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 6 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 8 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.

August 13, 2020 at 12:06am

#990638

On Merit

Meritocracy. Sounds good, right? Maybe think again.

https://aeon.co/ideas/a-belief-in-meritocracy-is-not-only-false-its-bad-for-you

A belief in meritocracy is not only false: it’s bad for you

Meritocracy has become a leading social ideal. Politicians across the ideological spectrum continually return to the theme that the rewards of life – money, power, jobs, university admission – should be distributed according to skill and effort.

Which, on the surface, certainly sounds ideal. Right? To each according to one's contribution.

Most people don’t just think the world should be run meritocratically, they think it is meritocratic. In the UK, 84 per cent of respondents to the 2009 British Social Attitudes survey stated that hard work is either ‘essential’ or ‘very important’ when it comes to getting ahead, and in 2016 the Brookings Institute found that 69 per cent of Americans believe that people are rewarded for intelligence and skill.

"Should be" is one thing. "Is" is quite another.

Although widely held, the belief that merit rather than luck determines success or failure in the world is demonstrably false.

Hey, I'm living proof of that.

This is not least because merit itself is, in large part, the result of luck. Talent and the capacity for determined effort, sometimes called ‘grit’, depend a great deal on one’s genetic endowments and upbringing.

Okay, but "everyone has different genetics and background" is so obvious as to be not worth pointing out in most cases.

Luck intervenes by granting people merit, and again by furnishing circumstances in which merit can translate into success. This is not to deny the industry and talent of successful people. However, it does demonstrate that the link between merit and outcome is tenuous and indirect at best.

I've been saying this for years: if hard work was all it took to become wealthy, there would be no poor sharecroppers.

But here's the important part:

In addition to being false, a growing body of research in psychology and neuroscience suggests that believing in meritocracy makes people more selfish, less self-critical and even more prone to acting in discriminatory ways.

You may think you have that high-paying position on Wall Street because you deserve it based on all the work you put in and sacrifices you made. You might even be right. You're still (probably) a douche.

The article goes on to summarize some of the science behind this idea.

The ‘even playing field’ is intended to avoid unfair inequalities based on gender, race and the like. Yet Castilla and Benard found that, ironically, attempts to implement meritocracy leads to just the kinds of inequalities that it aims to eliminate. They suggest that this ‘paradox of meritocracy’ occurs because explicitly adopting meritocracy as a value convinces subjects of their own moral bona fides. Satisfied that they are just, they become less inclined to examine their own behaviour for signs of prejudice.

I've noticed similar things (unscientifically, of course). People I've known who are convinced they are "good" act like bigger assholes than those who recognize that they're flawed.

As with any ideology, part of its draw is that it justifies the status quo, explaining why people belong where they happen to be in the social order. It is a well-established psychological principle that people prefer to believe that the world is just.

"That person is poor. It must be because they deserve to be poor. No need to do anything, then."

Anyway, obviously, the essay here is one person's opinion. I'm not so sure about the ultimate conclusion ("it ought to be abandoned both as a belief about how the world works and as a general social ideal") As with most things, I think there's probably a middle ground. I'd say, at the very least, we should try to recognize when we're just lucky, and, obviously (at least to me), stop thinking that the world as it is is just and fair.

That doesn't mean we can't try to make it more just and fair, though.

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