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!

a day 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

a day 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').

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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.

June 26, 2019 at 12:42am

#961545

Thriller

We’re taking a trip today to the southern most point in the entire United States. South Point on the Big Island of Hawaii is a popular daredevil destination where brave souls leap from the cliff into the churning ocean below.

Are you (or were you ever) a thrill-seeker? Do you visit carnivals or theme parks? What do you do that gets your heart pounding? Have you ever had a brush with death?

"Brave" is not a word I'd use in this situation. Nor is "souls," but that's not really the point.

Risky behavior is a part of life. It's probably an evolutionary thing, but you know by now how I feel about spurious "we evolved to be that way" arguments. Regardless of why we're like that, I've noticed that risky behavior, on its own, is not what's rewarded in humans. It's taking the right risks, and being successful at it.

Being successful at the wrong risks invites public scorn. People who drink "too much" based on some arbitrary standard, for example. We think nothing at all of the public health costs involved with people climbing rocks, jumping off cliffs or out of airplanes, hiking in the wilderness, participating in extreme sports, etc., but watch someone drink too much and all of a sudden it's all "I don't wanna pay for their life decisions."

Recently, there was this chick who got herself lost in the vast, unexplored wilderness of... Maui. Surviving for over a week on scraps and droplets, her disappearance made national news, and the entire country breathed a sigh of relief when she was found, leg broken, but alive. Weighed less, but hey, who doesn't need to lose some weight, right? Meanwhile, a dude disappeared on the other lobe of the same island. I only found out about this because I had a friend on Maui; it was nowhere in the news. Guy died. I guess men are just expendable and not worth the search effort.

My same friend is a hang gliding enthusiast and a scuba diver. You'd think both would be big on Maui, but it turns out that only the scuba thing is; most of the high spots on the island are part of the national park system, and there's a countrywide ban on hang gliding in national parks. This is silly any way you look at it; scuba is way riskier than hang gliding. It's too bad, because I'd totally take up the sport just to glide off Haleakala.

I tried scuba diving, but my ears just won't accommodate the pressure changes.

Point is, we as a species absolutely suck at assessing and managing risk. Hence, you get people who "hate" flying because the plane might crash, but don't even think twice about speeding to the airport if they're late; and you have orders of magnitude more chance of dying in a car crash (even without speeding) than you do of dying in an airplane mishap.

Are you (or were you ever) a thrill-seeker? Not really. Not out of fear, but just a matter of "what's the point?"

Do you visit carnivals or theme parks? I did quite a lot of that when I was younger. Loved roller coasters. But that was more because I was fascinated by the dynamics involved. I always did see the world from an engineering and physics standpoint.

What do you do that gets your heart pounding? Climb a flight of stairs. Okay, I'm kidding. I'm in pretty good shape, these days.

Have you ever had a brush with death? I've been close to dying at least three times, maybe four. Mostly from illness, though. There might be other times when I nearly died and didn't even know it. Life is inherently uncertain, and I'm not really afraid of death; I'm way more concerned about being maimed or being otherwise unable to function the way I'm used to. There was one time I was swimming in the ocean off the Carolina coast, and I found out the next day someone else had been munched by a shark somewhere close by; I'm not sure that really counts, though. Way more people are killed by cows than by sharks, and I've been around herds of future hamburgers.

No, I don't get thrills from taking risks, or even from beating odds. That may sound strange coming from a self-admitted gambler, but I don't gamble for the thrill. What excites me is learning new things. And, of course, feeling superior to the idiots who throw themselves off of cliffs.

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