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!

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.

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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