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 15, 2018 at 12:58am

#945587

R.E.M. Minders

I've written here before about my lovely experiences with sleep paralysis and its accompanying nightmares. But those are relatively rare; like most people, the majority of my somnolent experiences are just dreams.

When you think about it, dreams are weird. I don't just mean that the dreams themselves are weird, but also that the concept is strange.

As I understand it, we don't really know why we dream. Or sleep, for that matter. What evolutionary advantage is there to be unconscious for a third of the day? Or even more, if you happen to be a feline. The benefits to survival must outweigh the risks of being caught napping by a predator or a tax collector.

Sleep does seem to play a role in processing and organizing experiences, turning them into memories, and it's probable that dreams are key to that - a time for the brain to integrate the latest inputs into one's consciousness, or to discard the unimportant bits. Like a defrag routine for a storage medium, maybe, though it's not really accurate to compare organic processes with electronic ones.

Those of a more mystical bent might interpret dreams as messages. Perhaps it's a time for the subconscious to alert us to something important. I know that there have been many times I've gone to sleep after banging my head against a problem, for instance a spreadsheet solution that has eluded me, and woken up with the key to working it out. This is, I suppose, why sometimes you're urged to "sleep on it" when something's bugging you. But I've also woken up with ideas that turn out to be dead ends - though even failures are useful if you can learn from them.

I have to wonder, though, what role lucid dreaming might play in this kind of explanation for dreams, though. Lucid dreams, as I understand them, combine the knowledge that one is dreaming with the ability to control the dream. And just typing that sentence is enough to send me down the rabbit hole of a discussion about free will, which is not something I'm prepared to go into here just yet. For the sake of argument, let's say that one is in control when one believesone is in control.

For myself, I feel like I have some degree of control most of the time once I realize I'm in a dream. Usually when that happens I verify the hypothesis that I'm dreaming by willing myself to levitate. Since I can't levitate in real life, if I find myself starting to leave the floor or ground or whatever, I know I'm dreaming (of course, there are flying dreams that I don't initiate, but that takes the "will" out of the situation).

As for what I do then - well, let's just say this blog doesn't have a high enough content rating, and leave it at that.

I think it was Dave Barry who said something to the effect of "The holodeck [from Star Trek: The Next Generation] will be humanity's last invention." In their purest form, lucid dreams are like being in a holodeck simulation. Not only do I see and hear, but I have also engaged my other senses. The surroundings seem real. If you have control over not only yourself but your realistic surroundings, why would you ever go back to a world where random crap can happen to you, you have bills to pay, and the car needs new tires? I suppose some would, but not me. I'm always disappointed when I wake up from a lucid dream. Not that my life is horrible - far from it - but the dreams are just more interesting.

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