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 3 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 5 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.

September 3, 2020 at 12:07am

#992214

Everyone's a Critic

Does getting a failing grade on an elementary school art project count?

PROMPT September 3rd

Describe a time when your work was criticized. How did you react?

Well, of course my work gets criticized on a regular basis. I'm a writer on a peer review website. We all are, and it's part of the deal here. In fact, sometimes I wish for more criticism. "I love this!" is great to hear, yes, tell me that, don't stop, oh! But the only part of me that grows from it is my... ego. (What were you expecting me to say?)

But oodles of reviewing articles abound on this site, including one in my own port, so there's no need to belabor the point. I'm going to talk about my previous life as a professional.

Don't worry. I'm not going to go into details. The details of civil engineering, especially on the small scale that I practiced it, are incredibly boring. (That's a civil engineering pun. Boring? Because we sometimes bore to install.. oh, never mind.) So I'll write in broad generalities.

I worked on the design of a subdivision. Roads, lots, sewer, water and -- the important part for this discussion -- storm drainage.

Used to be you could design a subdivision and put in a pond downstream that caught all the runoff and released it slowly. But then there came issues with wetlands disturbance, and that sort of thing was frowned upon. So instead of one big pond that disturbed a natural stream, I needed to put in dozens of little ones to catch the runoff before it got to the existing streams. Again, I'm oversimplifying here, but I can already sense you nodding off. Bear with me just a little longer.

Well, this method was not only time- and math- intensive, but it was new to me and, more importantly, it was new to the people whose job it was to review the plans for compliance with local, state, and federal regulations.

Said reviewers tore me a new one for overcomplicating things.

How did I react? Well, my first reaction -- a private one -- was anger. What the hell was I supposed to do? I had to put in stormwater management, and couldn't do it in the traditional way, so I basically invented a new paradigm. Don't give me too much credit for that, though; I found out later that other people were already doing similar stuff, and I didn't know about it because, honestly, civil engineering advances on that small a scale aren't exactly fast-paced, so we didn't keep up with the literature like, say, computer scientists are supposed to.

So yeah, anger -- but it was short lived and, like I said, private. After taking a few deep breaths, I arranged a meeting with the reviewers and we hashed it all out like the professionals we all were.

Sorry there's no big denouement here; like I said, the details are ennui-inducing, and far in the past now. Hell, I don't know if the subdivision ever got built. The developer handled things after the plans were eventually approved, and this was just before the Bush recession.

I guess the point is that this is not the sort of thing I take personally -- provided it's not a personal attack in the first place. Getting back to writing, if I get a review that essentially says "this story sucks and here's why," I take it seriously. If I read, "you suck and here's why," well, that's another issue entirely. Fortunately, I don't recall any of that on this site.

On a related note, I went to see Tenet yesterday; Alamo had early access screenings. Once again, it turned out to be a private viewing. I was the only one in the theater. Great for avoiding the 'rona, probably not so great for Alamo's bottom line. Anyway, the movie was not nearly as clever as its makers thought it was. Now, I can't say "Christopher Nolan is a hack," because I've seen his other stuff and a lot of it is very good. But I, personally, would hate to put that much work, that much of my heart and soul into a movie, only to have someone (like me) with absolutely no talent for moviemaking say that it sucked. I suppose you have to develop a thick skin, like a politician.

I'm not sure I could do that, which is one reason I haven't aggressively pursued publishing. I mean, what if I succeed and some idiot calls me a no-talent hack?

I guess that ties into yesterday's entry, too.

Speaking of which, thanks for all the comments. I appreciate all of them, even the ones that disagree with me... because I can accept other points of view besides my own. I'm going to give today's MB to Charlie ~

, though, because of the math thing:

Math is probably one of the best examples of not people not believing in themselves. I don't know how many times in school I heard people say that math "just wasn't their thing" or "better you than me" when they found out my majors. I'll concede that some people have a natural aptitude for math, but it's absolutely learnable, just like any other language.

I think a lot of that falls on teachers though because the teachers in my formative years did not instill a lot of self confidence in us when it came to learning math. I do think you have to have a special understanding of mathematics to actually be able to present it in an understandable way. It seemed like my teachers gave up pretty quickly on students and allowed them to just go with the "I don't do math" mentality.

I mean, after all, math is the central metaphor for this blog. But also it got me thinking: the way other people are about math is the same way I am about art and music. I've tried both, with lousy results... but I love to look at art and listen to music. It's very frustrating to me that I can't create them the way other people seem to do so effortlessly.

Perhaps I just haven't tried hard enough. Or maybe I've just had it instilled in me from a very early age that I just don't have the aptitude for these things. So even though I (and Charlie) spoke about math blockage, I understand because I'm blocked in other areas.

Please keep the comments coming; I'll award another one of my free Birthday Week Merit Badges tomorrow, but mostly I just like feedback - positive or otherwise.

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