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-Merry Mischief Maker

about 7 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

about 12 hours 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

about 14 hours 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 ✍️

a day ago
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

a day ago
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

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

April 6, 2024 at 10:09am

#1067789

Ageless

I need to believe that this article, from CBR and brought to my attention by a friend, is satire... but I'm still not really sure.

10 Ways Calvin and Hobbes Has Aged Poorly
While Calvin and Hobbes is generally regarded as an iconic, timeless comic strip, there are some ways in which it hasn't aged especially well.

Imagine if I wrote something like "Why Shakespeare's plays have aged poorly," and cited things like archaic language, outdated customs, and now-obscure references?

Am I really comparing C&H to Shakespeare? Why yes. Yes, I am.

Any reader can still enjoy the strip the way Bill Watterson initially intended. The characters, setting, and circumstances always feel like something that could easily be replicated today.

Except that, admittedly, the characters don't have smartphones to stare at.

Like Cracked, CBR seems to be fond of countdown lists. As there are ten items, I'm not going to bore you with every single one of them.

10 No One Supports Calvin

Calvin's parents are often busy taking care of the house or working, leaving Calvin with few opportunities for emotional support... It is a lot of pressure for a little boy and would not be considered socially acceptable parenting today.

Leaving aside for now the open question of whether this is satire, that is kind of the whole point of the strip: that while all of C's physical needs are taken care of, he has Hobbes for everything else. Since Hobbes is only animated in his imagination, he's really supporting himself.

9. Calvin Gets Dinosaurs Completely Wrong

Dinosaurs actually had feathers, which contradicts Calvin's imagination

Now... everyone should know by now that I'm usually the first person to get all technical about stuff like this. I've done it here in this blog. I've done it in newsletters. Hell, I've done it in fiction writing. But dinging the strip for this is like ragging on Jules Verne for getting a trip to the moon all wrong, or on Ray Bradbury for imagining a Mars that's inhabited and inhabitable. Or, to keep it in the comics world, it's like ragging on Pogo because opossums don't hang out with cigar-chomping, bipedal, talking alligators. On top of which, it feels like the same thing as saying "an actual tiger would have eaten Calvin," neglecting the whole bit about it all being one little kid's imagination, and this is what first led me to believe that we're dealing with satire here.

Besides, I'm pretty sure not all of them had feathers. What we call dinosaurs were around for longer than they haven't been around, and they were quite diverse. Don't quote me on this, but I think they started scaly and at least some lineages, over the couple hundred million years or so that they dominated the planet, later evolved feathers, which are modified scales (so is hair on mammals). The few that survived the mass extinction event did have feathers, and their descendants are what we call birds.

None of which matters if you're a six-year-old kid with an overactive imagination.

7. Calvin's Babysitter Threatens Him

After all, Rosalyn regularly threatens Calvin for even the slightest offense. Staying up late or playing too loudly often led to threats of violence or a punishment of solitary confinement.

That's just the way we did things back then. Also, I can attest from personal experience that for a kid like Calvin, solitary confinement is hardly a punishment.

5. No One Tries To Stop Moe

In the modern world, students would receive instant support in the face of bullying, especially if the bully is physically hurting them or stealing from them.

From what I understand, lots of bullying still goes unremarked these days, too. True, it was probably worse in the past.

"Instant support," my ass.

4. Calvin Spends Too Much Time By Himself

This one, I took personally, having been an only child with no other kids usually around outside of school. That made me more self-sufficient and less needy, qualities that I consider part of my core personality. Would I be better at social skills otherwise? Possibly. But then I wouldn't be Me.

I've said in the past that Calvin was that kid I wish I had been, and hope I never have. The strip was a big influence on my decision to remain child-free, lest I end up with a Calvin to deal with.

3. Calvin Wanders A Forest Alone

Hey, so did I.

1. Calvin's Behavior With Susie Wouldn't Be Acceptable Today

I have only vague and sporadic memories of being six, but at that age, girls were alien creatures. Pretty sure they saw us the same way at the time. Cooties were scary.

In any case, most of these boil down to needing historical context. Lots of older works of art need this. I mentioned a few of them above. (And make no mistake; C&H was absolutely a work of art.) This doesn't negate the art; rather, it's an opportunity to learn more about changing cultural norms and even scientific findings.

And if this article really is satire, it's not very good satire.

3 Like • Comment

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