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.

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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.

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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.

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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.

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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.

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October 29, 2024 at 9:35am

#1079108

The Numbers Don't Lie

From Big Think, an article about how the numbers sometimes lie:

The “McNamara fallacy”: When data leads to the worst decision
Don’t make the mistake of blindly following quantitative metrics — whether you’re helping clients or looking for lunch.

Or do they?

In 2014, I took a weekend break to York.

You know what sometimes lies? Anecdotes.

York is a lovely city in the north of the UK, with an ancient cathedral, quaint cobbled roads...

Oh, hell no.

...and an interactive Viking experience.

Is that where you get hung by the ankles and forced to watch your clan murdered and village pillaged while you're upside-down?

The reason for the anecdote is quickly revealed: the author made the mistake of trusting online reviews whilst choosing a restaurant, Skewers, which turned out to have high ratings due to drunks lovingly appreciating their late-night-early-morning fare.

Let's be clear, here: This is not a failure of TripAdvisor. It's not a failure of the food establishment. It's certainly not a failure of the well-meaning drunks. The blame lies entirely on the author. I travel quite a bit, and I prefer to form my own opinions of places I visit, rather than relying on others' (sometimes real, sometimes not) experiences. Sometimes, it sucks, but usually, I'm pleasantly surprised. Either way, I get to write reviews of my own, which may or may not factor into someone else choosing to visit.

The story of Skewers is an example of the McNamara fallacy, and learning about it can help us all (especially the underprepared tourists among us).

In spite of my misgivings about the opening anecdote, I'm willing to read on.

Fortunately, the author again gets quickly to the point, which is how I know I'm reading Big Think and not The New Yorker:

The McNamara fallacy is what occurs when decision-makers rely solely on quantitative metrics while ignoring qualitative factors.

So, like, when you rely only on the studies that claim booze is bad for you, and ignore how good it makes you feel.

The fallacy is named after Robert McNamara, the U.S. Secretary of Defense during the Vietnam War, where his over-reliance on measurable data led to several misguided strategies where considering certain human and contextual elements would have been successful.

I'm no expert on war history, but claiming that different decisions "would have been successful" strikes me as arrogant. I'd weasel out with "might have been successful," instead.

The McNamara fallacy is not saying that using data is bad or that collecting as much information as you can is wasted time.

I'm just leaving this here lest someone be thinking, "What's the point in data, then?"

For example, it’s not uncommon for someone to deeply love a book that has few or no reviews on Goodreads. It’s possible to enjoy a restaurant or a movie in spite of what others say. Data is a great starting point, and a great many idiotic and dangerous things are done when we ignore data, but it doesn’t always make for the best decisions.

I'm very familiar with that assessment, anyway. People are different, and you might love something others hate, or vice-versa. Reviews and the like are aggregations, and blindly following them is roughly the same thing as blindly following the crowd. You can do that if you want, but I don't, because that way, I miss out on the joy of making my own discoveries.

If everyone liked the same things, there'd only be one beer, and that would be a boring world indeed.

I have a much bigger problem with the explicit comparison of "finding a place to eat" and "running a war where people die."

Still, I suspect the basic ideas of the fallacy are sound, and the article goes into other examples and applications thereof, which I don't feel the need to reproduce here.

Fear not, though; it's short, and there are very few actual numbers. So, a solid four stars out of five from me. Your experience may vary.

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