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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 17 hours 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

about 19 hours ago
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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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.

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