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 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').

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

November 20, 2018 at 12:42am

#945890

Better

I think I'm running out of shits to give.

It's not that I don't have strong opinions about things. Anyone who's read a decent number of entries here can figure that out. It's just that I no longer think it's my job to do anything about it, except maybe write.

Oh, I donate to certain causes, like most people do, I think. And I'm usually conscious of how my actions might impact the environment, but I won't usually do anything about it if it inconveniences me too much - because I've used up most of my shits.

Part of that is awareness of the proximity of my own mortality. I'll be dead soon enough, and with me will go whatever remains of my shits. Another part is the secure knowledge that no matter what I do - or what anyone else does - if we don't put colonies in space and/or on other worlds soon, we're doomed as a species.

But a big part, I think, is the realization that, inevitable doom notwithstanding, things are getting better regardless of what I do. It's not a smooth improvement - there are fits and starts, and sometimes it's a three steps forward, two steps back sort of thing - but there is a definite sense of ethical development over the years. It's a kind of cultural awareness that when something is wrong, eventually we overcome its inertia and begin the slow process of making it right.

These changes don't happen overnight. For instance, it was within my lifetime (just barely) that interracial marriage was illegal in my state. Not just frowned upon by conservatives and racists, but actually, written-into-the-law, illegal. Now, it's not, and we even have same-sex marriages and other things that would have been unthinkable when I was born - and make no mistake, these are changes for the better. After all, everyone should be subjected to the horrors of marriage, regardless of their race or sexual orientation. It's only fair.

Another thing that has received more awareness and attention is animal rights. While I think some people go too far (*cough*peta*cough*), there has been an evolution in the way we see nonhuman species, how we relate to them. Circuses have all but dropped animal acts. Zoos have moved from cages toward more natural habitats. Animal testing, where it's still done, is subject to ethical protocols and reviews. People become vegetarian or vegan, sometimes in a bid to protect the other animals on the planet. Places like SeaWorld align themselves more toward marine animal rescue and rehabilitation than exploitation. And in general, awareness has increased as scientists and researchers have found more and more commonalities between nonhuman lives and human ones.

That's nothing to do with me, though. I'll probably be eating a cheeseburger on the day I finally kick it. But I'll die secure in the knowledge that things will continue to get incrementally, if fitfully, better - until the giant meteor or supervolcano makes it all irrelevant.

4 Like • Comment • View 2 comments

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