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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 17 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.

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.

March 24, 2020 at 12:15am

#978997

Wisdom of the Waltz

Waltz is not sure what to make of this.

https://aeon.co/ideas/why-speaking-to-yourself-in-the-third-person-makes-you-wis...

Why speaking to yourself in the third person makes you wiser

Simple rumination – the process of churning your concerns around in your head – isn’t the answer. It’s likely to cause you to become stuck in the rut of your own thoughts and immersed in the emotions that might be leading you astray.

Waltz doesn't know anybody like that.

Instead, the scientific research suggests that you should adopt an ancient rhetorical method favoured by the likes of Julius Caesar and known as ‘illeism’ – or speaking about yourself in the third person

A lot of us are worried about illism now. Oh, illeism. Nevermind.

The idea is that this small change in perspective can clear your emotional fog, allowing you to see past your biases.

Or, just spitballing here, maybe focusing on changing a particular habit, such as referring to oneself in the first person exclusively, can help you break out of an emotional rut.

Grossmann’s aim is to build a strong experimental footing for the study of wisdom, which had long been considered too nebulous for scientific enquiry.

Good, because priests are just as important to an adventuring party as mages.

Grossmann found that these wise-reasoning scores were far better than intelligence tests at predicting emotional wellbeing, and relationship satisfaction – supporting the idea that wisdom, as defined by these qualities, constitutes a unique construct that determines how we navigate life challenges.

Waltz has known for a long time that intelligence isn't correlated with emotional wellbeing, or Waltz would be one of the happiest people on the planet.

The article goes on to describe some of the studies involved.

I find these emotion and relationship effects particularly fascinating, considering the fact that illeism is often considered to be infantile.

It always makes Waltz think of D&D.

Alternatively, it can be taken to be the sign of a narcissistic personality – the very opposite of personal wisdom.

Or, perhaps, narcissists are the only ones who have it right.

Previous experiments have shown, for instance, that rumination leads to worse choices in poker (hence why expert players strive for a detached, emotionally distanced attitude), and that greater emotional awareness and regulation can improve performance on the stock market.

Oh, like that's going to help NOW.

...actually, it probably would. Wisdom says "don't sell now that everything's tanked," but emotions are like "SELL, MORTIMER! SELL!"

It is notoriously difficult to increase general intelligence through brain-training, but these results suggest that wiser reasoning and better decision making are within everyone’s power.

Waltz is not holding out any hope that people will suddenly become wiser by referring to themselves in the third person. Or the second.

3 Like • Comment • View 2 comments

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