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 16 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 18 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.

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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.

March 11, 2020 at 12:03am

#977747

If You Don't Mindful

I've railed against this "mindfulness" crap in here before. Sometimes I know why I don't like something. Other times, it's a vague feeling and it takes another writer to put into words what my issues are. This is a case of the latter.

https://www.theguardian.com/lifeandstyle/2019/jun/14/the-mindfulness-conspiracy-...

The mindfulness conspiracy
It is sold as a force that can help us cope with the ravages of capitalism, but with its inward focus, mindful meditation may be the enemy of activism.

Prophesying that its hybrid of science and meditative discipline “has the potential to ignite a universal or global renaissance”, the inventor of Mindfulness-Based Stress Reduction (MBSR), Jon Kabat-Zinn, has bigger ambitions than conquering stress.

"Hybrid of science and meditative discipline?" Where's the "science?" Is it studies that show that meditation has beneficial effects? Yeah, we've known that for a long time. I'm not doubting that there's some benefit.

So, what exactly is this magic panacea?

Magic = not science. Pretty much by definition.

The accompanying feature described a signature scene from the standardised course teaching MBSR: eating a raisin very slowly.

This shit again. Be the raisin, Danny.

But anything that offers success in our unjust society without trying to change it is not revolutionary – it just helps people cope.

And here we get to why I have a knee-jerk reaction to "mindfulness."

Instead of encouraging radical action, mindfulness says the causes of suffering are disproportionately inside us, not in the political and economic frameworks that shape how we live.

The "suffering" bit betrays mindfulness's Buddhist roots.

Although derived from Buddhism, it’s been stripped of the teachings on ethics that accompanied it, as well as the liberating aim of dissolving attachment to a false sense of self while enacting compassion for all other beings.

Just as science without ethics is dangerous, religion without ethics is bleak and self-serving. I don't know enough about Buddhist ethical teachings to comment on the differences, myself, so I can't really speak to the truth of this quote.

Instead of setting practitioners free, it helps them adjust to the very conditions that caused their problems. A truly revolutionary movement would seek to overturn this dysfunctional system, but mindfulness only serves to reinforce its destructive logic. The neoliberal order has imposed itself by stealth in the past few decades, widening inequality in pursuit of corporate wealth. People are expected to adapt to what this model demands of them. Stress has been pathologised and privatised, and the burden of managing it outsourced to individuals. Hence the pedlars of mindfulness step in to save the day.

The way I see it, one of the aims of religion since there has been religion has been to preserve the social order by teaching people to accept their place in it, even if that place is unjust. "You are a slave because that's just the way things are. Pray to this god to help you understand why this has to be so, and to make peace with it." Periodically, of course, a new religion crops up that is revolutionary -- and here, by "religion," I'm including systems such as Marxism. The problem with all of these revolutions, of course, is that once they've gone on long enough, they become the establishment, and the establishment proceeds to enforce its own social order.

Christianity, for instance, was revolutionary when it began. Then it took over the Western world and started to enforce conformity, urging, for example, peasants to accept their place at the bottom of the social food chain.

The fundamental message of the mindfulness movement is that the underlying cause of dissatisfaction and distress is in our heads.

Once again, we have the same old refrain as with weight issues or financial distress: it's really your fault, and you're the one who needs to change your mindset.

Rather than discussing how attention is monetised and manipulated by corporations such as Google, Facebook, Twitter and Apple, they locate the crisis in our minds. It is not the nature of the capitalist system that is inherently problematic; rather, it is the failure of individuals to be mindful and resilient in a precarious and uncertain economy. Then they sell us solutions that make us contented, mindful capitalists.

Capitalism is another "religion" as I use the term above.

There's a lot more to the article I'm quoting here; it's fairly long, and I think makes some good points. Whether you agree or not, it's worth the read. Whether it's worth buying the book it's promoting, well, I'm going to remain agnostic about that; I know I'm not buying it any more than I'm buying a course in mindfulness.

Bottom line as far as I'm concerned: sometimes it's best to reject the status quo and be a revolutionary. But be mindful that in doing so, all you're really doing is establishing a new social order.

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