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 20, 2019 at 12:15am

#959306

Cycles

What time of day are you most motivated? Least motivated? For me, I’m most motivated and productive in the morning, and least in the mid afternoon around 2pm. What do you do to renew your motivation in those slumps?

As someone who finds motivation elusive at the best of times, I'm not sure how to respond to this one. But I'll give it a shot.

The idea that we're better off doing certain things at different times of day has some evidence for it. Here's an example:

https://www.outsideonline.com/2278086/timing-everything-and-everything-timing

If you consciously pay attention to timing, you can dramatically improve your performance, and you can dramatically improve yourself.

(This is probably the first and last time I'll ever link to Outside magazine. I mean... "Outside." Shudder.)

I was reading that because I did start to wonder - and therefore, I Googled - what's the best time of day to do a workout. According to that article, from what I gather from it anyway, the answer is: it depends on what you're trying to do.

It would make sense, then, if the same logic applied to other types of activity, including the relative inactivity of, say, sitting down to write something.

My work used to involve a lot of sitting around at the computer - much as my hobbies do now. Back then, when I was shoehorned into a roughly 8-to-5 schedule, I did notice a trend: I could get work done at anytime, including the many hours of overtime I pulled, but almost every day, around 3-4 pm, I'd get tired. Like, pass-out-in-a-coma tired. This wasn't the same kind of inertia that always clogged my mornings (because I'm not a morning person and I never did get a taste for coffee), but rather a shutdown, almost like my narcoleptic friend. But where she couldn't stay awake no matter how much she wanted to, I could usually force my eyes to remain open - but I wasn't worth a damn when I did.

On the other hand, if I was able, somehow, to nap around that time, I'd get a period of great energy going around 5-7pm - you know, about the time most office grunts like me had already gotten off of work.

But, I've found, I do my best work around midnight - and this is true even now; it's why I get these entries in around 12-1 am (I'm in the same time zone as WDC).

Because of that, I found it difficult to get to sleep then, and consequently, had trouble waking up between 7-8 am - and the cycle continued.

One of the things I promised myself upon retirement was that I'd sleep when I was tired - barring, of course, important events. This led to my current biphasic sleep cycle - which, probably not coincidentally, has me falling asleep for an hour or two in the late afternoon.

Even then, it varies. Sometimes I'm up until 4am. Sometimes I'm asleep by 2. Rarely, my afternoon nap turns into a four-hour snoozefest. But one constant is that I feel most creative and productive, writing-wise, in the hours immediately around midnight.

And my workouts? Noon-ish, usually. Probably not ideal according to that despicable link above, but that's about when I go, "You know, I probably should go ahead and get my gym time in before I get tired this afternoon."

I'll tell you what, though - it's been a few years since I could unfetter myself from the shackles of an office job, and I still revel in the luxury of being able to (most days) get up whenever the fuck I feel like it. And it is a luxury, I know - but when I'm up, I'm up. No snooze buttons, no missing alarms, no dragging ass into the shower... just up and showered and dressed and ready to do day stuff.

Just another of the great things about being me, I guess.

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