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!

about 5 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 7 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.

April 5, 2024 at 8:11am

#1067635

Plurality

This article, from Slate, has been, like a customer at the DMV, patiently waiting over four months to be called up. And it's 9 years older than that. But, despite the headline and date, it's not time-sensitive; grammar and spelling rules are timeless. Sort of.

How to Make Your Last Name Plural This Christmas Season

Nothing quells my Christmas cheer as quickly as a stray apostrophe. Every year they assault me.

Oh, you only notice them in December? Then you haven't been looking.

Usually it’s in the middle of an otherwise quaint moment: I am padding around my parents’ house, wearing pink slippers, sipping on some hot chocolate. Snow is falling outside the window, and Josh Groban’s Christmas CD is filling the downstairs with peace on earth and mercy mild. My mother is baking a pie. She’s about to ask if I want to lick the spatula (which, duh, I will).

Remarkable; 12-year-olds are writing articles these days, while other writers struggle to make money.

First, though, I find a stack of Christmas cards and begin to flip through them—pausing to marvel at how big so-and-so’s kids have gotten. And then I spot it: an apostrophe in a last name that isn’t supposed to be possessive.

This part, I absolutely relate to.

Gone is my Christmas cheer! All my glad tidings, replaced with fury.

Look. I get it. It pisses me off, too. But if that's all it takes to turn you into the Hulk, seek help. Maybe I overestimated her age, above.

Okay, okay, kidding. I recognize hyperbole when I see it. Usually.

This year I’d like to preempt the pluralization problems. It’s mid-November now, time to order Christmas cards again.

The holiday season starts earlier every year. It could be that, by waiting this long but not long enough to write about this article, I'm contributing to Christmas creep. But, given the prevalence of year-round Christmas stores in the US, I don't think my small contribution makes any difference.

I have created a brief guide to help you pluralize your last name. It is my humble attempt to preserve not only apostrophe protocol but also the dignity of the letter S.

You should visit the link just to see this chart. It's both informative and funny.

In summary, if you're writing in English, every surname adds either -s or -es to become plural, and an apostrophe doesn't belong anywhere near it, unless the name already has one like O'Brien. I have no idea how they do it in other languages.

Q: Why do people add apostrophes?
A: I have no idea.

Neither do they.

I have an alternative suggestion, though, since so many people misuse apostrophes, putting them in where they don't belong, or omitting them where they do belong:

Instead of "Happy Holidays from the Smiths," consider: "Happy Holidays from John Smith, Jane Jones, Braden Smith-Jones, Kaylee Jones-Smith, Fluffy, and Stinky. (Don't look at us like that. Braden was four when he named the dog.)"

It's not like you have to get the damn things engraved at a printer, these days, and get charged by the letter.

Besides, calling the whole household by one last name perpetuates the myth of the nuclear family, and enforces the hegemony of the patriarchy. We can't have that, now, can we?

I bypass the whole thing by remaining singular.

And not sending holiday cards.

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