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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Brandiwyn🎶

a day ago
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').

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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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

December 31, 2020 at 12:01am

#1001099

Lucky Streak

"30-Day Blogging Challenge ON HIATUS" [13+]:
31. Unlucky Day
Write about a time when you were particularly unlucky.

You know what?

No.

I don't believe in supernatural forces, but luck is a thing independent of that. It's random chance applied to circumstance. You can't affect random chance, but you can affect circumstance. It's like... if you're terrified of being killed at sea, you can change the odds of you being killed at sea by not going to sea. And yet, if you do go to sea, you still have pretty good odds of, you know... living through it. But it's still subject to chance. And if you move to, say, the Midwest to maximize your chances of not accidentally going to sea, well, there's always tornadoes.

Like everyone, I've had good things happen to me and bad things. I've had bad things that have turned out to be good things, and good things that turned out to be bad things. On balance, though, life isn't fair... and so far, it's been unfair in my favor.

So I don't want to dwell on the unlucky parts. It puts me in the wrong mindset for going into a new year, which, though I've ranted about the arbitrariness of January 1 here before, the reality remains that the world I live in uses the Gregorian calendar and there's no disputing that the odometer of that calendar rolls over tonight. And that's meaningful for a lot of people.

I just think all the folks who are going to breathe a huge sigh of relief just because the clock strikes midnight are going to be in for a rude surprise.

As usual, I could be wrong. But there's no guarantee that 2021 will be any better than 2020. Luck, remember?

This entry marks a personal achievement for me and, true to form, it is not one that I actually set out to achieve. Those, I inevitably fail at (hence why I'm not making any New Year's resolutions). But as of this entry, I have written a blog entry -- and, I trust, one that was meaningful in some way -- every day in 2020. You can verify this for yourself - just click back through the calendar there on the left, and you'll see that every month of 2020 is completely filled.

And so, in celebration of this achievement, here comes 2020's final...

Merit Badge Mini-Contest!

In keeping with the theme of "luck," I'm going to run this a little bit differently from previous contests. All you gotta do is comment, and tomorrow I'm going to pick commenters from this entry at random. For every five commenters or fraction thereof, I'll award one Merit Badge. So if only one person comments, they'll get one. If eight comment, then two people will get one. If 25 people comment, there will be five MBs. You're welcome to comment more than once, but only one of them will count.

If you're at a loss for what to comment, how about... what are your hopes for the New Year? Or... what are your favorite kinds of entries in my blog? Or, just tell me how awesome I am. It won't affect your odds of getting a Merit Badge, but it'll make me feel good. Hell, tell me I suck and it still won't affect the outcome. (Though, please do stick to the Content Rating, as always.)

As per usual, deadline is midnight tonight WDC time (Eastern Standard Time). Since that is the end of December 31, and I too am on Eastern Time, and it is New Year's Eve, I do not expect to be in any condition to post a blog entry shortly after midnight like I usually do. Just to be clear, I'll be staying home, not going out, but there will be alcohol. I will award the MB(s) and post an entry later that day, assuming the overindulgence doesn't destroy me.

Which, let's face it, would be a fitting end to this year.

And so, Happy New Year to all, and most of all...

Good luck.

9 Like • Comment • View 22 comments

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