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 an hour 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 3 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.

May 9, 2023 at 11:24am

#1049389

Alarming Trend

Sometimes, the random number generator points me to the same topic as the day before. Such is the clumpy nature of randomness.

Pretty sure this is the last article along these lines in my queue, so after today, we're probably done for a while.

Waking up at 5 a.m. every day could improve your life—here’s how to make it work for you

"Could" is absolutely the key word in the headline.

I will note that this article is from Fortune magazine, so I'm pretty sure what they really mean is "Waking up at 5 a.m. every day would improve your boss's bottom line."

Seize the day, we’re told.

We're told a lot of things, many of them contradictory. "Seize the day," or carpe diem in its maybe-original formulation, is meant to be an exhortation to live in the present moment with little to no thought of consequences or the future. Basically the opposite of "force yourself to get up at 5 am so you can worship at the altar of Holy Productivity."

The early morning wakeup has even become a TikTok trend...

I really should have stopped reading there. Once they mention that short-attention-span data-mining cancer of an app, I can be sure that nothing else in the article applies to me.

Not that I didn't already know that from the "wake up at 5" bullshit.

...coined the “five-to-nine before the nine-to-five,” where video montages illustrate a slow morning aesthetic of self-affirmations, workouts, and maybe even a head start into planning for the work day.

This is like the training montages in movies, which make it look like you can go from wimp to wolf with five minutes' worth of sweat.

“The pressure to be a morning person is pretty intense,” says Samantha Snowden, a mindfulness teacher at Headspace, the popular meditation app.

Everything about that sentence is pressure. Blood pressure. Mine. Higher.

For starters, getting up earlier can improve confidence, Snowden says, because it can feel like an accomplishment.

You know what else can feel like an accomplishment? Actually accomplishing something.

And if you can use those extra morning hours to make time for yourself in a way that calms you down, it can bolster productivity and make you feel less depleted by the end of the day.

Ahhh... I knew they'd get to the P word eventually.

You know what calms me down? Sleep.

Choosing to move up that alarm should not come at the expense of sleep.

I know I've said this before, but you're going to have the same number of waking hours every day, on average, no matter what time you get up. Unless you sacrifice sleep, as I did for many years, and I'm paying for it now—way more than I'm paying for smoking or drinking.

Prioritizing sleep means having good sleep hygiene, including waking up around the same time each day, limiting screens before bed, not consuming alcohol or caffeine in the evenings, and having a wind-down routine.

This is the first thing in that article that doesn't make me want to yeet something at the wall.

Not that I wake up at the same time each day, or get away from screens before bed, or refrain from alcohol or caffeine. I'm just saying it's decent advice, not that I follow it.

Snowden says you can spend 10 extra minutes slowing down (even walking a bit slower to the shower in the morning), not checking emails right away, and practicing a kindness message.

And there goes my blood pressure back up. Seriously. They say you can't feel it, but I definitely do.

Especially for the night owls, choosing to get up earlier won’t feel comfortable immediately.

It'll never feel entirely comfortable if you're a true night owl. Never. You can get used to it; we're pretty adaptable. But being an owl and being forced to be a lark is very similar to jet lag, only it never really goes away.

Source: me.

Now, look. I want to emphasize that I'm not knocking people who get up early, for whatever reason. You do you, as the kids say. Sometimes, we have to; I did for a long time. Some people are just naturally early risers, and that's okay, too.

What I object to is the incessant pounding in various media sources (usually ones with a pro-business agenda) that this is the One True Answer. People are different, and have different innate schedules; I'm not "lazy" because I prefer to sleep from 2am to 10am (or whatever my schedule is on any given day) instead of 9pm to 5am. I'm lazy for a lot of other reasons, but not that one. Still, I can acknowledge that more people prefer the latter, though I'd wager not so many among writers or other creative types.

I just want to stop the sleep-shaming.

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