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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Show Recent Comments

10 Most Recent Blog Comments:

tj is ready for spring!

about 9 hours ago
In response to "Failure Is Victory"

Just think, back in the mid seventeen hundreds, Ole Ben had himself an electric BBQ; he stunned the turkey, shocked himself and cooked the bird on a gadget called an electric jack. Personally, I think he should have used a pellet smoker.

2 Like

🌖 HuntersMoon

about 12 hours ago
In response to "Failure Is Victory"

"Yes, even I make mistakes sometimes."

You may remember, that once words are placed on the internet, they last forever. They WILL come back to haunt you.

3 Like

tj is ready for spring!

a day ago
In response to "Rebel Without a Clause"

It's just like this sometimes.

1 Like

🌖 HuntersMoon

Jan 12, 2025 at 9:58am
In response to "Revisited: "Squash Court""

tj is ready for spring!/Robert Waltz

Oh My God!

Now I understand the etymology of "Pun-ishment."

3 Like

Elisa: Snowman Stik

Jan 12, 2025 at 9:44am
In response to "Revisited: "Squash Court""

Ironically I'm eating scrambled eggs with zucchini as I read this entry.

As someone who has shopped for and cooked delicata squash, I can say that they're mostly seasonal. I occasionally see them in the summer, but they're far easier to locate in the cooler months. They haven't quite broken out the way zucchini has or even acorn squash. I can find acorn squash year round, although admittedly I don't eat it in the warmer months. Speaking of which, if I had to guess, I think pumpkins aren't sold year round because A. summer demand and applicability is low and B. size is a factor. Acorn squash is grown in similar areas/conditions as pumpkin, but it's a smaller size. Grocery stores can carry it year round in a smaller footprint compared to pumpkin even if fewer people are buying it in the warmer months.

4 Like

Robert Waltz

Jan 12, 2025 at 9:41am
In response to "Revisited: "Squash Court""

tj is ready for spring!

- if he has lesser relatives in the produce section, that must be why he's called Bigfood.

4 Like

tj is ready for spring!

Jan 12, 2025 at 9:38am
In response to "Revisited: "Squash Court""

Here in the north woods there are rumors of a giant squash that roams the woods; Sasquash! I've never seen him, but do, on occasion, come upon his lesser relatives in the produce section.

4 Like

tj is ready for spring!

Jan 11, 2025 at 12:48pm
In response to "Time Marses On"

I agree. When we finally get close to inhabiting Mars, I doubt anyone will be scouring through sci-fi books to find a design, asking, "How 'bout this, it looks pretty cool."

1 Like

Robert Waltz

Jan 10, 2025 at 4:24pm
In response to "Cell Mates"

Oooh, a demonstration of Waltz's First Law of the Internet: Any post criticizing someone's grammar, spelling, and/or punctuation will itself contain grammar, spelling, and/or punctuation errors.

I'm pretty sure I've read all of Heinlein's books and stories, with the exception of the naked money grabs his estate published posthumously. Some of them didn't age well, but taken in the context of their time, many of them are solid science fiction.

And I can't tell you how many times I've wanted to use "grok" in an entry but passed because I didn't want to overuse it. As the blog is nearing its end, I threw caution to the wind today.

Sum1's In Sacramento

Jan 10, 2025 at 4:13pm
In response to "Cell Mates"

@Jayne

Should I Play Grammer Police here, and tell you that you misspelled Henlein? He's one of my favorite authors. If I had the money (I do really), and a place to keep them (I don't), I would get his entire collection from the Heinlein Society. I have read many of his books and short stories, there are also many that I haven't read too.

1 Like

January 14, 2025 at 9:37am

#1082343

Failure Is Victory

Whaddaya know—it turns out that nobody's perfect, though some seem to be less perfect than others. Here's a Cracked article for examples, because I am in no way referring to recent real-world revelations.

The Most Embarrassing Failures of Famous Geniuses
They can’t all be winners, even from the biggest winners in history

Yes, even I make mistakes sometimes.

5 Benjamin Franklin’s Turkey Electrocution

At one point, he became convinced an electrocuted turkey would be tastier than a normal one, and in attempting to demonstrate this, he accidentally touched the electrified wire intended for the turkey and electrocuted himself instead.

As far as I can tell, this, unlike certain other Founding Father stories, actually happened. Except for the word "electrocute." That was unknown until a century or more later, when Edison coined the word in an attempt to make Nikola Tesla look bad (long story). It's a portmanteau of "electro-" and "execute," as in, one can only be electrocuted if one dies from it. As Franklin continued to live, he wasn't electrocuted.

4 Francis Bacon’s Frozen Chicken

Mmmm... bacon and chicken.

In 1626, Bacon was determined to prove that you could freeze and preserve a chicken by stuffing it full of snow.

Okay, fine, a hypothesis. Which he went and tested. This is science. Every scientist makes hypotheses that reach dead ends. So, in this case, it's a literal dead end, which is why this counts as a failure, and not merely a falsified hypothesis.

3 Thomas Edison’s Creepy Talking Doll

In 1890, the world wasn’t ready for Chatty Cathy.

It may be that the word "electrocute" from above was the only thing this guy ever actually invented, rather than stole from his employees. Point being, he probably didn't actually invent the talking doll, either. But he certainly marketed them, which led to people finding dolls creepy forevermore.

2 Mark Twain’s Ill-Fated Start-ups

Twain was one of the highest paid authors in 19th-century America, but it somehow wasn’t enough for him.

Some people are good at lots of things. Others, not so much. Thing is, you never know which you are until you try, and possibly fail.

1 Albert Einstein’s Cosmological Constant

When Einstein was forming his theory of general relativity, it was believed the universe was static, so when his equations kept predicting some wacky expanding universe, he added a term he called the cosmological constant to make them work.

I'm not sure this was such a failure. He seemed to think it was, but it's just an extra term in the equation, easy enough to take out. We did it in engineering school all the time; we called it F_f (read "F sub F"), for Fudge Factor. Let's also remember that this was back when people thought our galaxy was the extent of the universe.

I see it more of a lesson about examining your assumptions, even the unspoken ones. Which we should all be doing, scientist or not.

8 Like • Comment • View 2 comments

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