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 8 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 10 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 8, 2024 at 8:25am

#1065851

Law and Ordering Alcohol

Booze laws can be convoluted. You think they exist to promote public health and safety, and maybe some of them do, but mostly, the laws are about who does and does not get to profit from alcohol. (In order, that would be "governments and rich people," and "you.") Leave it to Cracked to find some of the weirdest ones.

5 Perplexing Laws Over Who Gets to Buy Booze

5. In New York, Bartenders Are Forbidden from Not Giving Alcohol to Pregnant Customers

Of all these examples, this one's actually the least weird.

In general, bartenders are allowed to refuse service to customers. In the case of an intoxicated customer, they’re legally bound to refuse service. But they are legally banned from refusing service to a customer because they're pregnant, as this may be “pretext for discrimination or as a way to reinforce traditional gender norms or stereotypes.”

For starters, the header is misleading. Bartenders aren't known for "giving" alcohol. They're known for selling it, and for patiently listening to sob stories in hopes of getting a tip.

Further, while late-stage pregnancy is usually quite visible, from my understanding, some of the worst and most (eventually) visible effects of drinking while pregnant occur during the first trimester, when it's not usually obvious.

And finally, we either need to assert that all adult humans are responsible for their own decisions, or go the way of places like Texas who exert government control over the pregnant ones.

4. The Very First Campaign Finance Law Was About Banning Alcohol Sales

*in the US

In 1811, Maryland banned liquor sales on Election Day, but the goal wasn’t just keeping the electorate sober in general. The goal was to keep politicians from buying voters alcohol — as bribes. As a result, this is remembered as the first campaign finance law.

Well, this one's not that strange, either. If I'm having a hard time choosing between two evil politicians, I would definitely vote for the one who bought me a beer.

3. Happy Hour Bars That Legally Must Give Discounts on Soda

During happy hour, bars slash prices on drinks, maybe selling you two for the price of one.

The laws I'm most familiar with are, obviously, those of my own state. I still don't fully understand them, but to the best of my knowledge, in Virginia, you can't do two-for-one promotions or ever give out free booze. We do get to enjoy discounts, sometimes, and a half-price beer is functionally the same thing without forcing you to buy two drinks.

This drives up business during times that are otherwise dead, and if they’re able to make a profit after halving their prices, that lets you know just how high the markup is on drinks the rest of the time.

Markups can be high, but this is not necessarily the case. Sometimes it's about ensuring the bartender has the opportunity to get tips so they're less likely to run off and work somewhere else.

France has it’s own anti-happiness law. They do have happy hour there (in France, they call it apéro hour), but since 2019, if a bar offers discounts on alcoholic drinks, they also have to offer discounts on soft drinks.

Sadly, the law we need is to fine people for misusing apostrophes.

2. A Special Car for Drunk Drivers

Also in France, if you get caught driving under the influence, they might suspend your license, leaving you unable to operate your car.

I mean, technically, no, not unless they also impound the vehicle and put you on the "no-buy-vehicles" list. But legally, sure.

If a vehicle moves at a top speed of 28 miles per hour or less, and has an engine with an output of no more than 5.4 horsepower, it’s a quadricycle, and in France, it’s known as a voiture sans permis, a VSP. That means you can drive it even if you have no valid license, so long as you’re 14 or older and have a few hours of recorded driving experience.

On the one hand, a top speed of 28 mph (or, as I'm sure they express it over there, 45 kmh) is going to reduce, though not eliminate, the hazards of drunk driving. On the other, you're pretty much limited to city driving, and cities there tend to have bars within walking distance, so what's the point?

1. The U.S. Has No Minimum Drinking Age, Actually

Yeah, technically correct. But effectively not.

Today, all 50 states set a 21 minimum. Even so, nearly every state offers some exceptions. In states like Wisconsin, for example, people of any age can drink so long as they’re with a parent or spouse who’s over 21.

Wisconsin's drinking culture is a truly awesome thing to behold.

In Puerto Rico (which is part of America, despite what that one song from West Side Story left entire confused generations thinking), the minimum drinking age is simply 18. Puerto Rico has highways, but they gladly reject 10 percent of the funds they could get. Some things are more important.

Me? I say lower the drinking age and increase the driving age. That way you get most of the stupid out of your system before you even get behind the wheel of a car.

Better yet, hurry up with the self-driving car thing. As you know, I'm alcohol-positive. But drunk driving? Let's just say I agree more with the French on that one.

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