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 24 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

a day 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.

December 19, 2019 at 12:08am

#971690

On Wealth and Economics

https://www.bloomberg.com/news/articles/2019-05-13/how-much-money-do-you-need-to...

Money is, apparently, a taboo subject. Few people admit to how much they make, let alone their net worth. And I'm no exception, even on the internet where I'm mostly anonymous.

So we rely on articles like this one, possibly comparing ourselves to the statistics presented therein. To me, this is akin to comparing your body to the models you see in magazines and movies.

How Much Money Do You Need to Be Wealthy in America?

Rich is relative.

Well, duh. Also, if your relatives are rich, chances are you are, too.

Merely having a net worth of $1 million, it seems, doesn’t mean you’re wealthy. In Charles Schwab’s annual Modern Wealth Survey, the amount people said it took to be considered rich averaged out to $2.3 million.

This, too, is hardly news (except maybe for the average value thing). It's been a long time since being a millionaire made someone feel "rich."

Even ignoring investment or other means of growing principal, a million bucks will give someone the equivalent of $50K a year for 20 years, or $100K a year for 10 years. Depending on where you live, this might seem like a lot, but the median household income in the US is somewhere in the mid-60s annually (yes, I looked it up; the exact value I found was $63,179 - and now you're comparing yourself to that.) And 10-20 years just doesn't seem like that long a time to me. Now, assuming investment and a commonly accepted safe withdrawal rate of 4%, one could eke out $40K a year on investments alone for at least half a lifetime.

Again, that may or may not be enough to live on if you live in a place like San Francisco or New York, but it's comfortable in the flyover states - where incomes tend to be lower.

More than three-quarters of [millennials] said their personal definition of wealth was really about the way they live their lives, rather than a discrete dollar amount.

I hate the generational warfare we've been fed in the US, and I consider generation names and their cutoffs to be entirely arbitrary, but this is one subject where there might be some value in assigning people to cohorts, so I'm running with it for now.

With 59% of the Americans surveyed saying they live paycheck to paycheck, instant gratification comes with a high price. While a strong economy and low unemployment are helping consumers stay current on their debt payments, the largest U.S. banks are seeing losses on credit cards outpace those of auto and home loans at a rate not seen in at least 10 years.

It's possible that the real problem here is debt, but I think it goes deeper than that. Most people don't go into debt for no reason. Sure, some are just bad with money, but there are externalities to consider. Health care is a big one, these days, but so are housing costs and the price of an education. And let's not forget advertising, which above all tries to sell a lifestyle. If you can just get a little more money, you'll finally be able to buy all the things you want and then you'll be content! Except you won't, because that big-screen TV you can afford keeps playing those ads...

Now, I'm not trained in economics, so take these observations with as much sodium chloride as you need (I do have some background in chemistry). But it seems to me that easy credit has the effect of pumping up the price of things.

Taking education as an example, it's well-known that tuitions have skyrocketed in recent decades, even when adjusted for general inflation (the measurement of which is a topic for a separate rant). This increase coincides with the general availability of student loans. The economic reason for this seems pretty clear to me: it's a manifestation of the classic supply and demand curves. If more people can afford higher education due to lending, and there are only a limited number of spots in higher education, that means the demand goes up while the supply is relatively constant. This causes the price to go up.

You can also apply this concept to housing, though it's somewhat different there because people tend to keep building dwellings (which kept me in dosh for most of my career). Back in the noughties, I saw that some lenders were starting to offer 60 year mortgages. The traditional mortgage had a 30 year term; a 60 year mortgage could, potentially, allow someone's monthly payments to be less than they would otherwise. So what happens? Price of housing goes up to compensate.

I wasn't smart enough (or perhaps I just didn't have enough information) to predict the ensuing crisis, followed by what's been termed the Great Recession, that occurred subsequent to this.

I don't know if they still offer 60 year mortgages. I'm not saying that this alone caused the housing crash, but it was a warning signal.

The situation with healthcare is different, but related. In that case, it's not easy availability of credit; there are no "medical loans" of which I'm aware. But insurance companies have deep pockets, and so hospitals can get away with charging $1000 for use of a tongue depressor. Consequently, if you don't have insurance, you're boned.

Hell, these days if you do have insurance, you're boned. Now that I'm going to have health insurance again, I have nightmares about things that could happen to me that they won't cover, meaning the exorbitant premiums I have to pay would be just money down the drain. And even though I'm not what you'd call poor, all it would take would be another heart attack to throw me into instant poverty, absent insurance coverage.

I've told my friends, if that happens, to just let me die. I'd rather die rich than live poor.

And when the bottom does finally fall out, the last thing most Americans will be thinking of is whether they qualify as wealthy.

Way to end the article on a positive note, there, Bloomberg. But you're right - we're boned.

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