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.

October 22, 2019 at 12:11am

#968243

StairMaster to Heaven

Buckle up.

https://waitbutwhy.com/2014/10/religion-for-the-nonreligious.html

Religion for the Nonreligious

Rarely has a domain name been so appropriate. "Wait but why?"

You go to school, study hard, get a degree, and you’re pleased with yourself. But are you wiser?

"Wiser" is an unmeasurable comparative.

You get a job ...[long list of career-related crap]... But are you happier?

Other things matter besides happiness. Take care of those and the happiness emerges naturally.

You do all kinds of life things—you buy groceries, read articles, get haircuts, chew things, take out the trash, buy a car, brush your teeth, shit, sneeze, shave, stretch, get drunk, put salt on things, have sex with someone^{[citation needed]}, charge your laptop...

Yes. We get it. We all do stuff. If we don't, we sit around and do nothing. While that's nice from time to time, even my cats get bored if they're not chasing something for at least a few minutes a day.

But as you do these things day after day and year after year, are you improving as a human in a meaningful way?

Going back to my cats, I don't see them being concerned about improving as a cat in a meaningful way. But okay, I'm human, and I think things and have some concern about the future beyond when someone is going to give me my next bowl of kibble, so I'll play along.

In the last post, I described the way my own path had led me to be an atheist—but how in my satisfaction with being proudly nonreligious, I never gave serious thought to an active approach to internal improvement—hindering my own evolution in the process.

That sounds like a personal problem to me. Even when I realized I was an atheist, I never stopped trying to improve myself (though I slowed down a lot when I was depressed). In fact, I worked harder at it because I realized that without accepting a higher power, it's all the more important to improve oneself. Also, stop confusing people with the "evolution" thing; it's distracting.

Society at large focuses on shallow things...^{[citation needed]}

Considering that the human mind is an ocean of complexity that creates every part of our reality...^{[citation nee- oh, hell, there's going to be more of these assertions without evidence, isn't there?]}

And then the author gets into some really weird and possibly misleading graphics about consciousness and enlightenment or whatever. Obviously, I can't reproduce them here; you're just going to have to go to the link to see what I'm talking about.

Suffice it to say that atheism in itself isn't any guarantee of intelligence or understanding. Neither, of course, is religion. Or spirituality. And this seems to be a weird kind of spiritual atheism, which is fine, but then they talk about "truth" and present speculation as fact.

I have to admit, I started skimming and looking at the MS Paint drawings. It seems to delve into New Age-y etheria, though, with all kinds of metaphors presented as fact.

Remember a few days ago when I found an article that asserted that the thing that most differentiates us from other animals is our ability to dive deep into metaphor? This is the dark side of that.

Thing is, though, I think the author makes some decent points. I'm sure you can find them yourself if you're up for it. But then he comes up with stuff like:

There are really two options when thinking about the big, big picture: be humble or be absurd.

Bit limiting, isn't it? I'd pick "be curious" or "be open-minded" as alternatives; neither are absurd or require particular humility. I mean, maybe in the sense that to realize you don't know everything is a kind of humility, I guess? Or it's just a facing of reality.

To me, complete rational logic tells me to be atheist about all of the Earth’s religions and utterly agnostic about the nature of our existence or the possible existence of a higher being. I don’t arrive there via any form of faith, just by logic.

Then you're not atheist; you're agnostic. By most accepted definitions of those two words. You even said the word. I mean, I don't care, either way, but this is confusing to people.

Labels sometimes just get in the way. So do bad MS Paint drawings.

2 Like • Comment

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