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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Lazy Writer est 4/24/2008

about 20 hours ago
In response to "Revisited: "Exercise""

I will be 75 in July. My stepdaughter gave me a bike because I wanted to ride it again. Well, I started off shaky and stayed shaky...then I fell off it...A passing motorist stopped to help me get up...I have NOT been back on that bike and will not ever again.

🌕 HuntersMoon

about 24 hours ago
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

a day ago
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

Kåre Enga in Udon Thani

Apr 25, 2024 at 2:58am
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

Apr 24, 2024 at 4:26pm
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

Apr 23, 2024 at 6:51pm
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

1 Like

Lazy Writer est 4/24/2008

Apr 23, 2024 at 3:31pm
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:14pm
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

1 Like

August 24, 2019 at 12:33am

#964620

Don't Be a Douche

Oh, here's one to have fun with.

https://www.inc.com/marcel-schwantes/these-31-life-hacks-can-improve-your-life-i...

I hate the phrase "life hack," because it's overused; but as this article is from 3 years ago, I'll forgive it. This time.

31 Ways to Improve Your Life in Just a Month
Imagine the possibilities of a life filled with these simple acts of joy that can be done in minutes. Here's a template to get started.

I've noted before that most people don't have "minutes." By the time you attend to basic bodily functions such as sleep; required activities such as work (and the associated commute); and dealing with life's little vicissitudes (such as kids), you're already at 24 hours.

Before this 31-day plan became a lifestyle, I first had to remove the tumors of self-gratification and hubris (trust me, the attitude ruling my life in the old days was all about "what's in it for me?").

While I'm all for self-gratification, I've rarely thought "what's in it for me?" outside of business situations. That is, sure, business transactions are two-sided, but social transactions are not, at least for me. Was I doing something wrong? I guess that's why I have a blog with 12 readers and not a column in Inc. magazine.

Now that you have a glimpse into your own plan, a fair warning: It's not about you. It took years for me to develop the understanding that life is really about giving, service, and meaningful relationships.

Really? Because I learned that at an early age. Again, I suppose that's why I'm comfortably middle-class instead of jetting around the world in my Gulfstream.

Day 1: Do something for someone else.

Do a "five-minute favor" for someone. Five-minute favors are selfless giving acts, without asking for anything in return from the people that you help. Examples of five-minute favors include: sharing knowledge, making an introduction, serving as a reference for a person, product, or service, or recommending someone on LinkedIn, Yelp, or another social place.

More proof that I'm doing it wrong - I don't need shitty advice columns for this.

Day 3: Stop striving to achieve.

We all have a tendency to work too much, lose our balance, and, ultimately, our joy in life. It's the unhealthy feeling that if we don't do something productive every day, we've somehow failed.

Okay, guilty. Or at least I used to be. I got over this when that contributed to a divorce.

Day 4: Put yourself in someone else's shoes.

Empathy and compassion are things you can develop, and it starts with thinking about other people's circumstances, understanding their pains and frustrations, and knowing that those emotions are every bit as real as our own.

I'm starting to get the impression that this author is a sociopath who is trying very hard to not be a sociopath.

Day 7: Give thanks. Your situation could be a lot worse.

I don't care what religion you come from, start your day by thanking your higher power for the things you take for granted. As it turns out, if you make more than $30,000, you earn more than 53.2 percent of Americans. If you make more than $50,000, you earn more than 73.4 percent of Americans. Feeling grateful now? Say a little prayer and give thanks, and then pray for the other 73.4 percent.

Really? Are we measuring success by comparison with other people? Also, all this talk about "higher power" sounds like it comes from AA. So, possibly a recovering alcoholic sociopath?

Day 10: Just. Say. No.

Truly happy people live a simple life. They have a simple schedule. They don't take on more than they can handle. They live according to their values and purpose. They have strong boundaries around what comes into their life. And they have no problem saying no. If it doesn't serve you, if it has little value, and if it doesn't make you better tomorrow than you are today--just ... say ... no.

Call me dense, but doesn't this contradict Day 1?

Day 16: Exercise for 15 minutes.

Achor also told Oprah that if you hate exercise, all it takes is 15 minutes of fun cardio activity, which is the equivalent of taking an antidepressant, but with a 30 percent lower relapse rate.

There is no such thing as "fun cardio." If you think there is, you're probably already exercising. Me, I finally got on board with the antidepressant qualities, but I still have to force myself to do it every day.

Day 19: Find something or someone that will make you laugh.

Humor helps you think more broadly and creatively.

Well, you're reading my blog, so hopefully you're already there on this one.

What would your life look like if you practiced some of these things everyday, extending this plan beyond a 31-day cycle? It just might help you live the life you've always wanted rather than settling for whatever comes your way.

Okay, look, I'm not saying this is bad advice overall. But really, this guy sounds like a douche. Or at least a recovering douche. Not unexpected, given the source. Maybe the real secret to happiness is: Don't be a douche?

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