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 23 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

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.

December 21, 2019 at 12:10am

#971798

Asleep Under the Sun

https://www.vox.com/2016/3/18/11255942/morning-people-evening-chronotypes-sleepi...

If you’re just not a morning person, science says you may never be

Sure, I'm a morning person. I'm at my best between about midnight and 4am. That's morning. And then I get my best sleep between 4am and noon. That, too, is morning.

Afternoons can be dreary.

Morning people and night owls are born that way. It's time we accepted that.

Yeah, right. People who think there's some moral superiority in waking up while it's still dark aren't going to change their minds. They have to feel superior about something, I suppose.

If Cassidy Sokolis ever needs to wake up before 11 am, she scatters three alarm clocks throughout her bedroom. Even then, she still often sleeps through the clamor.

Sounds familiar.

We all have a preferred, inborn time for sleeping. Science has validated the idea that there are "morning people," "evening people," and those in between. These are called chronotypes. And just like it’s rare for a person to be 7 feet tall, it’s rare for Sokolis to not be able to sleep until 3 am. We all have a chronotype, just like we all have a height.

Okay, but we can see your height.

And what's more, if we try to live out of sync with these clocks, our health likely suffers. The mismatch between internal time and real-world time has been linked to heart disease, obesity, and depression.

Oh, good; I can blame my biggest health issues on being forced to keep an 8-5 work schedule throughout my career. Score!

"Our clocks don’t run on exactly a 24-hour cycle," Gehrman says. They're closer to 24.3 hours. So every day our body clocks need to wind backward by just a little bit to stay on schedule.

I've heard variations of this in the past. As I recall, they studied this by keeping people in windowless rooms for extended periods of time, with no reference to timekeeping devices, and looked at what rhythm they fell into.

If that sounds like hell to you, believe me, I think so, too. After all, there's a clock on the computer. Going for days without a computer? The horror!

I have, on occasion, wondered what's up with the extra .3 hours (I've also heard .5 hours, but whatever). You'd think, maybe, "evolutionary holdover," but if anything, the Earth's rotation was faster in the distant past.

The second treatment is chronotherapy. Here, instead of pushing the body's clock backward it's wound forward. For a few weeks, a patient will go to bed two hours later every night until she reaches her desired sleep time. "It can work very well, but very few people have the absolute, total control of their schedule for the two weeks that it takes to do that," Gehrman says. Very few of his patients choose this option.

Pretty sure I've mentioned this sort of thing in this blog before, but I can't be arsed to look it up. I did do a sanity check to ensure I haven't linked this particular article before, and I don't think I have. In any case, I'm just quoting the bit above because of my confirmation bias.

I have also wondered what happens to someone, say, who lives on the east coast of the US and is a night owl, if they move to the west coast. With a three-hour time difference, can they keep an earlier sleep/wake schedule, or does the rhythm of daylight gradually reset them to night-owl-hood?

Simply put: Society favors early risers. Think no further than phrases like, "The early bird catches the worm."

My favorite rebuttal to that has always been, "The second mouse gets the cheese."

In any case, I know I've been on this subject before, because as a night owl, it's bugged me for a while. Not being a night owl - I love being up at night and sleeping through the boring parts of the day - but the general Puritanical scowling at anyone who doesn't rise with the accursed daystar.

I'm privileged, I know - I'm in a position where I can (usually) sleep when I want and get up when I want. Does it make me healthier? I don't know; I think the exercise does more for that. But it's just possible that I've been able to keep up with daily exercise for a whole year (an absolute record for me) because I'm not fighting an unnatural, for me, sleep schedule.

2 Like • Comment • View 2 comments

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