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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 7 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

about 12 hours ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

about 14 hours ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

a day ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

a day ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

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.

4 Like • Comment • View 2 comments

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