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!

a day 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').

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.

May 21, 2019 at 12:42am

#959365

The Derivative of Opinion

Write about an opinion you’ve had that has changed over the years.

There's been a few years, so there are quite a few opinions that fit the bill here.

One thing I've always tried to avoid in this blog is political controversy. It's hard to avoid completely, though, since everyone seems to have an opinion about something - and I'm no exception. What I don't want to do is get into arguments about issues like climate change, abortion, the death penalty, or military spending. We here at Writing.com tend to be more civil about these things than elsewhere on the internet; still, some of my friends here are on the opposite side of the political spectrum from me, and I don't want to lose friends over something over which we, individually, have little control.

However, my opinions, like I think most peoples', are either a) trivial or b) political. I mean, I could talk about how anchovies on pizza are an abomination against everything that is right and true in this land, and you could argue otherwise, but in the end it comes down to a matter of personal preference - and if we ordered a pizza together, we could settle the dispute amicably by getting anchovies on one half. So if I made this post about how I used to actually like anchovies on pizza, that would make it trivial.

I didn't, of course. Anchovies are, always have been, and always will be, an abomination.

Aversion to political discussion aside, I'm sure regular readers have figured out where I stand on most things. Some of you, like I said, feel differently. That's okay; if people didn't feel differently about things, there wouldn't be a reason to avoid the topics, would there?

My father always used to tell me that people shift to the conservative end of the spectrum as they age. I suppose, in a way, he's right, but it's not necessarily a political conservativism. You get to some age, and I think you just start liking things to be the way they are, or remember the "old days" with some fondness and want to go back there.

I'm not at that age, yet.

I used to fancy myself a libertarian. Mostly, this was the influence of Robert A. Heinlein, an author whose works I devoured when I was younger. The ideas of ownership and self-determination - well, those were appealing to my younger self. The individual, according to this philosophy, is at his or her best when left as unfettered by regulation as possible. One should always, the philosophy goes, do and think for oneself.

Then I read Ayn Rand and, thinking for myself, decided that libertarianism was bullshit.

Still, echoes remain. I maintain a belief in individual freedom of thought and conscience. When I hear about suppression of, say, religious belief, I get angry or outraged. There are, of course, trade-offs, as there are for everything - if your religion requires human sacrifice, for example, that interferes with others' rights. But choosing to worship your conception of God or Spirit or Whatever, harming no one in the process - that right should be sacrosanct. So to speak.

Or, like me, not worshipping.

But one of the core tenets of libertarianism is that of ownership. If you have ownership of a thing, that includes the right to, for example, destroy it. But, I eventually realized, there's a problem with this: if I put a toxic waste dump on my land, the effluent could seep into the water table and deprive others of a resource needed for life. Again, trade-offs are necessary. Exactly where and how we draw the lines on the trade-offs, well, that's what the arguments that I prefer to avoid are all about.

For instance, if you live in certain watersheds in the US (as a civil engineer, I see the world in terms of watersheds), it is illegal to capture rain in a barrel. That rain, left uncaptured, supplies rivers and aquifers to which other individuals have a claim. Now, we can argue all day about whether this is "right" or not, but that's not the point; the point is that we, collectively, have instituted rules about runoff because it's a necessary resource. Without those and similar rules, which are anathema to libertarians, things have a tendency to go very, very wrong. (They go wrong sometimes anyway, but that's not an excuse to throw out the entire concept of "rules.")

I've found that no one wants government interference, unless it suits their own ideas about what the government should interfere in. People who want "small government" often demand bans on alcohol or weed, for example.

And that's about as political as I want to get, and I wouldn't have gone even that far if it weren't for today's prompt.

Tl;dr: I used to be a libertarian, and then I grew up.

(To explain the title of this entry: in calculus, taking the derivative of a function is a way to determine its rate of change at any or all points.)

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