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 9 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 11 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.

February 22, 2024 at 8:54am

#1064670

Curiosity

Speaking of extraterrestrials...

Star Trek Starfleet insignia spotted on Mars

In case you were wondering what it would take for me to go through the process of incorporating an image into a blog entry (might have to scroll down), well, wonder no more. It helps when it's in the public domain. Less work. (credit: NASA/JPL-Caltech)

Like the infamous Face on Mars,

it's entirely possible that this particular example of pareidolia would go away, or at least diminish, under different lighting conditions or at a different angle or whatever.

Until then, let us Star Trek fans have our hour of glory.

From the linked "article:"

Amateur astronomer Scott Atkinson found the stone sculpture of the Starfleet insignia among a pile of rocks on the Red Planet's Mount Sharp (Aeolis Mons).

Just to be clear, Atkinson didn't travel to Mars or use a giant telescope; he apparently combed through the publicly available images from NASA's Curiosity robot. Well, they call it a rover, but come on; Mars is the only planet known to be inhabited solely by robots (to the best of our knowledge).

Curiosity itself (the robot) is a remarkable achievement. You can see more about it, and its images, here.

As of this entry, according to that site, the gadget has produced 1,165,040 images over 4105 sols, an average (because I can use a calculator) of over 280 photos/sol. Which would be a remarkable output even for a social media influenza.

Oh yeah, if you don't know, a sol is what they call a solar day on Mars. It's not too different from our own solar day: roughly 24 hours and 40 minutes.

More, all of those photos are transmitted, pixel by pixel, back to Earth. Truly a monumental achievement that this has been going on for, if I've done the math right, about 11 and a half Earth years, or roughly 6 Martian years.

Which is itself about four times as long as the original Star Trek series aired.

I wrote yesterday about the possibility of encountering alien life. But if we ever encounter technologically capable aliens, given our own history of exploring with robot probes before sending humans out there, it wouldn't surprise me in the least if our first contact was with one of their machines.

Those machines might send back videos of Star Trek, and since a sense of humor appears to be a requirement for technologically advanced life, they'll probably freak us out by showing up dressed as Ferengi.

And here I was planning on talking about how the image clearly shows indications of erosion, which is cool enough by itself without invoking science fiction. Oh, well. Sometimes I go in unexpected directions, which proves I'm not an alien robot.

Or does it?

3 Like • Comment • View 2 comments

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