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 24 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

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').

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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.

December 27, 2019 at 12:39am

#972088

Eschew Obfuscation

I know some people have issues with Inc as a source, but the content of this particular article intrigued me.

https://www.inc.com/glenn-leibowitz/the-single-reason-why-people-cant-write-acco...

The Single Reason Why People Can't Write, According to a Harvard Psychologist
This common affliction is behind so much unclear and confusing writing in the world today.

My biggest gripe is that the headline is misleading. It's not about why people can't write; it's about why people sometimes can't understand what we write.

These are questions Harvard psychologist Steven Pinker asks in his book, The Sense of Style: The Thinking Person's Guide to Writing in the 21st Century. They're questions I've often encountered --and attempted to tackle-- throughout my career as a business writer and editor.

My second biggest gripe is that this author seems to be shilling for Pinker's book. I have my own issues with Steven Pinker, but again, the points raised here can be valid.

Whenever I see writing that is loaded with jargon, clichés, technical terms, and abbreviations, two questions come immediately to mind. First, what is the writer trying to say, exactly? And second, how can the writer convey her ideas more clearly, without having to lean on language that confuses the reader?

Could it be that you're not the intended audience?

For Pinker, the root cause of so much bad writing is what he calls "the Curse of Knowledge", which he defines as "a difficulty in imagining what it is like for someone else not to know something that you know. The curse of knowledge is the single best explanation I know of why good people write bad prose."

Well... sometimes. Other times, it's because of a tenuous grasp of language and structure.

"How can we lift the curse of knowledge?" asks Pinker. "A considerate writer will...cultivate the habit of adding a few words of explanation to common technical terms, as in 'Arabidopsis, a flowering mustard plant,' rather than the bare 'Arabidopsis.' It's not just an act of magnanimity: A writer who explains technical terms can multiply her readership a thousandfold at the cost of a handful of characters, the literary equivalent of picking up hundred-dollar bills on the sidewalk."

Fair enough. Language, as I see it, can be like a compression algorithm. To follow the advice given here, I'll explain that a compression algorithm is what lets you turn what could be a huge file on the computer into a much smaller one, saving space. The tradeoff is that you then need to decompress it, unpack it, expand it, whatever, in order to be useful in an application or to read a text file.

To further follow the advice in the article, I'll give an example, albeit a simple one. To describe the position of a car or an airplane, we can use latitude, longitude, and elevation - as well as a time coordinate. To describe the position of a ball in a room, we can use three different distances along the x, y, and z axes (also in addition to time). We can also describe its velocity, or change in position with respect to time. The word "velocity" here is an expansion of "change in position with respect to time;" we say "velocity" to compress the longer phrase. Similarly, we can use "acceleration" to pack in "change in velocity with respect to time," or "change in change in position with respect to time with respect to time," which gets really unwieldy, which is why we say "acceleration" instead. Most people know what that means on an intuitive level, because we experience it on occasion (technically, we experience it constantly because of gravity, but that's irrelevant to the discussion).

Like I said, that's a simple example, even if the actual description of the relationship between position, velocity, and acceleration involves calculus. Calculus was invented, in part, as a compression algorithm.

But we take shortcuts like that all the time, as writers. A reader, encountering a new word, can choose to ignore it; or they can choose to glean some meaning from context; or (remarkably easy nowadays if reading on a screen) they can look up its definition. The definition unpacks it. But if you know the word to begin with, you already have the code internalized.

As easy as it is to look words up if they're on a screen, doing so repeatedly can be frustrating. So, yeah. As someone once said, "eschew obfuscation

."

See, that's funny because those are not common words and are thus themselves obfuscatory, and... aw, hell, once you explain a joke, it's no longer a joke.

Jokes are also compression algorithms. If you don't have the code to unpack them, you won't get the joke.

As Richard Feynman, the Nobel prize-winning physicist, once wrote, "If you ever hear yourself saying, 'I think I understand this,' that means you don't."

And see, right there? I know who Richard Feynman was. But if you didn't, the extraordinarily brief nutshell biography provided by this writer at least says, "Oh, Feynman wasn't just some schlub; he was a supergenius." Feynman was good at explaining physics to the masses; further, the thing he's most known for in nerd circles, the Feynman diagram, compresses complex and intricate quantum mechanics equations into simple scribbles.

Not that you (or I) are expected to understand these things, but the point is, I think, on some level, he knew that any language - English, or mathematics - was something that occasionally needed to be expanded in order to be understood, and compressed in order to facilitate communication.

I know I sometimes fall into the trap of assuming that because I know something, everyone else must know it too. But the polar opposite of that is assuming that your audience is ignorant (not necessarily stupid, just ignorant). Good writing, I think, strikes the perfect middle ground between getting your point across succinctly, and overexplaining everything.

There are also people who seem to just write to show off how brilliant they are. A lot of postmodern deconstructionist screeds do that, peppering the essays with words you can only learn after at least four years of liberal arts studies. As an example, I refer you to the following; more criticism of the following quote can be found here

In Terrorist Assemblages I propose a rapproachment of Foucauldian biopolitics and Achille Mbembe’s critique of it through what I call a ‘bio-necro collaboration’, one that conceptually acknowledges biopower’s direct activity to death, while remaining bound to the optimalization of life, and necropolitics’ nonchalance towards death even as it seeks out killing as a primary aim. I allege that it is precisely within the interstices of life and death that we find the differences between queer subjects who are being folded (back) into life and the racialized queernesses that emerge through the naming of populations, thus fueling the oscillation between the disciplining of subjects and control of populations. The result of the successes of queer incorporation into the domains of consumer markets and social recognition in the post-civil rights, late twentieth-century era, these various entries by queers into the biopolitics optimalization of life mark a shift, as homosexual bodies have been historically understood as endlessly cathected to death, from being figures of death (i.e., the AIDS pandemic) to becoming tied to ideas of life and productivity (i.e., gay marriage and reproductive kinship).

I don't know about you, but my brain stops about halfway through the first sentence and doesn't start up again. The author may, indeed, have good points to make, but any such are obscured by jargon.

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