's bottom line.

People who don't have the right kind of job pay more, because the risk isn't spread out over a group. Or they do without health insurance, which represents a more hidden burden on society.

In other countries (often labeled "socialist" by the right wing of this country), health care is provided in other ways, usually involving higher taxes.

If one were to back out their actual cost of health care, one would find that the individuals in those countries aren't paying any more taxes than we are; in some cases, less. Put another way, if you add what I pay for health insurance (via my company) to what I pay in taxes (between my company and personally), my total obligation is approximately that of, say, someone in Sweden, on a relative basis.

So to answer your question, what I think of "government-run" medical care is that if it provides access to coverage for more of the population, all of whom pay taxes of some sort or another, I'm for it. Provided it can be handled in such a way as to create less overhead than what we have now in the insurance-company-profit situation.

The individual's response included the postulate that it may be that a whole lot of people are tied to their soul-deadening, corporate jobs because said jobs provide them with health insurance.

Granted, a whole lot of people are in debt. I read statistics before this whole credit fiasco blew up in our faces that indicated that the average savings rate of Americans was negative - meaning that as a group, we have more debt than savings. I don't know if it's still the case, but it wouldn't surprise me. A person in debt has restricted freedom: if you owe, say, $200,000 (and between student loans, mortgage, car loans and consumer debt, this is not an unreasonable amount), your life is turned over to servicing those loans. While mortgage debt can be reduced or eliminated by selling the home, and student loan debt can, as I understand it, be deferred under certain circumstances, there's debt that you're stuck with, and a good part of your salary goes to paying that debt. You can't leave your soul-deadening job, or the creditors will come after you, and interest will continue to go up. Sure, there's bankruptcy, but that carries its own problems.

But what if you don't have significant debt, and maybe even some savings, and you just want to take your chances, strike off on your own, and, say, become a massage therapist or eBay retailer?

Not and keep your health insurance.

Is this why there's such a backlash against "socialized" medicine coming from the far right, which at times acts as if it's run by religious extremists, and at other times seems as if it's a puppet of the big corporations? Are there people - powerful people - afraid that if we suddenly had a parachute for folks who don't want to play the corporate cubicle game, they won't?

I think the health care system has many flaws, but one big one is how it's currently run by insurance companies. They dictate, rather than your doctor, what is "medically necessary." And I think that it's a direct result of insurance being in control of medicine that the cost of health care is so high.

Let me explain, briefly: As recently as a year ago, we had real estate values that were unsustainably high. I said it then, and with recent events, it seems I was right. But one reason why they were so high was a matter of affordability. With the advent of easily available interest-only loans, balloon mortgages, and low-teaser ARMs, a whole lot of people appeared to be able to afford houses that they otherwise wouldn't have - because their incremental cost (their monthly cost) was suddenly within their means as opposed to the way it would be with a traditional 15- or 30- year mortgage. This allowed sellers to artificially inflate their total costs, while still putting houses within reach of the average family.

And so we have the same situation with health care - they can charge more, so they do. We don't see the cost, even though we may get a bill for thousands of dollars, because on that bill it says your actual cost is, like, $20.

It's inflated. It's not sustainable. And it's going to bite us on the ass the same way that home-price inflation did.

It's not conducive to individual freedom.

I'm not saying people should rely on "the government" to pay for their health care, or that "the government" should provide health care. But the system as it stands now benefits only big corporations.

Yes, "socialism" is a loaded term, these days. The Republicans are using it as a pejorative, like "liberal." And, last I heard, the major sign that a country was turning "socialist" was that the banks get nationalized. Ummm... d'oh. But as I understand it, socialism is essentially the radical notion that individuals matter.

Isn't that what we're supposed to be all about?

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