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:

Write_Mikey_Write!

about 11 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 13 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.

June 14, 2020 at 12:16am

#985608

Oh, Pinions

"The problem with internet quotes is that you can't always depend on their accuracy" ~Abraham Lincoln

https://qz.com/1513176/john-stuart-mills-philosophy-shows-arguing-online-is-futi...

150 years ago, a philosopher showed why it’s pointless to start arguments on the internet

Wildly inaccurate facts and spurious arguments are unavoidable features of social media. Yet no matter how infuriatingly wrong someone is, or just how much counter-evidence you have at your disposal, starting arguments on the internet rarely gets anyone to change their mind. Nearly a century-and-a-half ago, British philosopher John Stuart Mill explained, in a few clear sentences, why certain arguments simply won’t go anywhere.

So despite the sensationalist headline, a philosopher made a point about arguments in general, and, quelle surprise, the point applies to arguments on a specific medium.

Philosophers, of course, are often wrong, but it's hard to argue with John Stuart Mill. Because he's dead.

Mill highlights the often overlooked reality that many opinions aren’t based on facts at all, but feelings. And so, contradictory points of information don’t shift emotionally rooted arguments, but only cause people to dig deeper into their emotions to hold onto those views.

Of course, MY opinions are entirely based on rationality and facts, while THEIRS are all emotionally based and WRONG.

And chartered psychologist Rob Yeung, whose book How to Stand Out emphasizes the effectiveness of emotions, rather than logic, in convincing others to agree with you, points to research showing that use of metaphors motivate people to make decisions.

1. What the neurological fuck is a "chartered psychologist?"
2. I've been saying that metaphors are powerful, but does anyone listen? No. They're brick walls.
3. "motivate" points to "use" and thus should be "motivates." Does no one edit anymore? Pay no attention to these editor-less blog entries.

There are often multiple ways of interpreting a single point of information and so, much though some people might like to think they’re right about everything, there are surprisingly few issues to which there’s an unequivocally correct opinion.

Except mine, of course.

Instead of seeking to convince others, we can be open to changing our own minds, and seek out information that contradicts our own steadfast point of view.

Now why would I go and do that? It might turn out that I'm wrong, which of course is impossible, and even if it were possible I could never admit it. So it must be that I've been brainwashed.

Maybe it’ll turn out that those who disagree with you actually have a solid grasp of the facts.

Unpossible.

There’s a slight possibility that, after all, you’re the one who’s wrong.

No, YOU are wrong, and probably a sheep-shagger too.

Okay, seriously, though, like most philosophy, this is not helping. Joking aside, yes, I could be wrong. But am I supposed to think that I might be wrong about issues like:

Science is the best tool we have for figuring things out, even if it's not perfect;

Racism is bad;

Vaccination saves lives;

Apollo 11 put two dudes on the actual Moon;

The Earth is roughly spherical

etc.?

I've run into arguments against all of these positions. Some of them, of course, come from trolls who are just trying to get a reaction, which I try not to provide. But some people actually believe the opposite of the above statments, and they're painfully, objectively, dangerously, wrong.

I also have opinions that, yes, could be wrong. Not only that, but I might have facts wrong. If so, I try to be open-minded when I get new facts. My problem in that regard is that I have a lousy memory and I tend to forget the stupid shit I used to believe. It's one reason I blog. But then I'd have to go back through old entries and cringe at old beliefs, and that's painful so I don't do it.

Still, I understand and respect that some people believe differently than me on, for example, the existence (or lack thereof) of a deity, or whether or not we're living in an advanced simulation (or that, effectively, these are the same belief). I admit that if a flying saucer landed near me and a little bald guy came out and asked me to take him to my leader, well, first of all, I'd ask if he could wait until we have a sane leader, but mostly, assuming I'm sober, I'd abandon my disbelief in sentient extraterrestrial life in a snap.

Unlike some people on the internet, I do seek out contradicting views, or, at the very least, I don't automatically dismiss them -- unless they're patently absurd, like the contradicting views to my bullet points above.

Still, nothing's worth the drama or hassle of arguing about it on the internet, or even, usually, in person. Just have a beer and relax already.

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