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 6 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 8 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.

January 2, 2022 at 12:01am

#1024005

Security

I should probably work on my satire skills. Just to be clear regarding yesterday's entry, I don't actually advocate summary execution of airborne miscreants, nor do I support drugging people en masse. Consider those "modest proposals."

Yes, I could use more satirical practice. But not today. Today, I'll attempt to address another "JAFBG"

[XGC] prompt without satire. Or at least without too much of it.

What is something you learned too late in life that you think others could benefit from knowing?

I'm tempted to keep any such revelations to myself. After all, when I was a kid, the adults in my life hit me with all kinds of advice that I scoffed at, only to find from empirical study that they were, after all, right. But I just had to see for myself, didn't I? I assume other young people would do the same with any advice I'd proffer.

But, in the spirit of doing a blog entry, and knowing that anyone who actually reads this isn't going to change their ways based on my personal experience, I'm willing to give it a shot.

There's actually plenty of stuff I learned too late, like "don't stick your dick in crazy" (which can, in fact, be adapted to "don't let crazy stick its dick in you"). I have not considered what happens when both involved parties are crazy. I imagine that would be a match made in hell. Which might explain both my former marriages, assuming I'm crazy. Because if I am, by definition, I wouldn't know it.

But I suppose the biggest thing I learned that I probably should have figured out earlier goes against all conventional wisdom, which probably explains why I didn't consider it earlier. This is that money does, contrary to popular belief, matter.

People tell you it doesn't, but I think that's one of those lies that they tell you so you don't feel so bad about being poor. But money is actually freedom.

I have a whole article about this in my article queue that I'll get to eventually. Some studies have apparently shown that one's happiness increases only up to a certain income level, after which you reach a point of diminishing returns.

Assuming they're valid studies -- and since any study that purports to measure "happiness" is necessarily subjective, that may not be a good assumption -- there are at least two problems with this:

1) If you're just measuring income, that's only part of the whole picture. I've known people with high incomes who spend nearly all of it. Of course that doesn't make them "happy;" it just buys them stuff and services. People need to stop conflating income with wealth.

2) Happiness isn't all that important. Security is. Incomes can be pulled out from under you with no notice, people can die or cut off contact with you, but if you have a stash of cash, you can weather a lot of downturns in life. The only way to accumulate such a hoard, barring lottery wins or other windfalls, is to make enough money to save it.

Consider some examples:

There's a wildfire heading your way. You've been told all your life that it's more important to be happy than to be wealthy. So now you're displaced, and you have nowhere to go and no way to get there, and have to rely on the assistance of strangers. Or, worse, family and friends.

Your beloved pet is ill and needs treatment and medicine. Without money, you're shit out of luck. More importantly, so is the pet. With money, you have options for them.

This is the US, and you've just been diagnosed with a dread disease (pick one, doesn't matter). Since your insurance sucks, you can either wither away and die poor, or, if you have money, you can pay for actual good treatment. To make matters worse, you've just lost your job because you're sick.

Paradoxically, things are more expensive if you don't have money. The late, great Terry Pratchett probably expressed it best with his "boots theory:"

The reason that the rich were so rich, Vimes reasoned, was because they managed to spend less money.

Take boots, for example. He earned thirty-eight dollars a month plus allowances. A really good pair of leather boots cost fifty dollars. But an affordable pair of boots, which were sort of OK for a season or two and then leaked like hell when the cardboard gave out, cost about ten dollars. Those were the kind of boots Vimes always bought, and wore until the soles were so thin that he could tell where he was in Ankh-Morpork on a foggy night by the feel of the cobbles.

But the thing was that good boots lasted for years and years. A man who could afford fifty dollars had a pair of boots that'd still be keeping his feet dry in ten years' time, while the poor man who could only afford cheap boots would have spent a hundred dollars on boots in the same time and would still have wet feet.

This was the Captain Samuel Vimes 'Boots' theory of socioeconomic unfairness.

(From his 1993 novel Men at Arms)

Apparently, though, we've decided as a culture that being poor automatically confers upon one the status of "virtuous," while being wealthy inevitably makes a person sinful. Some religions hold this as a core tenet, but a bit of thought is all it takes to disprove it: there are plenty of asshole poor people, and, yes, some wealthy people with qualities we'd consider good.

While it's true that the relentless pursuit of money has the tendency to make a person seem selfish and uncaring, part of that is that if people find out you have a lot of money they start wanting it, and if you give them all your money, or spend it on bullshit, then you're the one who will be poor. So you have to be selective, and sometimes even mean about it. Just ask a few lottery winners. The ones who tried to buy friendship ended up with no money and no friends; the problem isn't the money, but that the kind of people who waste money on the lottery aren't generally the kind of people who are good with money in the first place.

Also, there's nothing virtuous about begging rich people for a cut of their wealth. (No, I'm not going to allow this to turn into a debate about tax policy; fuck off with that shit.)

"But Waltz, money can't buy everything." This is trivially true, and that's why I'm saying it's important, that it does indeed matter, not that it should be the only thing in life. But while there are things money can't buy, in the immortal words of the Beatles, "but what it don't get, I can't use."

In the end, though, it doesn't matter how much money you have, only that it's enough for you. But having none or, as is the case for a lot of people, negative wealth in the form of debt, is a surefire way to insecurity.

In conclusion, on this subject, received wisdom is absolutely misleading, and I wish I'd realized that earlier.

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