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:

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.

1 Like

Adherennium Plotting something

a day 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.

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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.

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').

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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

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

August 23, 2019 at 12:31am

#964576

Consciousness

So, yeah, I took some time off to indulge vices (other vices than posting stuff here, anyway), but I'm back home now. And I have a difficult article to present today.

http://nautil.us/issue/47/consciousness/is-matter-conscious

Is Matter Conscious?
Why the central problem in neuroscience is mirrored in physics.

I'm of the considered opinion that most headline questions are answered "No."

What is physical matter in and of itself, behind the mathematical structure described by physics?

Math, too, has its limitations. Try as we might, for example, we can't precisely describe the shape of a cloud, or a tree, using math. Attempts to do so often result in something resembling a cloud or a tree, but not a particular cloud or tree.

Modern science has given us good reason to believe that our consciousness is rooted in the physics and chemistry of the brain, as opposed to anything immaterial or transcendental. In order to get a conscious system, all we need is physical matter. Put it together in the right way, as in the brain, and consciousness will appear.

Fair enough. But, again, try as we might, even if we could put everything together in the right place to recreate a human (or other animal), such an object would lack life. Life is still an elusive mystery in many ways, but we know it has to come from life, in an unbroken chain all the way back to the first life - whatever that was. Despite our certainty of our identities as individuals - as well as others' identities as individuals - there's a real, non-metaphysical, connection through time and space to all other life as we know it. That's why it will be so important to discover extraterrestrial life: is it connected in some way to ours, or was there a parallel to our own evolution?

My point is that "put it together in the right way" isn't as easy to do as it is to say.

If we were somehow granted knowledge of every physical detail and pattern in the universe, we would not expect these problems to persist. They would dissolve in the same way the problem of heritability dissolved upon the discovery of the physical details of DNA. But the hard problem of consciousness would seem to persist even given knowledge of every imaginable kind of physical detail.

This sort of thing always bugs me. Is it even possible to have knowledge of every physical detail and pattern in the universe? Magic 8-Ball says "No" - if only due to Heisenberg's Uncertainty Principle. And also because to fully model the universe, you'd need something at least as complex as the universe, which you can't have, because of the definition of "universe." Even if some of the wildly speculative alternative-universe hypotheses are true, you'd then have to model those, and the problem persists.

We take it for granted, however, that physics can in principle tell us everything there is to know about the nature of physical matter.

For previously unknown values of "we," anyway. Because I, and a whole lot of people who are smarter and more knowledgeable than I am, don't accept this premise.

There is already a tradition for connecting problems in physics with the problem of consciousness, namely in the area of quantum theories of consciousness.

Which collides with my personal tradition of dismissing, out of hand, any article that discusses "quantum theories of consciousness."

The article continues by going way over my head with the philosophy. I kind of get the impression that the author is trying very, very hard not to use words like "god" or "deity," instead mincing around the theological implications. But I could be projecting my own bias; I don't know.

Mind you, I'm not saying the article is wrong, or right, or doesn't have decent points. It just seems to me to be another approach to the "god of the gaps" issue - that is, there are always gaps in our knowledge, no matter how much we learn about the physical universe, and at that point you might as well ascribe the unknowns to an entity of great knowledge and/or power. Science cannot disprove the idea; again, the gaps are always there, and you can't prove a negative. It just seems like a cop-out.

Then again, I've thought that maybe the entire universe could be, itself, conscious; I mean, look at a map of the universe

and compare it to a neural network of the human brain

.

But that doesn't mean the two have any similarities beyond the superficial.

1 Like • Comment

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