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:

Elycia Lee ☮

about 5 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 18 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 21 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

about 23 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

March 18, 2020 at 12:16am

#978409

Rage of Aquarius

It's understandable, in this difficult period, to search for meaning and look for answers.

https://www.theguardian.com/lifeandstyle/2019/nov/06/i-was-an-astrologer-how-it-...

I was an astrologer – here's how it really works, and why I had to stop
Customers marvelled at my psychic abilities but was that really what was going on when I told their fortune?

I was a fortune teller. Every Sunday, I climbed the stairs of an old terrace house in Sydney’s historic Rocks district, to sit in the attic and divine the future. I would read Tarot cards or interpret horoscopes.

Remember what I said yesterday, about a holistic experience? Setting is important.

As a teenager, I’d devoured a book called Positive Magic. An instruction manual for witches, its central idea was that if you wanted something, and you had good intentions, you just told the universe and magic would happen.

And if magic didn't happen, then you didn't want it badly enough, or your intentions weren't good enough, or you didn't phrase it properly.

I also learned that all things are connected, and everything is a symbol of something else. Suddenly, I saw signs and omens everywhere.

All things are connected. That doesn't mean we can see, or exploit, the connections.

Although astrologers use Nasa data for their calculations, horoscopes aren’t a true map of the heavens. The Babylonians who invented astrology believed the sun rotated round the Earth; modern astrologers still use Earth-centred charts, as if Copernicus had never existed. That’s only the start of the scientific problems.

There's also this problem: because of precession, which is the rotation of the Earth's axial tilt, the astrological signs are off by about a month, or an entire sign. To expand on this: a "sun sign" is what's commonly known as your birth sign. For example, mine is Aquarius (though just barely; if I'd been born a day later I'd be a Pisces). Astrologically speaking, that means that the sun was crossing the 30 degree (that is, 360º / 12) section of sky associated with Aquarius. Aquarius is also a real constellation, but because of precession, the sun isn't actually in the constellation of Aquarius during Aquarius. I don't know if that means it's actually in Pisces or in Capricorn, and honestly I can't be arsed to look it up because it's irrelevant.

Full disclosure: I've been drinking, so I hope that was clear. My point is that these things were defined so long ago that they've disconnected from the astronomical world -- and that's not even getting into the more-than-dubious claim that your personality is defined by what house your Sun is in.

You forecast by combining meanings with planetary movements. Say Saturn, planet of restrictions, is about to transit the First House of self – your life will contract! You’re going to get more responsibilities than usual. Or maybe you’ll be denied the chance to take on more responsibilities. Or maybe a cold, critical person will come into your life. But anyway, it’s a good time to go on a diet.

Astrology is one big word association game.

But here's the thing: such games have their place.

I know that's going to sound odd coming from me, or even from Drunk Me. But bear with me and maybe I can make myself more clear.

Association is what we do, as humans and especially as writers. Whether "everything is a symbol of something else" or not, by finding these connections, we exercise our minds, and possibly find new meanings and new ways of looking at things. There is a danger, of course, of falling down a rabbit-hole, of believing our own thoughts even when it's unwarranted, but with the proper preparation, the process can open up new avenues of thought, maybe even stop us from thinking around in circles.

Metaphor is a tool. It's a bad idea to mistake it for reality, but approached with the right mindset, it can illuminate reality.

We charged A$50 an hour, a significant sum at the time, and I wanted to offer value. No fishing for clues from me – I printed a horoscope or laid the cards and started interpreting immediately, intending to dazzle the customer with my insights.

Half the time, though, I couldn’t get a word in. It turned out what most people want is the chance to unload for an hour.

And it's probably about as useful as psychological sessions.

I couldn’t say, “because my third eye is open”, so I reflected for a moment. Then it hit me. “You sounded happier when you said ‘photography’,” I said. My psychic teacher was right – the signals we pick up before conscious awareness kicks in can be accurate and valuable.

Here's the thing with so-called "psychics" and some of their readings: yes, the methods they use are bullshit; yes, some of them are pure scam artists. But there's a real art to reading people. And most folks are so wrapped up in their own problems that they miss the small signals from others; a "psychic" is really getting paid to interpret body language, microexpressions, and other subtle cues that even friends sometimes miss. I'm not saying people should consult psychics for advice and then follow it blindly, but if it helps you come at a problem from a new angle, hey, it just might be helpful.

Hell, sometimes all it takes to resolve a problem is to articulate it well enough to describe it to a stranger. The stranger doesn't even need to say a word. Maybe psychics should stick to being passive listeners.

On the other hand, separating the utter frauds from those who truly want to help is very difficult.

To conclude here, I was curious enough, after re-reading the article above, to see what astrologers might have to say about our current pandemic crisis. The result was... I want to say enlightening, but it's hard to be enlightened when your head has a dent from banging it on the desk.

https://astrologyking.com/coronavirus-pandemic/

I'm not quoting it here. See for yourself. But maybe put a pillow on your keyboard first, so it doesn't hurt as bad when you slam your forehead into it.

As always, I'm not going to begrudge anyone what psychological or spiritual comfort they might need in these troubled times. In the end, though, it's not religion or psychology or Tarot or astrology or prayer or ritual circles or meditation or anything mystical that's going to fix our current situation.

It'll be science.

But that won't stop people from insisting it was sky fairies, unicorns, or Saturn coming out of retrograde.

4 Like • Comment

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