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 5 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 7 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.

December 10, 2023 at 1:48am

#1060821

It's Not Beetlejuice

I know I just said I'd try to avoid articles while I'm traveling, but then I saw this one. Not random, but time-sensitive, and since I think it's cool, I'm sharing it.

Iconic Star Betelgeuse Will Temporarily Vanish From the Sky Next Week
In a rare cosmic event, an asteroid will eclipse Betelgeuse on December 11, causing a brief but significant disappearance of this bright, red star.

As is so often the case, the headline is misleading. The subhead, a bit less so. Fortunately, the article doesn't seem to be as sensationalist.

The bright star Betelgeuse, a prominent member of the Orion constellation, is set to be occulted by the asteroid Leona on December 11.

Of course I know Betelgeuse. As with many people, it was probably the first star whose name I knew. Well, other than that of the accursed daystar. It's not exactly pronounced "beetlejuice," but close enough to make kids cackle.

I don't think I've ever heard of Leona, though. A quick search tells me it was discovered over a century ago, and no one knows who it's named after. I'm betting it was a girl on whom the discovering astronomer had a crush, as the astronomer in question was very French.

This occurrence, expected to last no more than 10 seconds, will cause the star to temporarily vanish from sight, a phenomenon visible along a narrow path on Earth.

Of course, this is analogous to a solar eclipse, but the eclipsing body is further away from us than the Moon, and the star is WAY further away from us than the Sun.

The path of totality, where the occultation will be observable, stretches across a narrow band starting near Guadalajara, Mexico, and extends through south Florida, the Bahamas, southern parts of Spain and Italy, through to regions in Greece, Turkey, and central Asia.

So, not a lot of opportunity for most of us to see the "eclipse." The link includes a path map.

This event is not only a treat for stargazers but also a significant opportunity for scientists. Betelgeuse, a red supergiant approximately 550 light-years from Earth, is famous for its size, luminosity, and a penchant for confusing astronomers with sudden and unpredictable drops in brightness.

But mostly, it's famous for its name.

The article goes into why this is scientifically useful, but really, I'm just writing about it because it's so damn cool, and I wish I were in a position to see it.

Don't worry if it disappears, though. I'm sure it'll reappear if we all say its name three times.

8 Like • Comment • View 2 comments

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