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:

Write_Mikey_Write!

about 5 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 7 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').

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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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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.

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