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.

July 4, 2019 at 12:52am

#962012

Unleashed

Write about independence.

There are so many different angles to take this prompt, I’m eager to see your creativity!

Independence is a myth.

The word "myth" gets tossed around a lot, and it's one of those words that has different meanings, depending on context. It could mean a falsehood, or a partial falsehood, like the pervasive idea that glass is a slow-moving liquid (it's not; its only resemblance to a liquid lies in its noncrystalline, or amorphous, atomic structure; now you know). Or it could mean a foundational story, true or not, like the tale of Romulus and Remus and the she-wolf, or George Washington chopping down the cherry tree.

By "independence is a myth," I mean the latter.

Today is the Fourth of July, Independence Day here in the US, some 243 years since we told King George III to kiss our fat white asses. Said fat white asses proceeded to kick England's fat white asses and then continued to enslave and/or genocide a lot of non-white asses. So right there, it wasn't exactly "independence" for everyone.

To do this, we banded together in a union of, nominally, 13 former colonies or "states," united, each depending on the other 12 to help them keep their shit together. Interdependence, right there. Join, or Die.

Nowadays, of course, the UK is one of our closest allies, so I guess everything worked out in the end, to the extent that anything actually ends. We get to make fun of their silly accents, and they get to make fun of our fat multicolored asses. Freedom.

I suppose there's something to be said for self-determination, but really, how fine a sieve can we use for that? I mean, right now, a good half of the population of the country thinks our government is run by the wrong people. After the next elections, the other half might think our government is run by the wrong people. In theory, we all have a say in this; in practice, we have about as much control as a Brit does over who sips tea in Buckingham Palace.

Get any two people together, and you're going to have trouble deciding on pizza toppings, let alone how a country should be run. Get 300 million people together, and that problem is exponentially intractable.

So, we compromise, we cooperate, we argue, and we stare at a flag while saying words like "indivisible" and "liberty" and "justice," and pretend that these concepts have meaning.

Like "independence."

Funny thing is, though, if you pretend a concept has meaning for long enough, and write enough words about it, and argue for or against it, and even go to war over it... well, then it does have meaning. Could be a different meaning for everyone, but it's a meaning.

It raises it from idea to concept to story to myth - not falsehood, but foundation. Like how the story of George Washington and the cherry tree speaks to the core value of honesty, it gives us something to hold on to, to strive for, even if we can never really get there.

And we can't. We're all dependent, to some degree or another. No one is an island - and even if they were, an island is still dependent on a relatively constant sea level, without which it ceases to exist. (Yes, I just made a snide comment about climate change.)

But hey, we don't have hereditary royalty telling us what to do, so I suppose that's something. Of course, neither does England these days. Well, royalty, sure, but they don't really command anyone who isn't a Corgi.

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