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:

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').

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

September 24, 2022 at 12:02am

#1038061

Toxic Positivity

Today's article has about as much relevance to me as that going (shudder) outside one a couple of days ago. But other people probably need to see it.

Don’t insist on being positive – allowing negative emotions has much to teach us
Leaning into difficult feelings can help you find the way forward, according to a refreshing new wave of books, says Jamie Waters

As we all know, I really am positive: for example, I'm positive we're doomed.

Eight years ago, when Whitney Goodman was a newly qualified therapist counselling cancer patients, it struck her that positive thinking was being “very heavily pushed”, both in her profession and the broader culture, as the way to deal with things. She wasn’t convinced that platitudes like “Look on the bright side!” and “Everything happens for a reason!” held the answers for anyone trying to navigate life’s messiness.

"You have cancer? Look on the bright side! You'll be dead soon and won't have to worry about it!"

As for "everything happens for a reason," fuck that noise. I mean, sure, you can find reasons if you look hard enough, or make them up for yourself, but no, some things are basically a roll of the dice.

This stayed with her and, in 2019, she started an Instagram account, @sitwithwhit, as a tonic to the saccharine inspirational quotes dominating social media feeds. Her posts included: “Sometimes things are hard because they’re just hard and not because you’re incompetent…” and “It’s OK to complain about something you’re grateful for.”

I'd almost visit Instagram just to see that. Almost. Still, some things are hard because you're incompetent.

Goodman’s new book, Toxic Positivity, expands on this thinking, critiquing a culture – particularly prevalent in the US and the west more broadly – that has programmed us to believe that optimism is always best. She traces its roots in the US to 19th-century religion, but it has been especially ascendant since the 1970s, when scientists identified happiness as the ultimate life goal and started rigorously researching how to achieve it.

Ah. Now I begin to understand why I have such a strong negative reaction to positivity. Other than, you know, the whole "universal balance" and "symmetry" thing.

More recently, the wellness movement – religion for an agnostic generation – has seen fitness instructors and yogis preach about gratitude in between burpees and downward dogs. We all practise it in some way. When comforting a friend, we turn into dogged silver-lining hunters. And we lock our own difficult thoughts inside tiny boxes in a corner of our brains because they’re uncomfortable to deal with and we believe that being relentlessly upbeat is the only way forward. Being positive, says Goodman, has become “a goal and an obligation”.

Or as I like to put it, every silver lining has a cloud.

Toxic Positivity is among a refreshing new wave of books attempting to redress the balance by espousing the power of “negative” emotions. Their authors are hardly a band of grouches advocating for us to be miserable.

And just to be clear, I'm not advocating that, either. Going all the way in the other direction to overcompensate is a very American thing (see also: McMansions vs. tiny houses), but it's not my way. Besides, seeking out misery for the sake of misery also comes from religion.

The road to the good life, you see, is paved with tears and furrowed brows as well as smiles and laughter. “I think a lot of people who focus on happiness, and the all-importance of positive emotions, are getting human psychology wrong,” says Paul Bloom, a psychology professor at Yale and the author of The Sweet Spot, which explores why some people seek out painful experiences, like running ultra marathons and watching horror movies. “In a life well lived, you should have far fewer negative than positive emotions, but you shouldn’t have zero negative emotions,” adds Daniel Pink, the author of The Power of Regret. “Banishing them is a bad strategy.”

Or as the great philosopher put it, "If everything was cool, and nothing sucked, how would we know what was cool?"

It’s tougher making an argument for regret, which might be the world’s most maligned emotion, but Pink is game. From a young age we are instructed to never waste energy on regrets. The phrase “No regrets” is inked into arms and on to bumper plates and T-shirts.

I have no idea if it's real or a Photoshop job, but there's a picture circulating of some dude who got "No Regerts" tattooed on his skin.

“Regrets clarify what matters to us and teach us how to do better. That’s the power of this emotion – if we treat it right.”

I've spent my life toiling in the regrettium mines of Regrettistan, so I can dig it.

Telling others about it lightens the weight. Complaining is perfectly natural, says Goodman. And articulating it helps us pinpoint what it is that’s bothering us, because language converts this “menacing cloud” into “something concrete”, says Pink. That disclosure could be to a friend, therapist or total stranger.

Just first make sure the person you're kvetching to isn't a positivist. Or, you know, start a blog.

Your next step will likely depend on the nature – and severity – of the emotion. To help us sit with sadness, Russell advocates being in nature.

Does looking out a window count?

“It sounds a little ‘woo’, but there are lots of studies about the effectiveness of reading therapy and looking at a piece of art – and how music can change our moods,” she says. “Sad music can act as a companion when we’re feeling sad, rather than making us feel lower. I do think it’s liberating when you finally kind of surrender to it all.”

This is what I already do. Happy music makes me stabby. Sad songs make me happy. As they put it on a well-known episode of Doctor Who: "What's wrong with sad? It's happy for deep people."

Leaning into negative thoughts should ultimately leave you with a sense of fulfilment. While we might instinctively think that filling our days solely with joy and excitement is the dream, “if we want to live a meaningful and purposeful life, a lot of pain is going to be part of it”, says Bloom. “What I really want is for people to be able to enjoy the full range of the human experience,” adds Goodman.

Also, booze works for me. I know, I know, you're not supposed to say that, because, supposedly, alcohol is a depressant. But I've never been happier than when I'm sitting at a bar or on a patio, having a beer (or something else with ethanol), listening to depressing music.

Maybe I'm just weird. But I don't think I'm that weird.

I know I've shared this page before, but it's relevant here, and one of my favorite image series is Fitness quotes over pictures of drinking.

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