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 2 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 4 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.

August 28, 2021 at 12:02am

#1016259

Money Baloney

Thanks for the MB, Andy~hating university !

Today we're going back to that trusted source for all things financial, Cracked.

5 'Truths' About Business And The Economy That Are BS

Lots of people get money things wrong. Like when they insist they'll take home less money if they move to a higher tax bracket. Well, that might happen if tax laws change year to year, but no, otherwise that can't happen. Or when people conflate "income" with "wealth." Someone could make a million dollars a year and still have nothing left over; contrariwise, you could make 100K a year and save half of it.

But I think this article is more general than that.

Our lives are one long timeline of discovering everything thought we knew about money is wrong. Early on, we're shocked to learn fairies don't really hand out money for teeth but people do voluntarily exchange money for vegetables.

Actually, I knew that last bit from an early age, as we sold our surplus vegetables.

5. Everything People Think About What Companies/Billionaires Are Worth

In general, any given thing is "worth" exactly what people are willing to trade for it, usually denominated in some currency.

And what about those richest people in the world, going from net worths of $100 billion to more like $200 billion in no time at all. Where's all that money coming from?

Built on the backs of labor. Rise up and- oh, wait, no.

They're about stocks, and stocks are a special kind of made-up value. Like, you might say that all kinds of value are made-up, but stocks are super made-up, and a lot of people don't get this.

The article goes on to explain the stock market in what I consider simple terms. But fair warning; there is math involved. But in conclusion (for this first one, anyway):

But you don't have to get extra worked up over the terrifying spectre of unrealized capital gains. You're worrying about money that doesn't actually exist.

And that's the -- pun intended -- money quote. Capital gains aren't taxed until they're realized. We can argue about how much they should be taxed, but keeping one's money in the stock market isn't "evading taxes;" it's kicking the tax can down the road.

4. "You Can't Pay Employees $15 An Hour And Profit!"

"We won't make money if we pay employees more!" say companies. And the general populist response to this is "lol, fine." Higher wages are about letting employees live better, not about keeping profits high, right?

Again, the article goes into the math involved. But my own take on that nonsense: If you pay people more, they will have more income (duh) and they will be able to spend it on products and services, thus keeping the economy rolling.

3. "All Of Our Tax Money Goes Into Defense"

"We'd have enough money to pay for health care for everyone and wiping out debt and universal basic income and more if we only trim that ridiculous defense budget," people say. And the US does spend a lot on defense, more than the next several countries combined. But did you realize that the government spends more on Medicare than on the entire Department of Defense? Going by 2020 numbers, it's $917 billion on Medicare and $714 billion on defense.

And to be clear, we should spend money on defense. And we also should spend money on health and infrastructure. The issue is more complicated than I (or Cracked) can really grok.

For example:

So if you're wondering why Bruce Wayne wastes money on batarangs rather than solving crime by lifting all of Gotham out of poverty, maybe it's because his fortune can only go so far in a city of 10 million. Or, if you're wondering why Bill Gates doesn't just end world hunger tomorrow, well, we'd love to hear what your plan is for permanently feeding billions of people using only tens of billions of dollars. The government would probably like to hear that as well. They have tens of billions too.

2. "College Students Are So Narcissistic Today You Can't Hire Them"

The generation coming out of college right now harbors deep self-loathing. A constant need for approval doesn't signal a high opinion of oneself but rather the complete opposite. Many youngsters today don't even want to be alive. They're bitterly desperate to be hired and will even utterly debase themselves in the process.

Which is, like, the polar opposite of narcissism.

Every generation ends up complaining about "kids these days." We have short (and selective) memories.

1. The Myth Of "Self-Made" Fortunes

Few things get our eyes rolling harder than hearing a newly rich entrepreneur described as a "self-made millionaire" or "self-made billionaire." Because outside of the very rare exception (say, Oprah), these people didn't start from humble beginnings at all, did they?

That is, of course, a rhetorical question.

A self-made billionaire is someone who started their own business and grew their money to a billion, even if they had some money to begin with. If you start out with $1 million and grow it to a billion thanks to the company you start, you are a self-made billionaire.

It's a matter of definition. I figure if you generally grow your net worth, or at least have a plan when you don't, you're in good financial shape. I don't worry much about "self-made." The only time it matters to me is when someone claims to be "self-made" when they, as the saying goes, scored a home run while starting on third base. It sends the wrong message, among other problems.

On the other hand, if you put a bunch of cash into Bitcoin or other Dunning-Kruggerrands a while back, you may have a few million bucks on paper, but the above comments about unrealized capital gains apply.

And then there's the author:

For what it's worth, if I inherited $500 million, I would never manage to grow it to $1 billion or more. I would immediately spend the bulk of it creating Robot Island, and I'd deliberately run the place using a nutty business plan that loses money, as I build ever more elaborate singing animatronics. What do I care about becoming a self-made billionaire? I'm rich.

And you certainly wouldn't be writing for Cracked.

Anyway, the article's a good read, with the usual humorous take on things, but still largely factual.

4 Like • Comment • View 4 comments

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