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.

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.

November 11, 2019 at 12:52am

#969408

Spare Some Change?

PROMPT November 11th

Today, your prompt is one word: Transformation.

Everything changes and nothing stands still. -Heraclitus

The thing about those ancient Greek philosophers is that they lacked the 2000 or so years of history that we've experienced since they wrote their stuff. You might be tempted to say, "But Waltz, that quote survived all this time." Well, yes and no. No, because the original statement was written in Ancient Greek, and is thus subject to some of its meaning being lost in translation. It's not, as a physicist might say, time-invariant. Its meaning is also informed by one's cultural milieu. So even the quote about change changes over time.

And yet, there's something inviolably true about it.

Heraclitus also apparently was the originator of the nearly koan-like quote, "You could not step twice into the same river." (also on the wiki page linked above). I heard that quote a long time ago, without attribution. At the time, I was actively working in hydrology (the study of rivers and other water flow), so, being the technical-minded person that I was and am, I mentally changed it to "You can't step into the same river even once." Because between the time your foot touches the surface of the river and the time it touches the bed, the river has, in some sense, already changed - water has flowed out, evaporated, and/or seeped into the soil beneath; some has also, most likely, flowed or rained in.

Which doesn't change the basic fact, or the metaphor: that change is constant, inevitable, and defines existence. We may not always see the change - after all, rivers don't usually change in a very visible, macro kind of way in the few microseconds it takes to set foot in one - but it happens, anyway.

I think a most people feel a tension between the desire for change (preferably in a direction that benefits us) and the desire for things to stay the same, in defiance of all evidence and physical laws. That goes back to the "wish" entry a couple of days ago, doesn't it? What I mean is, maybe you want to win a major jackpot in the lottery, but what does winning the lottery actually look like? Ideally, you go on with your life as it was, only with more money, right? But you add that kind of money to a life, and it, necessarily, changes.

Life is change. Or transformation; I'm using a synonym here, and synonyms are an example of transformation. I'd even go so far as to assert that reality is transformation. Periodically, we humans (including another famous ancient Greek philosopher, Plato) come up with ideas about things that are "eternal" and "unchanging," but to me, that's how I know that something isn't real: it's conceptualized as nontransforming.

What is real is change, and the most pernicious lie ever foisted upon a gullible humanity is that what we see around us is illusion while "reality" is something unchanging. As I've noted before, this alters the definitions of "real" and "illusion" to the point where both words become meaningless.

There is, however, as far as we know (though our understanding could change) one thing that is constant, and that is the speed of light. Interestingly enough, this seems to be a true constant, for both light and matter, which are the same thing but transformed into one or the other. Every particle, every thing in the universe is always moving at exactly the speed of light - through spacetime. If it's not moving at the speed of light, that is, the constant c, through space, it's also moving through time.

What this means, if it means anything at all, eludes me.

I've mentioned in previous posts the predicted heat death of the universe. Billions or trillions of years from now (can't be arsed to look it up at the moment, and even that theory is subject to change as new data is interpreted), all thermodynamic processes stop, and we won't be around to see it because we are thermodynamic processes. But even then, change will still be occurring, in the form of fluctuations as a result of basic quantum uncertainty.

It is possible, and therefore, given the amount of "time" (which itself loses its meaning), inevitable, that such a fluctuation will bud another universe, like or unlike our own.

And so change continues. Eternally.

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