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 11 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 12 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 14, 2022 at 12:02am

#1028892

Grammar Police

I have mixed feelings about this one.

This Is Why; You Should Always Properly—Punctuate Your Social Media Posts’
An incorrectly punctuated Facebook post has brought on a potentially high-priced defamation suit.

Mixed, because on one side, I want to incentivize proper grammar, spelling and punctuation; on the other, I don't think the Grammar Police should use their power to screw the little people.

An Australian judge may make an example of sloppy punctuation, to the tune of over $117,000 USD in legal fees.

This article is from back in October, so it may have been settled by now, one way or the other, but that's irrelevant to the larger discussion. Also, it wasn't punctuation that was at issue.

Several outlets have reported that real estate agent Anthony Zadravic is now being sued for defamation for typing “employees” rather than “employee’s” in a Facebook post last year.

The original quote:

“Oh Stuart Gan!! Selling multi million $ homes in Pearl Beach but can’t pay his employees superannuation,” meaning an employer-subsidized pension fund. “Shame on you Stuart!!! 2 yrs and still waiting!!!”

There are, of course, several other problems with this post: there should be a comma after 'Oh.' Multi-million should be hyphenated. The $ should be written out as 'dollar.' There's another comma missing between 'you' and 'Stuart.' Starting a sentence with a digit is questionable, but most style guides specify that anything less than 11 should be written out; in this case, as 'Two.' How hard is it to spell out 'years?' This is Failbook, not Twatter. I'd be willing to excuse the triple bangs (two instances) as a casual stylistic thing meant to convey extreme shock.

But the actual point of contention, 'employees,' could be: employees, employee's, or employees'.

Each has a different meaning in the sentence.

As written, it implies that he can't pay his employees their superannuation. Awkward, and it's pretty clear on a casual reading that it's meant to be a possessive. But there are two possibilities for the possessive:

The second option asserts only a lack of funding for a single employee, presumably the author but that's not as clear as it could be.

And the third projects the idea that he couldn't pay the fund for ALL his employees.

So the grammar police are right: something needs to be done.

Last week, Judge Judith Gibson reportedly ruled that Zadravic’s employer Stuart Gan would be allowed to proceed with the suit because “employees” could suggest a “systematic pattern of conduct” of shortchanging staff, whereas Zadravic claimed he was only speaking for himself. Gibson also reportedly said that the suit might cost Zadravic up to $250,000 AUD, or roughly $180,000.

However, I'm not sure that even the most hardcore Grammar Cylon would consider $180K to be a reasonable penalty for even the most egregious error in forming a possessive.

Well, maybe the most hardcore. There is the temptation to also string the offender up by his toenails until he learns the difference between it's and its; there, their, and they're; and you're, your, and yore.

Temptation, but still, probably too far. After all, you can't practice your grammar if you're hanging upside-down. You'd be too busy howling in agony. Though the pooling of blood in the brain could cause you to lapse into a comma.

Another Australian court has lately gone off the rails with social media defamation policy.

Is it really "going off the rails?" Or is it enforcing some level of needed accountability?

Last month, the High Court of Australia upheld a ruling in favor of a former teen detainee who sued media companies over commenters’ ridicule under news posts showing him hooded and strapped to a chair in a detention center. The court found that media companies (or anyone who so much as runs a Facebook page) for commenters’ speech; as a result, some outlets and the Tasmanian premier have restricted comments, and CNN has blocked Australian readers from its Facebook page.

Bit of both?

Seems more reasonable to hold the people making the speech accountable, and not those merely providing a platform. But maybe it'd be too much work to search out the true identity of OzMan69, only to find that they're hiding behind seven proxies and besides, they live on a troll farm in Kazakhstan.

Amusingly, that last quoted paragraph contains a grammatical error; it's missing a phrase. Sue them in Australia! Props for correct use of the semicolon, however.

And now, maybe some unfortunate soul goes to apostrophe jail.

Where they'll be mercilessly teased by a whole gang of purveyors of fine fresh produce.

Now, Waltz's First Rule of the Internet states that any post correcting someone's spelling, grammar, and/or punctuation will itself contain spelling, grammar, and/or punctuation errors. So I've tried to go through and proofread this entry, eliminating any of my own mistakes. Did I succeed?

Well see.

4 Like • Comment • View 4 comments

March 13, 2022 at 12:01am

#1028825

It's a Gas

I want to help make the world a better place. Okay, well, that's not entirely true, but I want to at least not make it a worse place.

We need to talk about your gas stove, your health and climate change

Articles like this one make me stop giving a shit, though.

Americans love their gas stoves. It's a romance fueled by a decades-old "cooking with gas" campaign from utilities that includes vintage advertisements, a cringeworthy 1980s rap video and, more recently, social media personalities. The details have changed over time, but the message is the same: Using a gas stove makes you a better cook.

I couldn't care less about those ads, rap videos, or especially social media personalities. Gas stoves are simply superior.

But the beloved gas stove has become a focal point in a fight over whether gas should even exist in the 35% of U.S. homes that cook with it.

Let's just use less efficient electricity instead. Some of which comes from gas.

If you have an electric stove, the energy for cooking may come from fossil fuels, but the combustion happens at a power plant far away, Kephart says. "When you have a gas stove, that combustion is actually occurring right in your kitchen — you can see the blue flame down there," he says. "There is no smoke-free combustion."

And?

This reminds me of the whole California Proposition 65 thing. I'm sure it means well. But as with "this product contains chemicals known to the State of California to cause cancer," there are problems.

First, say you have Substance A, which has a 0.0001% chance of causing health issues, and Substance B, which has a 10% chance of causing health issues. You lump these two together, and it's misleading as hell. Sure, you can say "But I only want products that have a 0% chance of causing health issues," but you can also say "I want a pet unicorn."

Second, if you put a warning label on everything, pretty soon it's going to be utterly ignored. It becomes background.

Point is, I don't care if the combustion comes from my gas stove or a gas-powered plant, it's still going to happen, and I don't run the gas stove 24 hours a day.

Well, my local power mostly comes from nuclear, so I guess that's better somehow? (Keep in mind you have to consider waste disposal in the equation).

There is no hood over Kephart's stove to vent the pollution outside. Instead, like many Philadelphia row houses, there's an old room fan high up in a wall. It vents outside, but even after Kephart turns it on, NO2 levels remain high. Kephart says that's because the fan is about 6 feet away.

This is an argument for venting gas stoves, not for eliminating them. These days, houses are way more insulated than they used to be, in order to save on energy in general. When I was a kid, the house was drafty as fuck. No chance for NO2 or CO2 or our farts to accumulate inside.

In the absence of federal oversight, California is taking action.

Because of course.

To encourage more people to ditch natural gas, environmentalists are focusing on the gas stove. At first it may seem like an odd choice because other gas-burning devices in the home consume more fuel, notably furnaces.

I'm going the other direction. I'm putting in more natural gas appliances. I bought the house with a gas furnace nearly 30 years ago. When it came time to replace the water heater, I switched to gas. Then I changed the electric stove to gas, which of course improved my overall quality of life. I put in a gas fireplace. Then I added a gas-fired emergency generator, which required upgrading my gas meter.

Now, I want gas lights for my deck. They'd look cool.

I'm sure my idiotic city council will pull some shit about increasing the price or cutting me off, but until they do, I'm going to live in my own little methane utopia.

I'm not going to rant too much more about this, but I will point out the major flaw in the "anti-gas" movement, which is that right now, they burn that shit off at the source because there's a surplus and they can't sell it fast enough. Burn it in Oklahoma and waste it and still warm up the climate, or burn it here and at least make me warm before it contributes to climate change? Hm, tough choice.

In other words, you can pry my natural gas from my cold, dead fingers. And they will be cold.

I saw a movie in the theater on Thursday but I keep forgetting to review it. So here it is:

One-Sentence Movie Review: Cyrano

Part stage play adaptation, part 80s music video, part Renaissance painting, this film celebrating the utter foolishness of humans is about 45 minutes longer than it needs to be.

Rating: 3.5/5

3 Like • Comment • View 5 comments

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