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.

May 31, 2021 at 12:02am

#1011036

Memorial

Tomorrow, I'll resume posting links to discuss, and sometimes entries into "Journalistic Intentions"

[18+] (which I suggest checking out; the prompts are really open-ended and I'd love to see others' takes on some of them). So consider today kind of a break. A holiday, if you will.

Interesting word, "holiday." In British English, it means "vacation." Here in the US, it's usually some sort of observance, commemoration, official day off for some people (but obviously not everyone).

The word itself is, as one might guess, derived from words meaning "holy day." As I've noted here too many times to count, words change over time, but I find it useful to know their origins sometimes.

I urged yesterday, "Tell us about your Memorial Day plans!" So it's only fair, I suppose, that I do the same. Memorial Day is, by most definitions of the word, a holiday, a secular one observed in the US (though of course different religions can interpret it as they see fit) to commemorate primarily those who have died in the service of the Armed Forces of the US. We have other days for honoring those who have served, or will serve.

In practice, it's a day for cookouts, going to a beach, setting off fireworks (really, we Yanks don't need much excuse for that), or getting together with friends. It's the unofficial start of the summer season here, so summer, outdoor activities predominate. Some find this disrespectful of the purpose of the holiday, I suppose. I'm not going to take sides either way, but I will point out that such activities fall under the nebulous heading of "freedom," which is what they say service members fought for.

Me? I hate crowds even when there's not a pandemic going on (and make no mistake, there still is), so I generally stay home and drink. Coincidentally, that's what I do most days, but when I drink on Memorial Day, I remember my dad and my uncle (not his brother, but my mom's).

Neither of them died in battle, but both served in WW2. And I say they didn't die in battle, but with my uncle, that's a technicality. I knew him, growing up, as a big, scary, ominous presence who wasn't entirely sane.

Not that he was dangerous. I mean, sure, he had been a soldier, but when we got into that war, it's not like he had much of a choice. He had been a gentle man (that space between those words is entirely intentional), and remained so. He was just... not all there. Unpredictable. My aunt, his sister, had to take care of him while running a business, and she did her duty, too. They called it "shell-shock" then, but I suppose he had what would be called PTSD now.

That's what happens when you're a Jewish American soldier marching through Germany, seeing your comrades die in battle, and learning the first-hand truth about what was going on at Dachau.

I know from his letters home, which I've read but shan't quote here, that he did, indeed, die in that war, even though his body lasted another 45 years.

So that's what I remember on Memorial Day. Take it as you will. I won't shame anyone for grasping what enjoyment they can from the holiday, or for doing the opposite and commemorating it in silent solemnity, or anything in between. Like I said... freedom. It comes at a high price, and it's not only the soldiers who pay it.

Mini-Contest Results

Thanks for all the comments! I couldn't be arsed, after all that above, to use the Virtual Dice, so you'll just have to take my word that the choice is random. LeJenD'

gets the Merit Badge for today. There will be other opportunities, hopefully ones less depressing.

Just a couple of quick notes:

SandraLynn, Victoria Day is the "first official weekend of non-winter" in Canada? I didn't know there was a "non-winter" in Canada. Okay, I exaggerate. I've been to Toronto and Vancouver (different years) in July, though, and they were complaining about 80F being a heat wave. That's not a heat wave. That's a mild spring day.

Don't get me wrong; I love Canada. Beautiful country, when it's not covered by s**w.

Happy birthday to Turkey DrumStik!

Write_Mikey_Write!, you honor your friend.

Everyone else, again, thanks for your comments, and however you observe the day (or not), have a good one.

7 Like • Comment • View 2 comments

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