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!

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 20, 2019 at 1:01am

#964457

Idle Ideals

https://www.theguardian.com/technology/2017/may/08/virtual-reality-religion-robo...

The meaning of life in a world without work

Okay, I'm thinking, this should be good. I haven't done "work" in about a decade, and I almost never miss it.

As technology renders jobs obsolete, what will keep us busy? Sapiens author Yuval Noah Harari examines ‘the useless class’ and a new quest for purpose

"Usefulness" is overrated, anyway. So is "purpose."

The same technology that renders humans useless might also make it feasible to feed and support the unemployable masses through some scheme of universal basic income. The real problem will then be to keep the masses occupied and content.

I'm not going to turn this post into a discussion of UBI. Just know that I'm sure that it has positives and negatives, and I'm even more sure that here in the US, we will fuck it up royally.

People must engage in purposeful activities, or they go crazy. So what will the useless class do all day?

Must we? Must we, really? I guess if that's true, then I'm officially crazy. Well, that shouldn't come as any surprise.

One answer might be computer games. Economically redundant people might spend increasing amounts of time within 3D virtual reality worlds, which would provide them with far more excitement and emotional engagement than the “real world” outside.

Ah. Well, that explains me.

This, in fact, is a very old solution. For thousands of years, billions of people have found meaning in playing virtual reality games. In the past, we have called these virtual reality games “religions”.

Ooooooh! BURN.

Muslims and Christians go through life trying to gain points in their favorite virtual reality game. If you pray every day, you get points. If you forget to pray, you lose points. If by the end of your life you gain enough points, then after you die you go to the next level of the game (aka heaven).

Okay, look, I'm no fan of organized religion, but even I think this may be taking things a bit too far.

The idea of finding meaning in life by playing virtual reality games is of course common not just to religions, but also to secular ideologies and lifestyles. Consumerism too is a virtual reality game. You gain points by acquiring new cars, buying expensive brands and taking vacations abroad, and if you have more points than everybody else, you tell yourself you won the game.

Okay, now you're stretching. Fine, I can see that some people think of things this way, but I'm not convinced.

In the end, the real action always takes place inside the human brain. Does it matter whether the neurons are stimulated by observing pixels on a computer screen, by looking outside the windows of a Caribbean resort, or by seeing heaven in our mind’s eyes? In all cases, the meaning we ascribe to what we see is generated by our own minds.

Saved. Now the author has come back around to what I've been saying for a long time: there is no meaning to life, save that which we impose upon it.

In any case, the end of work will not necessarily mean the end of meaning, because meaning is generated by imagining rather than by working. Work is essential for meaning only according to some ideologies and lifestyles. Eighteenth-century English country squires, present-day ultra-orthodox Jews, and children in all cultures and eras have found a lot of interest and meaning in life even without working. People in 2050 will probably be able to play deeper games and to construct more complex virtual worlds than in any previous time in history.

And that is an interesting point. I've never been on board with the Protestant work ethic, partly because I've never been Protestant, and partly because to me, the only purpose of work is to accumulate enough money to live with all the basic necessities and a few luxuries (with the luxuries becoming more and more important with time).

Problem is this: Back in the 1970s, when computers started to insert themselves into our daily lives, promises were made. These promises took the form of something like: "As computers become more prevalent, our productivity will increase. Paperwork will become a thing of the past, and we could work for no more than 10 hours a week."

Oh, you sweet summer disco children... how wrong you were. Okay, maybe paperwork has become more electronic, but instead of maintaining our 1970s productivity with a reduction in time spent working, we've expected greater and greater productivity... and, worse, we're expected to work more than the standard 40 hours a week to get shit done.

So I take projections like this with a grain of salt. Society, especially a society as rooted in Puritanism and work-worship as ours is, won't let its members have significant downtime. "Idle hands do the devil's work," as the saying goes. No, this utopian vision of the future will never pan out.

Besides, we probably won't be around long enough to give it a chance to do so.

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