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:

Write_Mikey_Write!

about 5 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 7 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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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