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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

October 24, 2022 at 12:03am

#1039622

Writer's Bl

Yesterday, Sumojo

said, "Never mind the language lesson, I’m more excited about your new car! Can we see a picture of said vehicle? Model, year etc. Oh, don’t forget the colour."

Here you go.

I couldn't think of another introduction to today's article.

The 5,000-year history of writer’s block

Pretty sure 5000 years ago, "writer's block" was the literal block of stone they chiseled into.

Ann Patchett, who has written eight novels and five books of nonfiction, says that when faced with writer’s block, sometimes it seems that the muse has “gone out back for a smoke.”

You know, I get the whole "muse" thing. It's a useful metaphor. But that's all it is: a metaphor. Like that cartoon devil on one shoulder telling you it's okay to eat pizza, and the cartoon angel urging you to have some nice warm kale instead. In reality, that's all you.

It doesn’t matter whether you’re an award-winning novelist or a high schooler tasked with writing an essay for English class: The fear and frustration of writing doesn’t discriminate.

Yeah, it kind of does. Or else some people wouldn't be more susceptible to block than others.

My most recent book, “A Writing Studies Primer,” includes a chapter on gods, goddesses and patron saints of writing.

Oh, look, another book ad. Well, considering that the only free articles you can find these days are ads, I'll take it.

When conducting research, I was struck by how writers have consistently sought divine inspiration and intercession.

Still all you.

The first writing system, cuneiform, arose in Sumer around 3200 BC to keep track of wheat, transactions, real estate and recipes.

Especially recipes for beer.

Originally the Sumerian goddess of grain, Nisaba became associated with writing. She was depicted holding a gold stylus and clay tablet.

See? Grain is the basis for beer.

Writing was all about communicating with the gods, and the Greeks and Romans continued this tradition. They turned to the nine daughters of Zeus and Mnemosyne, known collectively as the Muses.

There is, in my mind, a big leap from using writing to keep track of harvests, and to communicate with the gods. Perhaps the author makes a better connection in the book.

Gods and other legendary figures of writing are not limited to Western civilization.

Gee. You think?

In India, writers still invoke the elephant-headed Hindu god Ganesha before putting ink to paper. Known as a remover of obstacles, Ganesha can be especially meaningful for those struggling with writer’s block.

Even Westerners can use elephants to overcome writing obstacles. In my case, pink ones.

In Mesoamerica, Mayan culture looked to Itzamná as the deity who provided the pillars of civilization: writing, calendars, medicine and worship rituals.

While I'd heard of Itzamná, some of those associations were new to me. Still learning.

In Christianity, patron saints are exemplars or martyrs who serve as role models and heavenly advocates. Various groups – professions, people with a certain illness and even entire nations – will adopt a patron saint.

"We're monotheistic." "But what about Mary and the saints?" "Mon. O. The. Ist. Ic."

St. Brigid of Ireland, who lived from 451 to 525, is the patron saint of printing presses and poets.

Yeah, tell me she didn't inherit associations from her namesake deity

Some writers may think supernatural figures seem a bit too far removed from the physical world. Fear not – there are magical objects that they can touch for inspiration and help, such as talismans. Derived from the ancient Greek word telein, which means to “fulfill,” it was an object that – like an amulet – protected the bearer and facilitated good fortune.

I have a magical object I can touch for inspiration and help. it's called a computer. Hey, any sufficiently advanced technology, and all that.

Today, you can buy talismans drawn on ancient Celtic symbols that purport to help with the writing process.

They certainly help the sellers with the eating process.

One vendor promises “natural inspiration and assist in all of your writing endeavors.”

But not, apparently, their own writing endeavors. Especially those requiring consistent parts of speech.

Another supplier, Magickal Needs, advertises a similar product that supposedly helps “one find the right word at the most opportune moment.”

I have one of those too. It's called a thesaurus. ...okay, confession: I don't actually own a thesaurus (or if I do, it's buried under a stack of other books). I use online ones like normal people.

Others turn to crystals. A writer’s block crystals gift set available through Etsy offers agate, carnelian, tiger eye, citrine, amethyst and clear quartz crystals to help those struggling to formulate sentences.

I know I get snarky about these things, and maybe a touch cynical. But when it comes to things like writing, it's often the case that if you think a rock will help, then the rock will help. I mostly just object to people taking advantage of the gullible.

To me, it’s no mystery why writers have sought divine intervention for 5,000 years.

Because seeking divine intervention is kind of what humans do. Whether it has any basis in reality or not can be argued, but what can't be argued is that religion predated writing.

Sure, tallying counts of sheep or bushels of grain might seem like rote work. Yet early in the development of writing systems, the physical act of writing was exceedingly difficult – and one of the reasons schoolchildren prayed for help with their handwriting. Later, the act of creation – coming up with ideas, communicating them clearly and engaging readers – could make writing feel like a herculean task.

I see what you did there with "herculean."

I imagine that, like any new technology, the whole "writing" thing must have seemed like high magic, same as when wielders of bronze weapons encountered iron-wielders. And there was probably pushback against it from that era's equivalent of conservatives: "By the gods, this newfangled 'writing' thing is going to destroy our ability to memorize things. And probably society."

Sure, that society did eventually collapse, though that was probably not because of writing. But I wouldn't be a bit surprised if the last person to leave the dying city of Sumer muttered, "I told you writing would lead to this."

The romantic image of the writer in the garret doesn’t do justice to the tedious reality of churning out words, one after another.

I want a garret.

No matter how accomplished a writer, he or she will inevitably struggle with writer’s block. Pulitzer Prize−winning author John McPhee, who began contributing to The New Yorker in 1963, details his writer’s block in a 2013 article: “Block. It puts some writers down for months. It puts some writers down for life.”

To be fair, if I had to write for The New Yorker, I'd have writer's block too. At least until I touched the talisman; that is, the contract promising me money.

I’ve even wrestled with this article, writing and rewriting it in my head a dozen times before actually typing the first word.

Every writer is different, but that's what I do: mull things over before writing. But at some point (usually about five minutes before the deadline) I start typing, and when I do that, generally the words just happen. Sometimes even in a logical progression.

If you've made it this far, congratulations—and yes, I was absolutely joking about the Lamborghini. No, it's another Subaru, not a midlife crisis car.

Speaking of writing...

One-Sentence Movie Review: Black Adam

Everybody involved in this movie, from the executive producer down to the second assistant to the 3rd unit studio janitor (except for Dwayne Johnson, who is always awesome) can at this point restore honor to their houses only by performing seppuku.

Rating: 0.5/5

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