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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 10, 2022 at 12:02am

#1040465

The Far Side

Well, it missed the full moon by a couple of days, but this one finally came up from my queue for me:

Why the Moon’s two faces are so different
The far side of the Moon is incredibly different from the Earth-facing side. 63 years later, we know why the Moon's faces are not alike.

Article has a helpful picture of both hemispheres of the moon up at the top, and lots of other pretty pictures scattered throughout.

The Moon, by far, is the brightest object and largest object that’s visible to human eyes in Earth’s night sky.

I was about to object to this until I realized that it means that it appears the largest, not that it's the largest object, in absolute terms, that we can see. That would probably be a star somewhere, depending on your definition of "object."

So to satisfy my urge to be pedantic, I'll point out that one can often see the moon in the daytime sky, as well, during certain phases when it doesn't appear too close to the sun.

With even an off-the-shelf pair of binoculars or the cheapest telescope you can find, there are two main features about the Moon that you can’t miss:

That it's made of cheese and it's round?

In addition, because the Moon’s orbit is elliptical, moving faster when it’s closest to Earth and slower when it’s farthest away, the face of the Moon that’s visible changes ever-so-slightly, a phenomenon known as lunar libration. Even though this means, over the course of many months, we could see up to a total of 59% of the Moon, it wasn’t until 63 years ago, when the Soviet spacecraft Luna 3 swung around to the far side of the Moon, that we got our first pictures of the far side of the Moon.

Because of this, most of the features of the far side are named in Russian.

Although it wasn’t very impressive in terms of image quality, it was remarkable for an unexpected reason: the near side of the Moon appears vastly different, in terms of both cratered features and maria features, from the far side that always faces away from us. This discovery came as quite a shock, and for decades, even as our imaging and understanding of this elusive side of our nearest planetary neighbor improved in quality, we lacked an explanation as to why this difference existed at all.

The rest of the article explains just that, and it's pretty cool, not only for the explanation, but for the observations, deductions, and science that got us there. Which is fascinating, but there's not a lot of point in rehashing it here.

Then:

No matter how wild or unusual your idea may be, if it has sufficiently strong explanatory power to account for what we observe, it just might be the necessary idea to solve whatever puzzle it is that you’re considering.

Until some better observations come along, of course, and change everything we think we know. But that, too, is part of the awesomeness of science.

2 Like • Comment • View 1 comment

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