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:

Turkey DrumStik

about 3 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!

a day 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

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').

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

January 20, 2019 at 12:46am

#950038

Historical Witness

What event from history do you wish you could have witnessed?

So many options...

I'll assume here that I'd be a kind of fly on the wall - unobserved myself, unable to affect anything, and unable to be affected by anything other than photons and sound waves. I mean, what's the point in observing if you can't observe? I'm also thinking this means singular events, not protracted ones such as wars.

This means that I wouldn't be able to pre-emptively kill anyone whose impact on history is negative, like the classic Hitler or the pure evil of Carrie Nation.

So, the first thought that came to mind was to observe the first moon landing (as per above, I'd be unaffected by the extreme heat or vacuum). It was, after all, not only the most significant thing we've ever done as a species, but the most significant thing I can imagine doing (we might set foot on other worlds in the future, but the moon was the first). Then I'd be able to tell all the retards who think the event was faked that it wasn't. But then I realized - these people are retards. They wouldn't believe me, anyway, any more than they do now. They're not worth my time, and besides, the event was well-documented anyway.

That got me thinking about conspiracy "theories," so of course the next thing I thought of was the assassination of JFK. But that's depressing, and besides, there were already hundreds of witnesses as well as video footage, and there's still controversy surrounding it, so why would I expect to see what actually happened any more than anyone else there? Besides, again, no one would believe me, so I'd only be indulging my own curiosity - and honestly, I'm not that curious about it.

Consequently, I figure it needs to be something that I'm personally curious about, but feel no need to share with others. If I take "history" literally, I'd be limited to the past 10,000 years or so; before then, there were no written records - hence, "prehistory." A shame, because I think it'd be fun to observe the presumed impact event that ended up wiping out most of the dinosaurs, or the moment some human first tamed fire.

Then there are the events that supposedly took place during historical times, but are of questionable historicity and/or the subject of contradictory accounts: Washington chopping down the cherry tree, the birth or death of Jesus, Alexander cutting the Gordian Knot, etc. Those are cans of worms I don't really feel like opening; again, we run into the idea that I'd have to satisfy my personal curiosity without feeling the need to convince others of what I saw/heard. Or didn't see or hear, in these cases. At some point, it doesn't matter whether an event really happened or not; it makes no difference to how we think about it.

So I think it depends on when you ask me. I'd have had a different answer last year, and will probably have a different answer next year. Hell, I'll probably have a different answer tomorrow. But right now, today, this article caught my eye, as it concerns two of my most favored historical figures:

https://theculturetrip.com/north-america/usa/new-york/articles/what-you-need-to-...

Given his longevity, bookishness, and outstanding oddness, it’s not terribly surprising that one of Tesla’s few friends was the writer Samuel Langhorne Clemens, alias Mark Twain.

Having met at a New York social club to which both men belonged, Mark Twain credited Tesla with curing his debilitating constipating[sic] with an oscillator that vibrated his bowels until he narrowly made it to the restroom in time to experience his remedy.

That. I'd pay to see that.

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