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!

about 8 hours 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

about 10 hours 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.

January 23, 2024 at 10:21am

#1062853

Irresolute

This article, from The Guardian, is a couple of years old, though I doubt that matters. It's a New Year's article, so if you've failed at your resolutions (statistically, you have by now), maybe this can replace them.

Be bad, better – from anger to laziness, how to put your worst habits to good use
Forget new year resolutions and stop striving to be someone you’re not. It’s time to embrace your messy, imperfect, soft-bellied self

While much of it aligns with my personal philosophy, I have some objection to the wording in the headline—the point shouldn't be that you're being "bad" or "good," just you. I think.

We have the Babylonians to blame for making the new year a festival of self-flagellation – although their resolutions were more about appeasing gods than weight loss or cutting back on booze.

I'm finding it difficult to tell the difference there. Either way, it's a sacrificial rite.

John Wesley, the founder of Methodism, rekindled the idea in 1740, with an annual new year’s service of resolution – his included a promise never to laugh, which might explain why we also choose punishment over joy.

And yet people look at me funny when I say something bad about Protestant "morals."

What if, instead of being motivated by guilt and shame, we leverage our worst habits to serve us better? By being intelligently, purposely lazier; less mindful, disorganised, slower (and with a bit of self-compassion), we might actually be more successful, productive and happier – but on our own terms.

Yeah, that might or might not fly in England, but almost certainly won't in the US. We're all about guilt and shame here. You are bad and you need to do penance.

I'll be skipping a lot of this, because it's long and I'm lazy, but a couple of highlights for me:

Be brilliantly lazy

Kendra Adachi, podcast host and author of The Lazy Genius Way, is a productivity expert who isn’t necessarily into making people more productive.

And that's a lazy way to promote your books.

Learn to love negative emotions

When your cat dies and someone breezily says, “Never mind, you can get a new cat,” that’s toxic positivity. “Two things go wrong with toxic positivity,” says Robert Biswas-Diener, positive psychologist and author of The Upside of Your Dark Side. “One is relational – when your friend comes to you wanting support, what they want and what you offer has to match, but often it doesn’t.” When we try to cheer someone up who actually just wants to be heard, “toxic positivity feels invalidating”. We’ve all felt this – when a parent or partner wants to solve our problem instead of letting us talk about it.

Someone says "Never mind, you can get a new cat," or something similar, to me, then after they complain about where my fist went, I can say, "Never mind, you can get dentures."

And oh, look, more book promotions.

Here's the thing: I admit I'm one of those whose first response is to try to solve the problem. It's either why, or because, I'm an engineer. Maybe both. I try to resist this, but in the moment, I don't always. Someone comes to me with a problem, my brain just goes "must... solve... problem." But one thing I don't do, because I'm terrible at it, is cheer someone up.

I do try to make them laugh, though. Which isn't the same thing.

Be moderately disorganised

According to the book A Perfect Mess: The Hidden Benefits of Disorder, “neatness and organisation can exact a high price”.

ALL the book promotions!

Problem here is finding a balance. I'm too much on the "disorganized" (I'm spelling it American) end of the scale, but I'm entirely too lazy to do anything about it.

Freedman and Abrahamson point out that Albert Einstein’s desk – among many others – was always in “stupendous disarray”, so next time someone officious tells you to tidy up your desk (or your sock drawer), remind them that if one of the greatest thinkers of all time thrived in a semi-shambles, then so can you.

Um, yeah, no, Einstein trap there. What worked for him won't work for us regular schlubs.

Be less mindful

“Traditionally, meditation was never thought of as something to create a moment of calm in the middle of crisis,” says Dr Julieta Galante, a neuroscience researcher at Cambridge University, who studies the upsides and downsides of mindfulness and meditation. “That is a western repurposing of it.”

To be blunt, it's nearly pure copium.

Work less

“We now have a century of research that shows overwork is counterproductive,” says Alex Soojung-Kim Pang, a four-day week campaigner and author of Rest: Why You Get More Done When You Work Less.

Surprise! Another book.

Anyway, that's easy to say, and will often fall on angry ears, for good reason. This isn't on us. It's the system that needs to change, the one that requires people to work a lot just to maintain food and shelter.

The next one I'll highlight (still more book promotions) is the one I liked best, though:

Abandon meaningfulness

“The pursuit of meaning is not fundamentally a bad thing,” says Wendy Syfret, author of The Sunny Nihilist: How a Meaningless Life Can Make You Truly Happy. “If you want to spend your whole life in a monastery, or pursuing enlightenment, good for you. But meaning has become so intensely commodified that now everything around us has to have meaning: your job has to be meaningful, your relationship has to be groundbreaking and every single consumer product is presented as life-changing – I saw a pack of tampons the other day that said, ‘This box is a revolution.’” When looked at this way, it’s obvious that meaning is a construct – and if absolutely everything is meaningful, then, arguably, nothing is.

There's a kind of poetry in finding meaninglessness in meaning, and meaning in meaninglessness. I'm impressed.

As usual, many more suggestions appear at the link, each with its own lovely book someone's trying to sell. I've said it a hundred times, but for any newcomers or forgetful folk: I'm not going to shun an article just because it promotes books, not here on a site promoting reading and writing.

The article, as I said, came out as a New Year's thing, but to me it works better as a "You failed your resolutions; now what?" kind of thing. Which just so happened to come up for me at random in January.

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