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 7, 2020 at 12:12am

#987434

You're My Inspiration

Hey, it's this "inspiration" thing again.

Also, yesterday's Mini-Contest winner announced below.

PROMPT July 7th

What do you do when you are out of inspiration or ideas?

I look at prompts, of course.

Well, that's for blog entries and most of my fiction and poems. When I'm not in a blog challenge month, I have web links I can post here and comment on. Keeps me busy.

I have written a few stories and poems that weren't prompted, but the bulk of my work comes from "Invalid Item"

or other contests with their prompts.

Occasionally, very rarely, someone else will write something that makes me want to create a thing. I mention this because it just happened recently: "Procrastination"

[E]

That particular one was ironic because I jumped right into writing it, instead of my usual thing which is to wait until later, after which I forget about it. In other words, I didn't procrastinate writing a limerick titled "Procrastination."

You know what the toughest thing for me is? Newsletter editorials. There are no prompts, only the topic of the newsletter - for me, Fantasy or Comedy. Don't get me wrong -- once I settle on a subject, the words usually flow well enough. But I've been editing both newsletters 13 times a year for about 13 years, now, and sometimes I agonize over what to write about right up until the deadline. Okay, sometimes past the deadline; see the above poem.

And yet, having done it for 13 years, I'm sure I've duplicated earlier ideas more than a few times. That's usually okay; with more experience comes different perspectives on the subject.

What's also hard for me is coming up with prompts in the first place. I mean, if it weren't, I'd write more, right? Instead, about once a week, I have to come up with something for other people to write about. Generally it's about whatever's on my mind right at the moment; the problem with that, these days, is that my mind generally cycles between "pandemic" and "beer." The former is overdone, and the latter excludes people who prefer to keep their writing below 13+. I'd rather be inclusive with the Cramp.

So yeah, inspiration comes to me rarely and fleetingly, so I try to catch it when it zips by. But, as any professional writer (e.g. Not Me) might tell you, it's not about inspiration or ideas; it's about doing the work. And that's where I fail.

Mini-Contest results!

Some great answers yesterday, where I asked for examples of when presenting both sides to a controversy can be a Bad Thing. Since I mentioned my newsletter editorials above, I had some vague memory of writing about this sort of thing in a Comedy editorial -- and behold, I managed to find it: "Both Sides"

I felt that the closest thing to what I was talking about was the comment provided by Charlie ~:

I mean, a good example of "false both-sidedness" is vaccines. We have scientific evidence that vaccines work with little to no harmful effects, especially when taking into consideration all the good they do. When you present one side as someone saying, "Vaccines are pointless! When is the last time you met someone with [insert disease that has been mostly eradicated due to vaccines]" it's like, yeah, that's the ENTIRE point of vaccines and it's proof that they work.

Those situations annoy the hell out of me because the one side isn't actually presenting anything reasonable or sensible. Their side is, "I feel like I'm special and shouldn't have to protect my children or the rest of society from illness."

Admittedly, it didn't hurt that this was one of the specific topics I had in mind when concocting my "flat-earth" hypothetical example. But as usual -- I appreciated all of the comments, and you'll all get another chance in a few days. But this week's Merit Badge goes to Charlie ~

14 Like • Comment • View 11 comments

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