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:

Write_Mikey_Write!

about 6 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 8 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.

August 23, 2020 at 12:02am

#991386

Fur the Fun of It

I haven't been to a barber in months, so this article caught my attention.

https://www.smithsonianmag.com/science-nature/why-did-humans-evolve-lose-fur-180...

Why Did Humans Lose Their Fur?
We are the naked apes of the world, having shed most of our body hair long ago

As usual, I have issues with parts of the article, but pointing them all out would be tiresome. All I'll say right now is that almost all of the evolutionary discussion here is pure speculation and hypothesis... and that the question of "why" in the headline might be misleading.

Millions of modern humans ask themselves the same question every morning while looking in the mirror: Why am I so hairy?

Well, for me, as I said, the answer is "I haven't been to a barber in months." I suspect a lot of men my age are more asking themselves, "Where did all my hair go?"

Evolutionary theorists have put forth numerous hypotheses for why humans became the naked mole rats of the primate world.

And few of them are testable.

Scientists aren't exactly sure, but biologists are beginning to understand the physical mechanism that makes humans the naked apes. In particular, a recent study in the journal Cell Reports has begun to depilate the mystery at the molecular and genetic level.

Even the Smithsonian mag can't resist a bad pun.

The article proceeds to explain some of the genetic findings, which I can't really comment on but I'm assuming are current science. But that answers "how," not "why."

I'll come up with an analogy here. Could be relevant, could be not; I don't know. Cats can't taste sweetness.

. This is, apparently, due to a genetic mutation way back in their lineage. And it's not just your house panther; it seems that all felids have this mutation. ("But my cat loves ice cream and cake!" "The article I just linked addresses that. Also, don't feed your cat ice cream and cake.")

But does that mean that there was a reason for it -- a "why?" No. And whether it's the mutation that makes a cat a cat, or being a cat is what resulted in the mutation, well, that's above my pay grade. But it appears to connected to why cats, unlike most mammals, are obligate carnivores. ("But my cat eats grass!" "Yes, and then she throws up on your bed.") So it seems to me with the hairless thing in humans. A mutation turned out to be beneficial for our early human ancestors' survival.

With a greater understanding of how skin is rendered hairless, the big question remaining is why humans became almost entirely hairless apes. Millar says there are some obvious reasons—for instance, having hair on our palms and wrists would make knapping stone tools or operating machinery rather difficult, and so human ancestors who lost this hair may have had an advantage.

Okay, this is one place where I am going to quibble here. Operating machinery wasn't, I believe, much of an issue a million years ago. Evolution didn't plan for the industrial revolution; evolution doesn't "plan." This is symptomatic of my issues with the evolutionary guesswork these people do. Of course, it's important to make such guesses, but there's nothing substantial to back it up. And again, they might have cause and effect switched.

There's more of this guesswork in there; I won't belabor it fur-ther.

Now, I wonder if I can borrow some of my housemate's ponytail bands?

3 Like • Comment • View 3 comments

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