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

June 12, 2009 at 5:42pm

#654258

Virginia is for Loving

Almighty God created the races white, black, yellow, malay and red, and he placed them on separate continents. And but for the interference with his arrangement there would be no cause for such marriages. The fact that he separated the races shows that he did not intend for the races to mix.
-Leon Bazile

It was 42 years ago today when the Supreme Court kicked my home state forcibly into the 20th century.

I didn't know it at the time. Neil Armstrong had not yet walked upon the moon, the sight of which is my first memory. I was only a year old and my parents lived in Baltimore, which was in, perhaps, a slightly more progressive state.

http://en.wikipedia.org/wiki/Loving_v_virginia

Basically, in the late 50s, a white man married a woman of mixed race descent - said mix including African genes. They tied the knot in DC, which was apparently also slightly more progressive, then moved back to Virginia - not far at all from where I (arguably) grew up. Virginia, at the time (less than fifty years ago!) held marriage between a man and a woman illegal - unless they were of the same "race."

The above quote illustrates the legal basis of this law. I'm sure, at the time, it seemed perfectly reasonable. An argument, that is, based on a theology (which is not supposed to form the basis for laws in this country, but creeps in anyway) that is itself suspect. Hell, if that argument were any more circular we could use it as a frisbee.

People who claim to know what God did or did not intend scare the living crap out of me. If God had, for example, intended for us to fly, he would have given us the brains to develop airplanes. Oh... wait... yeah. Nevermind.

When said people are making or interpreting laws, they scare me so much that I feel like running into the relatively safe, comforting arms of a clown.

Anyway... Loving v. Virginia, like I said, kicked us into the 20th century. 67 years too late, perhaps, but it happened. It took even longer to shift public opinion to general agreement, but hell, I know damn well that there are still pockets of resistance to the radical idea that the human race is one race. One such pocket, in a racially-motivated rage, killed a security guard the other day, not far from where that case was decided.

According to that Wikipedia article, for whatever it's worth, the last miscegenation statute wasn't repealed until 2000, just missing overlapping into the 21st century (which actually began in 2001).

The idea that the government has no business regulating such things is, apparently, newer than I thought. But we seem to be getting there.

Marriage is one of the "basic civil rights of man," fundamental to our very existence and survival.... To deny this fundamental freedom on so unsupportable a basis as the racial classifications embodied in these statutes, classifications so directly subversive of the principle of equality at the heart of the Fourteenth Amendment, is surely to deprive all the State's citizens of liberty without due process of law. The Fourteenth Amendment requires that the freedom of choice to marry not be restricted by invidious racial discrimination. Under our Constitution, the freedom to marry, or not marry, a person of another race resides with the individual and cannot be infringed by the State.

So... how're those gay marriage bans working out for y'all?

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