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:

Kåre เลียม Enga

about 5 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

about 23 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

March 31, 2024 at 8:32am

#1067220

Revisited: "Heartbreak"

You know what today is, right? Oh, sure, it's April Fools' Eve, of course, but being Sunday, it's time to revisit an older entry. This one was from five years ago, and follows my usual comment-on-an-article format: "Heartbreak"

The BBC article

is still up, as of today. Remarkable, considering how many episodes of early Doctor Who the BBC destroyed to make room for newer things. Maybe they learned their lesson.

To summarize, there was some evidence that a particular heart condition is at least partially caused by stress or strong emotion.

Me: The more I learn, the more I think that the commonly stated dichotomy between mind and body is bogus.

I would probably phrase this differently, now. Also, throughout my entry, I seem to have used "mind" and "brain" interchangeably, which I think I'd be more careful to avoid, these days.

My understanding of the internal workings of a person or other animal is rudimentary, but from what I've heard, the brain is a physical, identifiable organ located mostly in one's head. It does lots of stuff, including stuff we don't understand, but it seems to be responsible for, among other things, telling your feet to walk and your chest to occasionally fill with air. The "mind," however, is a far more nebulous concept. I seem to recall doing another entry about that at some point in the not-so-distant past; it involved René Descartes, who asserted that because you can't put a set of spatial coordinates (Descartes invented coordinates) on the mind, but you can on the brain, that there's no way that one can act upon the other. Might as well link it, too, for reference: "Mind, the Gap"

In addition to walking and breathing, the brain is sometimes fairly good at thinking. More in some people than in others, but "thinking" is a legitimate activity, even if it doesn't look like much of an activity to other people. But many people think (pun intended) that it's the mind doing the thinking. Once you're done thinking, often, you do some physical activity. For example, writing what you just thought. We draw that separation between "thinking" and "doing" because we love categorizing things, like cats and dogs being different species, or Pluto not being a full-fledged planet after all.

Point being, a lot of the seeming contradictions of mind/body dualism, such as in the last entry I linked, go right out the window when you consider that the mind is a product of the body. It's probably more complicated than just being conjured up by your brain, but the brain is a body part too. Descartes' assertion that the mind cannot influence the body is, therefore, wrong. Brilliant guy, but in this case, he was working from a shaky premise.

Anyway. The original entry, from up top, ends with the literal video version of Total Eclipse of the Heart, and I'm pleased (state of mind and body) to see that the video is still in existence. I have a whole entry on that song (the actual song, not the parody) coming up at some point; it's in my queue and will pop up at random.

3 Like • Comment

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