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:

tj-Merry Mischief Maker

about 15 hours ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

about 18 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

about 21 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

a day ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

May 21, 2024 at 7:17am

#1071456

Habit Your Way

Here's a book ad disguised as a self-help article from GQ for me to snark on.

How to Kick Your Bad Habits (And Why That's More Important Than You Think)
Plus: Why goals don’t work, and why your to-do list is wrecking your ability to do anything.

Hey. HEY! You're doing it wrong. Send me money and I'll tell you how to do it right! Until the next guru comes along to tell you that you're still doing it wrong.

A few years ago, we reached peak you're-doing-it-wrong with a video on how you're opening a banana from the wrong end. Since that, I've felt free to ignore any attempts to convince me I'm doing anything the wrong way.

You might not spend much time thinking about your habits. They are, after all, mindless.

No, they're really not. They may be comfortable, but they're not mindless.

James Clear, on the other hand, has made something of a living on it.

Lots of people make their living preaching that "you're wrong and the only way to be right is to follow me and give me money." Some of them are literally called preachers. It doesn't mean they're right.

One of his big takeaways is a bit unsettling when you consider all the habits you’ve sworn to kick (but haven’t), and all the habits you’ve really been meaning to start (but haven’t): habits, multiplied by time, equal the person you eventually become.

I get around that by not swearing to kick or start habits. Do this, combined with accepting yourself as you are, and your life becomes simpler and happier. You won't even be tempted to give what little money you have to people who are trying to convince you that there's something wrong with you and only they can fix it.

That advice, by the way, is 100% free.

In a year, the difference between a person who does 10 push ups a day and a person who eats one bag of Doritos a day is that one person has done 3650 push ups and one person is sad.

No, the difference is that one person has sore arms and pecs, and the other person is following their bliss.

Besides, this creates the illusion of a dichotomy: while it might be difficult to eat a bag of Doritos while doing push-ups, there's absolutely nothing that says you can't do both in one day.

The rest of the articlead is in interview format, and I won't quote most of it. But at one point, the interviewer poses:

But if you think about it, I feel like we're so often controlled by sort of nudges that we aren't even conscious of.

Yes, and one of those nudges is ads.

The way I see it, if you think you want to do something (or refrain from doing something), and you continue to do (or not do) it anyway, then you don't actually want to do it, and you should respect that.

Now... this may seem hypocritical of me, since I've been on a 4+ year daily streak in both blogging and language learning. Those are habits I picked up, and sometimes I go out of my way to practice them. I'm not saying we shouldn't try new things, or improve ourselves. It's the pressure that we're somehow lacking, and we should feel bad about ourselves, that I object to.

So I make it a habit to identify that pressure when it presents itself.

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