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 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.

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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.

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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.

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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.

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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.

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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.

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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

January 17, 2019 at 12:21am

#949810

Rage of Aquarius

Do a bit of research on your Zodiac Sign. What are the associated characteristics of people with your sign and how do you exhibit those characteristics? https://www.horoscope.com/zodiac-signs

Aquarians: Most likely of all signs to consider astrology nonsense.

But hey, I don't judge. It's like religion - if it makes you feel good, helps you try to become a better person, and you don't try to force me to accept its premises, we're fine.

Thing is, I've actually studied astrology. There are aspects to it that are interesting, mostly the folklore and the math involved. Well. Interesting to me.

There's a lot more to it than just the sun sign, which is what you usually get in the pop culture "horoscopes." For the purposes of this discussion, we can ignore the "precession" thing, which means that all the traditional star signs are off by pretty much a whole month compared to their astronomical definitions.

"Sun sign" refers to the segment of the zodiac that the sun is in at a certain time. Usually this refers to someone's birth date, so when someone says "I'm a Pisces," that means the sun was in Pisces at their natal event.

But, in this system, the sun is far from the only influence. The moon sign has a meaning as well, usually taken to be the more hidden side of a person's psyche. So, for example, you can be a Capricorn with a Libra moon.

Consequently, as you might guess, the other planets (yes, the sun and moon are defined as "planets" in astrology) also have an effect. Mercury, Venus, Mars, Jupiter, and Saturn have been known since antiquity. The discovery of other planets - which, again, includes things that aren't defined as planets by astronomers - led to their inclusion in the astrological canon: Ceres, Vesta, Uranus, Neptune, Pluto.

According to astrology, your personality is influenced at birth by the effects of each planet as it relates to the sign it's in.

But wait, there's more.

There are other effects, reinforcements or contradictions, caused by certain alignments of the planets with respect to each other. If you're born on a new moon, for example, it's called a conjunction between the sun and the moon. Full moon - opposition. You can also have, say, Jupiter conjunct (or opposite) Mars.

Obviously you can never have Mercury or Venus opposite the Sun.

And just to make things more complicated, there are also squares (planets about 90 degrees apart) and triangles (planets about 120 degrees apart). Also, rising signs (the zodiac sign on the eastern horizon) and so on.

Oh, and then there are retrograde planets - those times in the Earth's orbit when a planet appears, from our point of view, to be moving backwards. They're not, of course - it's strictly a frame of reference thing. Those mean something as well, usually the opposite of what the planet normally means. Like - Mercury represents change (among other things), but when it's retrograde, it represents stagnation. (Mercury retrogrades happen on a regular basis.)

A "horoscope," then, compares the alignments at a given time with the alignments at a person's birth, and outlines the influences on one's life at that given time.

Because all of these things relate back to the moment of your birth, you need to know your temporospatial coordinates at that moment - time, latitude, longitude. Only then can you achieve a full description of your personality, needs, desires, predilections, and kinks. Even then, it's subject to interpretation and unknowable factors (say, maybe there's a planet yet to be discovered out in the Kuiper belt).

Now, I've never known the time of my birth, though I can pin down the coordinates to within a degree of latitude/longitude. So one time, I asked an astrologer to use my personality to backtrack what my rising sign must have been, which would reveal, to within 2 hours or so, what time I was born. At least - that's how it should work, right? If your rising sign affects your personality, then you should be able to infer the rising sign based on your personality. This guy knew me pretty well, so I figured it was worth a shot.

And - you probably know where I'm going with this - it could have been any of 8 of the 12 signs.

Astrology is bunk. But it's fun bunk.

So let's see what this sun-sign-only website has to say about me.

Strengths
Progressive, original, humanitarian, independent.

Oh, gosh, that couldn't possibly apply to anyone but Aquarians.

Likes
Fun with friends, fighting for causes, helping others, intellectual conversation, a good listener.

Nothing in there about beer.

Dislikes
Limitations, being lonely, broken promises, dull or boring situations, people who disagree with them.

Find me someone who likes broken promises. Further, I revel in dull or boring situations, and I absolutely do not dislike people who disagree with me (only people who are assholes about it, and, to be fair, I also dislike people who agree with me and are assholes about it).

Best environment
Any gathering of people to exchange ideas.

You know, I'm fine with that from time to time, but what part of "introvert" did you not understand? My "best environment" is being home alone with a bottle of scotch, a cigar, my cats and my computer.

Charismatic marks
Good looks, beautiful eyes, angular faces, thin build.

Oh hey, there's something to this astrology thing after all! Sign me up!

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