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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Turkey DrumStik

about 7 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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').

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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

November 21, 2018 at 1:23am

#945957

Final Frontier

I was born with Star Trek.

Once, I read somewhere that what would become known as The Original Series was greenlit by the studio the same month I got shoved into the world, and I always imagined it happening on the same day. The show made its TV debut later that year.

The series ran no more than three seasons, so if I saw any of them when they originally aired, I don't remember. There's not much I can recall from the first three years of my life, and what memories exist from that time are highly suspect. The first thing I do remember with any certainty was my mom waking me up to view the first moon landing, an event whose significance I would only appreciate much later in life.

Similarly, I'm sure I didn't fully appreciate Star Trek when I was a kid. I remember enjoying the action and the aliens, and wanting very much to be Spock, and I watched the reruns whenever I got a chance, but it was only later that I began to appreciate its nuances and social commentary.

It was still later before I began to cringe at what passed for special effects. Part of that was me becoming more discerning, but part of it was being spoiled by advances in SFX technology. They did the best they could with what they had, I think, and if you watch the contemporary episodes of Doctor Who (Patrick Troughton, Second Doctor era), well, those are even more primitive. But that never really mattered to me - what mattered was the story. At least the Enterprise wasn't powered by lava lamps. Seriously, something in Doctor Who was powered by lava lamps and I can't even.

Since then, of course, we've been treated to numerous other Star Trek series and many movies, some of which were even good. But I can't overstate the impact the original series had on my life. It wasn't just that it inspired me to read more science fiction, or write it, but it was also the vision it had of interpersonal relationships, the spirit of exploration and discovery that is so central to all of science, and the understanding that it is merit, not privilege, that should determine one's position in life. Perhaps as importantly, it provided an optimistic view of the future: hope, not despair. As I am prone to despair and cynicism, I need that sometimes. Sure, it's unrealistic, but when has realism been a prerequisite for a good story?

I'm by no means alone in my appreciation for the show, of course. That imagined world has influenced our real one in significant ways, and also insignificant but very funny ones like when William Shatner's character on Boston Legal opened his flip phone and it chirped. Life, as they say, imitates art.

It's had its ups and downs over the years, but it endures. And isn't that what every creator hopes for?

5 Like • Comment • View 7 comments

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