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 an hour 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 3 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.

November 4, 2023 at 11:44am

#1058881

Brew-tality

Inspired by "Invalid Item"

, though this particular observance wasn't in the suggested prompts. The first Saturday in November is officially:

Learn to Homebrew Day

One question I get asked on a regular basis, once someone discovers that I'm a beer snob (which happens pretty quickly, but not as quickly as some people tell you "I'm a vegan."): "So, do you brew your own beer?"

No.

Not just no, but HELL no.

I mean, sure, the process is fascinating, and as with most work, I could watch others do it all day. And I'm glad some people are into that, because that's how we get brewers at craft breweries. So, of course, being curious, I looked into it, and discovered that home brewing is approximately 1% enjoying the product; 3% brewing; and 98% cleaning and sterilizing.

Yes, I'm aware those add up to >100%; I'm using hyperbole to emphasize just how much I hate, loathe, and despise cleaning.

Why do all that work when I can go to a bar or brewpub and, using the fruits of my other labors, purchase a delicious golden beverage that's the result of other peoples' labor? It's not like diamonds, coffee, smartphones, clothing, avocados, or chocolate; you can be fairly confident that craft beer in the US isn't brewed using slave or child labor.

Still, I wanted to take a moment to appreciate that here in the purported Land of the Free, we have the freedom to brew our own beer, if we want to put in the really outrageously tremendous amount of work involved. It wasn't always that way.

I'm not just talking about those dark times known as Prohibition, either. Once Prohibition was lifted—an anniversary I also celebrate, coming up next month—it took a while before home brewing was legalized.

By "a while," I mean "nearly half a century."

Before Prohibition, brewing had mostly been done at home, though industrialization facilitated commercial breweries more and more as time went on. Many, if not most, of the Founding Fathers of the US were brewers (or their spouses were; brewing, like cooking, was considered largely a woman's job). The entire idea of the American Revolution was fomented (as opposed to fermented) in pubs, so beer is in our DNA, which made Prohibition that much more of a slap in the face to Lady Liberty.

Amendment XXI to the US Constitution, passed on December 5, 1933, repealed the earlier amendment XVIII that created Prohibition. Yay. But home brewing didn't become legal until 1978, 45 years after the repeal of Prohibition. Thanks, Jimmy!

Which is not to say that people didn't break The Law on a regular basis, but the legalization of homebrew led inexorably to our craft beer landscape, which remains the primary reason I travel. Last I checked, which was five minutes ago, there were nearly 10,000 breweries in the US. I'm not even sure how many of those I've visited and/or sampled brew from; my records are spotty, as you might expect after I've had a couple. It's also complicated by the fact that, on average, every week, we lose one and gain two.

With that much beer to choose from, why bother making my own?

No, if I'm going to do work, it's going to be important work, like figuring out how to visit the 9000 or so that I still haven't.

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