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:

Turkey DrumStik

about 5 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

December 6, 2018 at 12:33am

#946926

Expert Opinion

I wanted to share this, not because I have a stake in the topic being discussed, but because my lack of a horse in the race means I have some tangential thoughts to share.

https://www.nytimes.com/2018/11/27/well/family/the-fallacy-of-the-i-turned-out-f...

The column is ostensibly about child-spanking. To summarize, a soi-disant "parenting expert" (

) from Down Under asserts that spanking should be a criminal offense (or offence or however they spell it in Oz), is nonplussed when a bunch of people disagree with him, and then goes on to demolish what he sees as their arguments.

I'm not overly familiar with the political situation in Australia, but first of all, it seems to me that "this practice is ineffective and probably dangerous" shouldn't automatically translate to "this practice should be a criminal offense." It's dangerous in itself to turn everything that we don't like into ban laws; also, a law of this sort strikes me as being especially easy to abuse. It's a basic idea in a so-called free country that not everything is either "permitted" or "banned." Some bad ideas can be handled with social pressure. For instance, I dare you to speak out against public breast-feeding. Go on. I triple-dog-dare you. There's no law against speaking out against public breast-feeding, nor, by our right to freedom of speech, should there be. But I guarantee that you will face a typhoon of scorn and ridicule that will make you wish you had never been born, let alone breast-fed.

I also wanted to point out that there is a such thing as making things too safe, for kids as well as for adults. Take away all the world's sharp corners, foam-pad all the walls and floors, and rope off anything that might pose even the slightest hazard to people, and that's what people will come to expect. They'll be wired to think that if something might be dangerous, there will be a barricade. And that if there is no barricade, it can't possibly be dangerous. Or, worse - you rope off areas of both minor and major hazard, so people get complacent about ignoring barricades. And that's how we get things like people sliding into the Grand Canyon

or boiling to death at Yellowstone.

Life, in short, is full of risk, and there may be paradoxically more risk in protecting people all the time than in allowing for the occasional bump on the noggin. Or, to stay more on topic, the occasional light swat on the tuchis.

Unfortunately, if you don't foam-pad everything lawsuits happen, so the world just gets more and more covered with the illusion of safety.

Getting back to the article I originally linked, however, I'll just point out one fallacy that the author didn't even consider, which is the survivorship fallacy. "We didn't have car seats when I was a kid and I'm still alive." He covers that as "anecdotal fallacy" but doesn't go far enough. It doesn't take into account the loose kids bouncing around in station wagons who didn't make it to adulthood. Such a logical fallacy doesn't even require hazardous situations. You see it all the time in articles with such titles as "The Five Traits of a Successful Business Leader." The implication is that anyone with these traits will be successful, and completely ignores the vast majority of people with those traits who are standing in line at the food bank.

Well, that's enough for now. But I did want to say, one more time: "parenting expert."

4 Like • Comment • View 2 comments

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