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.

October 27, 2019 at 12:02am

#968470

Sleep Now

https://www.inc.com/jeff-haden/how-to-turn-yourself-into-a-morning-person-backed...

Note the timestamp on this blog post. I don't know if that qualifies as irony, but it might qualify as comedy.

How to Turn Yourself Into a Morning Person, Backed by Science
Even if you're convinced you're a night owl, and hate waking up early.

Backed by SCIENCE!

First things first: Getting up early is not a prerequisite for success.

As much as I want to believe that, it is if your job starts... wait for it... early. If it does, and you don't get up early, I predict you will not be successful.

Even though The Wall Street Journal says that 4 a.m. may be the most productive time of the day, the most successful people wake up and start work whenever the (heck) they decide is the best time for them.

4 am might very well be the most productive time of the day; if so, I stay up for it pretty regularly.

But still: Even if you're a committed night owl who loves to wake later in the day and work late into the evening, you may not have that luxury. Maybe you have clients in other time zones. Maybe you run a business that requires you start your day early.

Or maybe you're a peon with no control over when work starts and ends. Remember yesterday I talked about working for a surveyor? If it was light out, we'd work. And if you recall, it was a summer job. The accursed daystar rises disgustingly early in the summer.

But it was good experience for my later career.

If tomorrow is your first day of shifting to an earlier start time, don't try to go to bed early tonight. Just go to bed when you normally do. Sure, you'll be tired tomorrow, but that's OK. Natural fatigue will help you get to bed a little earlier that night, or the next night.

In time, your body will adapt -- as long as you don't shift back to your night owl ways on the weekends. Shifting back and forth results in an endless cycle of sleep schedule resets.

I'm... not sure this is the best advice.

2. Exercise first thing.

Wait... wake up early AND exercise?

My dad used to wake me up with "Rise and shine!" I'd spit from underneath the warm covers: "Pick one!"

"Moderate intensity aerobic exercise improves mood immediately and those improvements can last up to 12 hours," says one of the researchers. "This goes a long way to show that even moderate aerobic exercise has the potential to mitigate the daily stress that results in your mood being disturbed."

Can't argue with that. My anecdata agrees.

6. Start every day with something you really want to do.

This directly contradicts #2.

I've said it before and I'll say it again: I am absolutely not convinced that becoming a morning person is the key to superiority. Maybe the key to feeling superior; I wouldn't know. But we only have about 16 hours to get stuff done that isn't sleep, and to me it doesn't matter when those 16 hours occur.

But if you have to shift your sleep schedule, the best way I know to do that is not to work backwards. That is, don't just try to go to sleep at 10 and get up at 6 (for an eight-hour sleep cycle). Instead, take some time, if you can, and work it forward: go to bed an hour later, and really try to sleep for exactly 8 hours. Stay up for 17 hours, not 16. Repeat the following night, and again until you have the schedule you need.

So, like, if you're used to going to bed at midnight and getting up at 8, and you need to start getting up at 5 for a shitty commute or some such, you can't just try to sleep at 9 and call it a night (pun intended). No, find a way to take the time to shift forward. First night: Hit bed at 1, set alarm for 9. Make yourself stay awake until 2. Set alarm for 10. Stay up to 3 am. Set alarm for 11. And so on, until you hit your 9pm-5am target.

Of course, lots of people can't take the time off to do that, what with family and work commitments. But it's still the best way I know to reset one's clock.

"But Waltz, you just said you're a night owl!" Well, yeah, but when I travel I like to wake up earlier, use the daylight. It's not my natural schedule, but I'm adaptable. To an extent.

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