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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Brandiwyn🎶

a day ago
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').

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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

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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

October 17, 2022 at 12:02am

#1039313

Working Easy

What the hell is hard work, anyway?

The law of reversed effort: The harder you try, the harder you fall
There are many things in life that cannot be improved with greater effort. Sometimes, life requires that you step back.

Is it meant to mean great physical exertion, as in the hard work of keeping a farm going? Or something mentally difficult, like proving something in mathematics? Neither is any guarantee of success.

You’re lying in bed, staring at the ceiling, listening to cars go by. You have no idea how long you have been like this, but it must be a few hours, at least. Go to sleep, you tell yourself. Just close your eyes and: Go. To. Sleep. You shut your eyes tight, force your body to relax, and wait for the blissful slumber to come. But, nothing happens. More minutes pass and… nothing happens. It’s 3 a.m., and you’re still staring at the ceiling.

I learned how to deal with that years ago. No, drinking isn't involved.

We have all been in this situation. Try as we might, it is nearly impossible to consciously will yourself to sleep. Sleep comes to those who let their mind wander and focus on anything other than sleep itself. Count sheep, control your breathing, listen to an audiobook, or whatever — so long as it turns your mind from wanting to sleep.

Or I'd get up and do something else, which would make me tired, so I'd go back to bed, which would keep me awake, so I'd get up and do something, which would make me tired, etc. Then I'd fall asleep at my desk all day.

That's my body telling me I'm a night owl.

This is a common and familiar example of the “law of reversed effort.”

Yet, whenever I make an effort, I get tired.

The Law of Reversed Effort was first coined by the author Aldous Huxley, who wrote:

“The harder we try with the conscious will to do something, the less we shall succeed.

“Proficiency and the results of proficiency come only to those who have learned the paradoxical art of doing and not doing, or combining relaxation with activity, of letting go as a person in order that the immanent and transcendent unknown quantity may take hold.”

I guess it worked for him. He sold a few books, as I recall.

But, there’s a spiritual or holistic way of viewing the “law of reversed effort” as well. It’s something that has a much longer history than Aldous Huxley — it’s the Daoist idea of “Wu Wei.”

I was going to say that makes sense and I understand it, but it doesn't and I don't. I'm pretty sure that's what Taoism is all about though, so I guess I'm doing it right. Which means I'm not.

To surrender to a greater power — or a nobler, righteous one — is not an act of cowardice. It is an act of profound wisdom. There is nothing praiseworthy about swimming in a storm or punching a bear in the face. There is wisdom in knowing our limits, in embracing humility, and even in being pushed around.

This is the meaning of Wu Wei. It is not some lazy torpor, or an excuse for a duvet day and Netflix binge.

Sure it is.

That’s nice, you might think, but how does that actually translate to real life? The problem with a lot of philosophy of this kind is that it rather leaves us no better off than before.

And usually worse off.

Anyway, here's why I decided to share this:

Writing: For an author, there is nothing so terrifying as the blank page. If you have been told you have to write something, especially on a deadline, the mind often can go into meltdown grasping for something — anything — to write. It’s much better to let ideas come and write them in a notebook so they don’t get lost.

I'm sure that works for some people. Me, I do freewrites. And since I started doing these entries on a daily basis, I think maybe one in 400 entries gave me writer's block. It's even more trite than Tao, but sometimes you "just do it."

Stress and anxiety: We all get stressed about things. All jobs involve bottlenecks and crunch points. Life has good days and bad days. But when we obsessively run things over in our heads, we actually make anxiety worse. There is a reason why “mindfulness” is such a breakaway phenomenon, and why Headspace is a $250-million business.

Yeah. That reason is marketing.

Conversations: When it comes to how we talk to people, less really is more. A bad conversation involves you talking too much and your “listening” consisting of simply waiting to talk again. Yet research shows that active listening gives more “conversational satisfaction” and leaves the partner feeling more understood.

Oh, sorry, were you saying something?

Perhaps it is time to step away from what you are doing and enjoy Wu Wei or inaction. After all, if I tell you not to think of pink elephants, there’s only one way to do it.

Drink until they go away.

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