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 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 5, 2023 at 9:07am

#1058937

Revisited: "60 More Years!"

For anyone coming in new here, I like to highlight random past blog entries on Sundays. Why Sundays? Well, why not? I like to talk about how things, their context, my opinions, etc. have changed since I wrote the original entries.

Today's comes from a very long time ago indeed: June 3, 2008. "60 More Years!"

For context, as the entry notes, we here in the US were in the middle of the chaos of an election year.

I don't talk about politics much in here. I used to mention it more, in that time period, but at least I wasn't overtly taking sides in that entry.

Me: In other words, five more months of whose hair is better-groomed and which candidate's grandmother or great-grandmother was a flapper during the Roaring Twenties, because, of course, that makes a huge difference in who should be president.

Turns out that my grandmother, who was around 30 years old at the time, was a flapper during the Roaring Twenties. I don't remember if I discovered this fun fact before or after I wrote that blog entry.

But enough about politics. Let's take a closer look at one of those flyover states: South Dakota, land of... land of... um. What the hell is in South Dakota?

At the time, I'd never set foot in South Dakota. Between then and now, I've visited maybe three times? I don't mean "drove through," though I did, but I also sightsaw. Last time, I was stranded there for three days, but I covered that fun incident when it happened in 2021. However, during none of my trips have I visited the main subject of that 2008 entry. Not yet.

Ah, yes. The Crazy Horse Memorial which, as it happens, was begun sixty years ago today

I question my grammatical choices at the time, but whatever. From math, I deduce that the onset of construction of the memorial was over 75 years ago now.

There follows a raw link which, not surprisingly for a 15-year-old link on the internet, is well and thoroughly broken. Apparently, it was a retrospective on the 60th anniversary of the monument's official beginning. If you're not familiar with this enormous project, here's the Wikipedia link.

There's not much else to the blog entry, but I will say this: upon re-reading this today, it was too easy to interpret the last lines as a tribute to General Custer. While it's difficult to remember much about something I wrote a decade and a half ago, I'm certain that wasn't my intention. I should have phrased that better.

I'm tentatively planning another one of my cross-country road trips for the end of the month and on into December. Maybe, if the weather works in my favor, I'll pay a visit to the Crazy Horse Memorial.

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