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:

tj-Merry Mischief Maker

about 7 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

about 12 hours ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

about 14 hours ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

a day ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

a day ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

April 18, 2024 at 10:15am

#1069054

Cash Stash

This Guardian article should be uncontroversial.

‘Outdated and misleading’: is it time to reassess the very concept of money?
It’s regularly being created and destroyed – and economic models that don’t reflect that fact are not even slightly useful

...not.

The article itself, which begins with the simplistic concept of banking enunciated in the well-known movie It's a Wonderful Life, isn't as amenable to out-of-context quotations as some of the stuff I post here. I haven't seen that movie in decades, by the way, but I do remember the scene in question: George Bailey (James Stewart) dealing with a potential run on the bank.

When you borrow money and your bank credits your loan account, the account balance is created anew, “from thin air”, not from or in relation to existing deposits or other existing money. And as you repay the loan principal, the money created at the time of the loan gradually disappears, reverting to its previous form of airy nothingness.

I'm no economist money talker person, but to me, that just sounds like someone figured out that negative numbers exist and can be operated upon just like positive numbers.

Your deposit account is a liability for your bank and, as a depositor, you are no more than one among many of the bank’s unsecured creditors.

Leaving aside for the moment the mistaken assumption that everyone has enough unspent money to even have a deposit account, this has been the way it works for at least as long as I've been alive. Which, of course, doesn't mean it's the only way to do things.

In normal times a promise from a private bank is nearly as good as a promise from a government or a central bank. But in a crisis the promise is worth much less, and can be worth as little as nothing at all.

That may be true in the UK, where this source is based (I don't know), but here, we have federal deposit insurance for that sort of thing, as a result of the Great Depression.

If money is as unreal and ephemeral as they claim, though, a run on the bank shouldn't make a difference. The bank, or its insurance agency, could just conjure more out of the same "thin air" that they do when you get a loan. That it does make a difference tells me there's something incomplete about this article's assertions.

Details aside, the US and UK banking and government systems aren't all that different from each other where finance is concerned, so I'm not saying this article is completely useless on my side of the pond.

There's a bit more I can't really quote out of context, then:

Crucially, society as a whole needs to think differently about the nature of money –possibly by first discarding the term itself. “Money” encompasses a range of phenomena that have intrinsically different purposes and risks.

Well, yeah. I've known that for a while. Look up money supply.

Oh, wait, no, I just did that so you don't have to.

The article's conclusion:

A new financial conversation would make the causes of inter-generational inequities more explicit, and it would allow the community to rethink allowing “too big to fail” banks to earn, year in year out, enormous risk-free profits.

There may be ways to accomplish these seemingly-worthy goals without drastically switching to what appears to be a Star Trek economy, which, in truth, would never work without access to near-unlimited free and clean energy. Not that it's ever properly explained in the Star Trek universe, except for maybe holding up the capitalistic Ferengi as counterexamples to it.

The truth that no one seems to acknowledge is that our current system requires an underclass. Okay, that might not be a "truth," but only my opinion, and, like I said, I'm not an expert. The underclass doesn't have to be a particular demographic, though, sadly, it has been. Gotta have people desperate enough to eat and otherwise stay alive, so they'll take the crappy but necessary jobs that almost no one would freely choose.

That's not freedom. That's economic coercion. But any suggestions to improve it are above my pay grade.

Pun intended.

6 Like • Comment • View 1 comment

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