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 17 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 19 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.

March 19, 2020 at 12:13am

#978494

Bulk Smash!

I'm sure you'll all use all that toilet paper. Eventually.

https://nymag.com/intelligencer/2020/01/buying-in-bulk-when-is-it-a-mistake.html...

When Buying in Bulk Is a Mistake

I like the way Andrew Tobias talks about bulk buying in The Only Investment Guide You’ll Ever Need: You can think of each bulk purchase as an investment, with a return equal to the savings you accrue by not paying a higher unit price for smaller purchases. If you lay out $110 today to buy a 12-bottle case of wine you’ll drink over the next month, instead of buying bottles one at a time for $10 each, that’s like earning a $10 return on a $110 investment in one month. On an annualized basis, that’s a rate of return well over 100 percent — far better than the stock market.

Stuff and nonsense.

For starters, as much as I like wine, I recognize that it's not a necessity. "If you lay out $110 today to buy a 12-bottle case of wine," then you have spent $110. While I agree that this is better than spending $120 on 12 bottles of wine, it is not an investment; it's an expenditure.

Also, this article is from January. Everything is far better than the stock market now (at least in the short term).

But that analysis relies on a few assumptions, the most important being that you will use the items you purchase. If you let products spoil, or you decide you don’t like them anymore halfway through the box, or if you forget what drawer your huge package of batteries is in, then you’re not getting as much value out of your bulk purchase as you had planned.

That's not the most important assumption; the confusion of "spending" and "investing" is. But going with it, using the wine as an example, if you're drinking a case of wine every month like the article stipulates, sure, that $10 you're saving is probably worth it. But contrary to popular belief, most wines don't get better with age; they spoil just like any other consumable. If your consumption is one a month, then by the time you get through the case, the last bottle may well taste a bit off.

And let me just throw another curveball at the bulk-buying thing: poverty. Assuming you're poor but still want wine (for which you can get a better deal than $10 a bottle), you probably don't have $110 sitting around just begging to purchase wine. Maybe you have $10 left at the end of every paycheck, if you're lucky. So you can only buy one bottle at a time.

In other words, it may be cheaper overall, but people living paycheck to paycheck simply don't always have that luxury.

Fortunately, toilet paper doesn't spoil. It does suffer from overabundance losses, though -- if you have a lot of something, you tend to value it less, so you waste more of it. This can be overcome with discipline, but given all the loo roll memes floating about, it looks like at least some people bought 20,000 rolls of bumwad and proceeded to cover their house with some of them because it was there and we have plenty. This is more than just a waste of money (unless you consider your 15 seconds of internet fame to be an effective deployment of capital); it's a slap in the face to people who legitimately needed Charmin and the stores were out because of your fear and greed, you fucking monsters.

Time is money, and if you’re tying up your money in bulk products for many years instead of weeks, a 10 or 20 or even 100 percent return may no longer look attractive compared to keeping your money in the bank or the stock market.

Again, dubious math. Money certainly has a time value, but, in defense of bulk purchases that do make sense, there's this thing called "inflation." It means that most products increase in cost over time. Not all, of course, and not always smoothly, but it might make sense to stock up at lower prices. The trick there is being right about the price increasing. And then, of course, when you finally do run out, the price has gone up anyway.

Author goes on to examine the additional problems with bulk purchases for someone living in a closet in Manhattan, which I will excuse because this is, after all, a magazine called New York (not to be confused with the stylistically problematic New Yorker).

I did get a good per-ounce deal on the pound of Tellicherry peppercorns that I bought a decade ago and will probably still be working through during the Donald Trump Jr. administration.

Oh, hey, an optimist.

I think I've found the secret to getting something actually published: take a thing that people are doing, and put out an argument for doing exactly the opposite. Maybe I can do that with the Marie Kondo crap that this author keeps referencing. As we're clearly headed for another Great Depression, the time is probably right to tout the value of keeping things around for when you need them.

Prime takeaways here:

Spending is never investing. The sooner you internalize this, the better.

Thinking before you buy is a good idea whether it's bulk or not.

Apparently, some magazines will pay actual money for crap articles. Maybe even in bulk.

2 Like • Comment • View 1 comment

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