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:

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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

JayNaNoOhNo

Nov 5, 2024 at 1:16pm
In response to "Anomalies"

*Toronto, Canada submits a sternly-worded petition and disputes the center of the universe claim.*

2 Like

August 4, 2023 at 10:28am

#1053629

Leisure Seizure

I distinctly remember, when computers started taking over in the 1970s, predictions of how much less work we'd have to do, how much leisure time we would be granted because of these great labor-saving devices. How we'd be able to devote the extra time to creating poetry, art, philosophy...

‘Can I just … rest?’: guilt, the four-day working week and what to do with the fifth day

Around the world, the four-day week is inching towards grasping distance. For many, I am living the work-life balance dream. Living in some imminent utopian future in which, in return for decades of increasing productivity for relatively little increase in pay and in light of a global epidemic of stress, the four-day work week becomes the next great shift in the way we structure work and leisure.

Except that, from what I've heard, it's a shift from five 8-hour days to four 10-hour days.

This is OK, say economists and employers, because actually productivity does not decline when workplaces shift.

Whew. I'm so glad Holy Productivity is safe.

You know what would be utopian? John Maynard Keynes predicted a two-day work week.

Work two days, fish five. For the same standard of living. Like most economists, he was wrong, but it sure did sound good—to everyone, apparently, except the ones getting rich off other peoples' labor, and they're the only ones whose opinion counts.

Because as that fifth day opens up, one starts to feel the weight of its infinite potential and the questions start to pile up. What will you achieve on that fifth day? Can you justify non-work on a weekday? Can you stop work seeping to that day?

Maybe the only thing you need to achieve is non-achievement. Maybe you don't have to justify anything.

Not so long ago, being able to pursue or enjoy leisure for its own sake was a marker of your elevated place in the social hierarchy. Now our busyness, the demands on our time, writes sociologist Jonathan Gershuny, has become a key signifier of social status.

Oh, good, more proof that I'm on the very bottom of the social hierarchy.

Eighty years ago philosopher Bertrand Russell, in calling for a four-hour working day, suggested “a great deal of harm is being done in the modern world by the belief in the virtuousness of work”.

Work is its own punishment.

I would steadfastly not work-work – take no calls, check no emails – but I could not escape the feeling that I must be accountable for that time. The idea of rest came with a spectre of guilt.

I know this is just one person writing, but to me this sounds like a massive social problem.

A 2019 study, The Rest Test, led by psychologist and BBC radio presenter Claudia Hammond, found that for many people the prospect of resting was associated with anxiety and guilt.

Look, I come from a culture structured around guilt. My mom was a travel agent for guilt trips. No matter what I did as a kid, I was always "supposed" to be doing something else. I grew to hate the concept of guilt.

And so, I planned my day with precision.

In other words, you worked.

For no purpose other than my enjoyment of it. I leave the house and, from exactly 12:15pm to 1:15pm, I go to an exercise class I love.

Why would anyone need another purpose? Well, okay, I almost never do anything for just one reason. Exercise can be enjoyable and good for you.

So can rest.

Yet here we are, in the age of the most magnificent machines. Burnt out.

It could have been different. But social attitudes tend to change slowly, if at all, and there's that continued emphasis on the need to justify your existence by producing something for others.

I'm glad I got off that treadmill. To me, work was always about obtaining sufficient resources for myself that I would never have to work again. Which doesn't mean I never do things considered "work," but I do them on my own terms.

And yes, I know some people find work satisfying and meaningful. I'm not ragging on that at all. People are different. But what this author talks about sounds more like neuroticism than meaning.

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