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.

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').

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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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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

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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.

2 Like

October 3, 2021 at 12:03am

#1018576

Irrational Primacy Effect

Usually, people cling to the first thing they hear about a subject, ignoring or dismissing any further evidence. This is called the Irrational Primacy Effect, and it's a form of confirmation bias.

Nobody Really Wanted To Drink The Kool-Aid: 4 Big Misreported Historical Moments

It's worst, I think, when the first thing you hear already speaks to your inherent biases (and we all have them). Then, when you get evidence to the contrary, your first reaction is often, "bullshit."

I'm not going to go through this item-by-item, but it's a Cracked link, so at least there's some humor sprinkled in, which (for me anyway) helps to soften the blow of being told what I thought I knew is wrong.

History is so complicated that it’s arguably a minor miracle the average person knows as much as they do. Sometimes dumbing down complex events is enough to help us get the point of how it’s shaped the modern world; you don’t need a PhD to understand that the Nazi were bad. But sometimes simplifying accidentally leaves everyone with the completely wrong impression...

All of the things at the above link are subject to misreporting and simplification, and I think it's important to get a fuller picture, including facts learned later on in the process. Science and knowledge are supposed to correct as well as refine old ways of looking at things, as the section about Galileo hints at.

And we see it all the time, even now. Perhaps the worst example, both in terms of outright falsehood and deleterious effect on society, is the reported link between childhood vaccines and autism. Leaving aside for the moment that believing this is basically saying, "I'd rather have a dead child than an autistic one," which is a big slap on the face to anyone on the spectrum, the entire report was completely fabricated in order to sell a book. Not only is there absolutely no link between vaccination and autism, but the person who perpetrated the hoax in the first place ended up losing his license. I won't name him, because I don't like giving oxygen to frauds, but you probably know who I'm talking about.

And yet, you still have people clinging to that complete falsehood. They might say, "why take chances?" Well, because the alternative is sick or dead kids, and more epidemics. They might say, "But my kid was fine until he got vaccinated, and then he became autistic." If true, that's coincidence, not causation.

There are, of course, likewise falsehoods being spread about other, more recently developed vaccinations. This is not a matter of people having different opinions; it's about some people simply clinging to the wrong set of facts.

None of us should feel bad if we fall victim to the irrational primacy effect, though; it can -- and does -- happen to everyone. But once you know about it, you have a mental check on it. Not that every bit of contradictory information suddenly changes the truth, so a healthy skepticism is still required.

So yeah, if you want humor, read the link. I'm entirely too frustrated with willfully ignorant idiots to even make jokes right now.

Mini-Contest Results!

Thanks for all the ideas for replacing the utterly despicable "revenge bedtime procrastination" label. Honestly, I thought all of them were better than the original. I'm going to award the one by Write_Mikey_Write!

for its appropriate acronym: TIMES UP. (See yesterday's entry's comments for what that is, and for the other good ideas.)

I'll do this again soon, so there will be more chances at a Merit Badge!

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