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.

May 3, 2021 at 12:01am

#1009557

Getting Salty

PROMPT May 3rd

What was the best thing that happened in your life over the weekend? Looking at the week ahead, what are your goals and how will you motivate yourself to achieve them?

Ha ha. Very funny. It is to laugh.

Let's take this one step at a time, shall we? Some of this is going to require some background, even if you've been following along here.

What was the best thing that happened in your life over the weekend?

In the Before Time, one of my favorite activities was to travel around and visit breweries. Sometimes I'd visit breweries incidentally whilst traveling for other purposes, but most of the time the breweries were my destination.

Each of them has its own take on the holy nectar. Some are focused on hop-forward beers like IPAs; I don't generally like those. Others avoid IPAs in favor of other traditional styles. Some like to go their own way, inventing new styles or fresh takes on old ones; others tend to stick to the basics. There's a lot of variety, is what I'm saying. So each one presents a new experience. One thing I've come to expect from them is the tasting flight: some number, usually four to six, small samples of beers of your choice.

Now, certainly there are styles I gravitate to, like the Belgians and Imperial stouts. But I'll try almost anything. I say "almost" because some brewers, in a misguided effort to stand out, do completely weird shit. I mentioned hemp beer here a couple of days ago, but that's not the weirdest, or the worst, experiment I've seen. One brewer famously brewed beer using yeast that he'd extracted from his beard (all male brewers have beards; it's the law). Yes, they called it Beerd, but that atrocious name isn't the only reason I'm never going to go anywhere near that brew.

But with those vanishingly rare exceptions, like I said, I'll try almost anything. Some I'll really like, some I won't; most fall in between. I purposely try styles I don't generally like so I at least know what the beer is "supposed" to taste like, and also because, in the words of the great philosopher, "If everything was cool, and nothing sucked, how would we know what was cool?" I like variety, so with two exceptions, I don't generally go out of my way to purchase the same beer over and over. But when the pandemic hit, my ability to visit breweries was severely curtailed. Fortunately, governments, in a rare case of getting something right, relaxed restrictions on beer delivery and shipping. So instead of me going to the breweries, they were able to come to me... at least on a limited basis.

Enter BeerDrop, a beer subscription service out of Colorado, home of many a delicious brew. Think of it as a "beer of the month" club, where you can get five different beers shipped to you every month. You have some control over the styles. I'm not shilling for them, by the way, but I have to explain the system somehow. Basically I get two cans each of five different beers from, usually, five different breweries; most are from Colorado, but I've also had beer brewed in other states (they are all, as far as I've seen, American beers).

So it's a bit like getting a tasting flight, only with full-size beers and not limited to one brewery. So far, so good. And I've generally liked every beer they've sent me.

Until this weekend.

Last week, April's shipment came in but, contrary to popular belief (that I foster), I don't drink every day or when I have something to do like a newsletter or a blog entry... usually. So I finally got to crack a couple open on Saturday, when I had some help drinking them. The first one I tried was nice, a Vienna lager that was nearly as good as the one a local brewery makes. But the second one... how shall I put this... sucked. It, too, was a lager (and this entry is already far too long to go into the technical difference between lager and ale), but the brewery mixed in lime and salt.

I don't like limed and salted beer on a good day. Sometimes a bit of lemon in a wheat beer can be nice, but lime imparts a completely different experience. As for salt, the only reason to salt your beer is if the beer sucks in the first place, and if that's the case, why drink it? (Salt and lime are, of course, perfectly acceptable with agave-based booze like tequila and mezcal.)

In this one, though, the ratios were all off and the beer was just plain disgusting. I think maybe they were trying for stealing the Bud Light Lime-a-Rita crowd; I don't know, because that particular abomination is one of the ones I'd steer entirely clear of. Hell, I don't even know if they still piss out the stuff.

So. After all that. Why is this under "What is the best thing that happened in your life over the weekend?" Well, because I've had some mediocre beers in the last year, but this was the first one that I actively hated. And this was good because, as per the Beavis and Butt-Head quote above, it helps me appreciate the rest of beer that much more.

Looking at the week ahead, what are your goals and how will you motivate yourself to achieve them?

Allow me to laugh once more: Ha.

Of course, I do have some goals. Obviously, this challenge is one of them. I don't really need motivation for it; having done a blog entry every day for nearly a year and a half now, it's more like a habit.

Same with French lessons on Duolingo. Every day, 613 days and counting. It's just something I do.

Later today (Monday), weather permitting, I'll make my traditional weekly trip to a local taphouse. That's a goal. It's motivated by some new beers they've acquired since last week's visit.

Wednesday? D&D night. Videoconferencing and online gaming capability are some of the best inventions of the tech age. Probably won't drink that day; D&D requires some mental focus. Except for that one time I was playing a Drunken Master character, when scotch was my role-playing aid.

Thursday is Three Notch'd Day for me (that's a nearby brewpub); during lockdowns, I'd get stuff delivered from them, beer and food, every Thursday. Now that I can venture forth once more, and they have bar seating again, I'll go there. The only motivation I need there is to get off my ass and walk the two-mile journey -- but, again, beer is a great motivator for me.

I generally judge "Invalid Item"

on Fridays. So call that a goal. It's a commitment for me, so no other motivation is necessary.

Saturday is Zoomies day. A bunch of us get together on Zoom every other week and talk about writing, life, and the writing life. Please feel free to join us; watch my newsfeed for details. (That is an advertisement. But it's free.)

Other than that? Keep myself fed on the days I'm not visiting restaurants. Sleep. Take care of the cats. Continue my chronological rewatch of every Star Trek episode and movie. Try to choke down the second can of the horrid lime salt lager.

What? It's not like I can allow myself to waste it.

By the way...

Yesterday, we talked about the sense of smell and how it can trigger memories. Well, today, I found this article

talking about the science behind the sense of smell, and I think it's really damn interesting. So there it is if you want to read it.

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