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 8 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 10 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.

July 6, 2020 at 12:14am

#987334

The Edge of the World

Oh boy, here we go.

First, the good news: Time for another Win A Merit Badge Mini-Contest! Details below.

And now, the news!

PROMPT July 6th

Is there such a thing as “unbiased reporting?” (Consider not just journalism, but storytelling - is it possible to tell a story without bias?)

The one thing that bothers me the most about "unbiased reporting" is when, in an effort to show "both sides" of an issue, a reporter has to call in people who believe the opposite of what the story's about.

In an effort to avoid too much controversy here, I'm going to use an extreme -- and pulled straight out of my ass -- example.

First, the part that hasn't been in my colon:

GPS uses satellites to fix the three-dimensional position of a receiver. That is to say, with enough satellites (four, if I recall correctly), you can pinpoint a receiver's location on or above the surface of the planet. This generally translates to latitude, longitude and elevation, relative to the equator, the Prime Meridian, and the official Mean Sea Level datum, respectively.

The actual inner workings are irrelevant (and beyond me), but basically, this requires the following at a minimum:

the existence of satellites in orbit;

the existence of a receiver;

a (nearly) spherical Earth, with reference points for the coordinates I mentioned above;

extraordinarily accurate timing, provided by atomic clocks in the satellites;

the latter of which is adjusted for relativistic time-dilation effects.

Yes, the satellites are moving fast enough relative to the surface of the planet that relativistic corrections have to be made. Otherwise, instead of accuracy in the general vicinity of a meter or so, your reported position could be off by on the order of a kilometer - not nearly enough for the things we use GPS for, like finding the nearest Starbuck's or searching for buried treasure.

And now the rectally retrieved part:

Imagine a news story about, I dunno, hypothetically, replacing the GPS satellites with updated ones.

A reporter relating this story might have to present "both sides." So she writes about the creation of the new satellites, the software updates necessary in the receiver units, and so forth.

And then, for "balance," she has to find a flat-earther so they can explain how the entire GPS thing is a massive hoax; there can't be satellites because the Earth is flat; and it's all a conspiracy by Big Positioning or something. As a bonus, the moron might even throw in something about relativity being "obviously" wrong, and that time is invariant with speed and position.

Writing your news article like that has the advantage of appearing unbiased and presenting different sides to the issue. More importantly to the publisher, it generates controversy, causing fact-free idiots to share it with each other and crow about how they've exposed something They don't want the rest of us to know; and causing non-stupid people to share it with each other to point and laugh at the idiots. This drives page clicks and increases advertising dollars.

Mission accomplished.

There's only one glaring problem: flat-earthers are completely and utterly wrong, and giving them a platform in the same piece that attempts to explain how physics actually works does science a Dark Ages level disservice.

End of anus extraction. But I'm sure you can think of other issues, real ones, where the facts are presented alongside amateur and ignorant speculation

The only appropriate thing to do with fact-deniers is to completely ignore them, and yet... we don't. We give them false equivalency. When a scientist is interviewed and says one thing, and another scientist is interviewed and says another thing, well, then, yes, by all means, present both sides -- but even there we should be aware that people without an appropriate education are in no position to draw their own conclusion as to which, if either, is right. But when a scientist takes one position, and John Bumpkin with no background in the subject matter at all takes the other, just fucking leave Bumpkin right out of your story. He's irrelevant.

Now, look, it's true that flat-earthers exist. Whether any individual one is trolling or really believes their ignorant bullshit is above my pay grade, but even if you present their "side" of things with all kind of disclaimers about "actually, science has established to near-100% certainty that we live on a globe," you're wasting your and your readers' time. But most of the time they don't even do that, leaving it to the reader to do the fact-checking.

Which they don't, which, again, leads to the ignorant belief gaining the appearance of being on the same level as the science.

So by attempting to show that they don't have bias, they create a new bias: one that is taking us down a path to a fact-free world. Has taken us. Hell, the city lights are looming on the horizon, and we just passed a sign that reads "Welcome To Postfactopolis, population 7 billion."

I'll just say one more thing about bias in reporting, though, and this is relevant to fiction-writing as well. Okay, two more things. Well, a few more things, but they're all related.

It is impossible to present all the facts.

Some things are known to a greater degree of certainty than others.

Even if you report nothing but facts, you're going to demonstrate bias by choosing which facts to report.

Presenting too many facts leads to boredom and tl;dr.

Conclusion: No, there is no such thing as unbiased reporting... and logically, there cannot be.

So, today's mini-contest is this:

Give me an example of false-both-sidedness. This can be real, or, in the spirit of the "fiction writing" part of the prompt and my fictional GPS explanation above, you can make it up.

The response I like best will earn its author a Merit Badge that I feel is appropriate, tomorrow. Deadline, as always, is midnight WDC time.

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