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 14 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 16 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.

April 19, 2022 at 12:01am

#1030967

Save Game

Today's link will probably only make sense to people in the US. Or, more accurately, some people in the US.

401(k) Plans No Longer Make Much Sense for Savers
The inherent extra return participants enjoyed for many years has almost disappeared because of changes in tax laws and high fees.

So if you don't know what the hell this is all about, congratulations.

The tax advantage of a 401(k) depends on four factors, all of which have changed dramatically since 1980 to the detriment of 401(k)s.

I'm certainly not going to argue the numbers here; they seem right. I do wonder if one can truly extrapolate the average situation here, as the author does, to give advice to everyone.

Nevertheless, the reason I'm sharing this article isn't to nitpick it, but to point out something that I should formalize into a Law of Life, though I'm not quite sure yet how to boil it down to bumper-sticker philosophy (or even if I should)

That observation is this:

Any advantage the ordinary person might enjoy will promptly be socially engineered out of existence.

There's also this bit from the article:

Reducing taxes on 401(k)s will cost the government money, but it could be a good investment if it results in retirement security for more middle-income households.

Except that it would be an even better investment for the government to ensure that people don't retire, so they can continue to pay higher taxes until death.

When I was a kid, saving was considered a virtue. Not that I was any good at practicing that or any other virtue, but at some point, most likely coinciding with the 1980s, we collectively decided that saving was for suckers, and to keep the economy going, we should spend, spend, and spend some more. And then borrowing money became easier for most people, so the economy retooled itself again. Can't afford something? No problem, just go into debt. We offer easy finance terms!

Younger people today don't expect to ever retire anyway, so why bother saving anything?

This, of course, plays right into the hands of our economic overlords.

So how do you get ahead? Well, I don't know. Some people never will, though it's important to hold out a carrot of hope to them, usually in the form of a lottery. Even without state-sponsored gambling, though, a lot of it comes down to luck.

The whole deal with any kind of retirement account, at least here in the US, is a trade-off: do I pay taxes now, or defer them until later? Deferring them until later requires that you make some assumptions about "later." What will inflation look like? How will tax brackets be handled, and where do I expect to be in them? Will the government cave in to short-sighted efforts to increase the capital gains tax rate? Will all of this effort be in vain because we're going to get nuked anyway?

If you asked these questions in 1982 about 2022, you'd have gotten the answers wrong. Back then, we all figured we'd either be radioactive craters by now, or populating a Mars colony. Or maybe even both.

The reality, as always, turned out to be way worse.

5 Like • Comment • View 4 comments

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