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!

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 26, 2019 at 12:13am

#972043

Lead Me Not Into Temptation, For I Can Get There On My Own

Well, Christmas is over, which means New Year's Day is coming up fast. Which means, for many people, resolutions. Which means failing at them at a breakneck pace.

So today's blurgle is about self-control. Or, rather, lack thereof.

https://www.vox.com/science-and-health/2016/11/3/13486940/self-control-psycholog...

The myth of self-control

Psychologists say using willpower to achieve goals is overhyped. Here’s what actually works.

Incidentally, I am so not looking forward to dealing with the gym after next Tuesday. It'll be stuffed full of amateurs taking up space on the machines, and I expect to spend as much time waiting for a damn weight machine as I do exercising. Maybe for the first two weeks of January, I'll go at like 2 am. It's a 24 hour gym and even the most hardcore resolutionists probably won't swarm the place until 5.

After those two weeks, it'll be less crowded. I figure by my birthday in mid-February, it'll be back to normal.

As the Bible tells it, the first crime committed was a lapse of self-control. Eve was forbidden from tasting the fruit on the tree of knowledge. But the temptation was too much. The fruit was just so “pleasing to the eye, and also desirable for gaining wisdom,” Genesis reads. Who wouldn’t want that? Humanity was just days old, but already we were succumbing to a vice.

I've come to understand that particular story as a metaphor for prehistoric human awakening - that moment when some human looked at the stars, maybe, and instead of seeing stars, saw potential.

But that's not really relevant.

“Effortful restraint, where you are fighting yourself — the benefits of that are overhyped,” Kentaro Fujita, a psychologist who studies self-control at the Ohio State University, says.

Lately, I've envisioned temptation this way: you're underwater, about to drown. You can see light above you; it's the surface. Every move you make is, ostensibly, under your control: arms pushing at the water, legs kicking, even holding one's breath. But really, these motions aren't willful, even if under other circumstances, they are. You can, perhaps, direct them, but except in the most rare of circumstances, you can't force yourself to stop. You will claw your way to the surface, no matter what.

That feeling, that desperation, is hijacked when a wannabe ex-smoker sees a cigarette, or when someone like me sees a pizza.

Speaking of which:

A recent national survey from the University of Chicago finds that 75 percent of Americans say a lack of willpower is a barrier to weight loss. And yet the emerging scientific consensus is that the obesity crisis is the result of a number of factors, including genes and the food environment — and, crucially, not a lack of willpower.

Every day, on my way to the gym and back, I have to pass a Krispy Kreme. Twice. So far, I've talked myself out of visiting.

Many of us assume that if we want to make big changes in our lives, we have to sweat for it.

But if, for example, the change is to eat fewer sweets, and then you find yourself in front of a pile of cookies, researchers say the pile of cookies has already won.

Now that you've talked about it, it's already already won.

“Our prototypical model of self-control is angel on one side and devil on the other, and they battle it out,” Fujita says. “We tend to think of people with strong willpower as people who are able to fight this battle effectively. Actually, the people who are really good at self-control never have these battles in the first place.”

My own personal angel is battered from being flicked off of my shoulder whenever temptation strikes.

1) People who are better at self-control actually enjoy the activities some of us resist — like eating healthy, studying, or exercising.

I don't like eating healthy, but I've arranged things so that I either have to do it, or do work to not eat healthy, and I'm allergic to work. With exercise, it's a bit different for me; I despise exercise but I like how it makes me feel afterward, so I try to focus on that. It's the same sort of idea as beating one's head against the wall because it feels so good when one stops.

2) People who are good at self-control have learned better habits

I think daily exercise has become a habit for me, now. But I'm terrified that as easily as I learned the habit, I could fall out of it.

“People who are good at self-control … seem to be structuring their lives in a way to avoid having to make a self-control decision in the first place,” Galla tells me.

That tracks. One of the main reasons I get groceries delivered - apart from the aforementioned work allergy, and I consider driving to the store, pushing a cart down the aisle, standing in a checkout line, driving home, schlepping the bags inside, and putting everything away to be work - is because at the grocery store, I inevitably find myself drawn magnetically to the Oreos display, and the cookies have already won.

Yeah, delivery is more expensive, but I figure between the lack of Oreos and lower health care bills, I'm still ahead.

3) Some people just experience fewer temptations

People are different from each other. Who knew?

4) It’s easier to have self-control when you’re wealthy

Yeah... I'm going to need some more convincing here. Take the Oreos again for example (no, really - take the damn Oreos away from me). I'm not poor, so I don't even look at the prices on the Oreos display shelves; I just grab one or six packages. Someone for whom money is tight might look at the prices and then go deeper into debt to obtain them. Assuming they're not one of the lucky ones for whom Oreos aren't a temptation at all, as per #3 above.

Another intriguing idea is called “temptation bundling,” in which people make activities more enjoyable by adding a fun component to them. One paper showed that participants were more likely to work out when they could listen to an audio copy of The Hunger Games while at the gym.

What little I've read of that series is written in present tense. If I have to read more than a short story in present tense, I get angry and then I want a donut. I'm not saying it's a bad story - I saw the movie and enjoyed it well enough - but reading, or listening to it, would be counterproductive to me.

But then, I listen to boring science / math / philosophy lectures while working out, which is probably other peoples' idea of Hell, so whatever works.

“Because even if the angel loses most of the time, there’s a chance every now and again the angel will win,” Fujita says.

He's going to have to recover from his injuries, first.

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