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 2 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 4 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 6, 2021 at 12:05am

#1022813

Astronomically

I just wanted to share this link at some point, and today's the day it came up. I've always loved astronomy, and I found this informative. Also, it's timely right now, because there's a comet

that you might consider viewing, if you don't mind the cold.

How to Find the Best Stuff in the Night Sky From Absolutely Anywhere
A beginner’s guide to admiring stars, planets, and satellites—no mountaintop or fancy gear required.

My utter hatred of being cold is one thing that kept me from pursuing professional astronomy. As you'll see in this article, sky watching is better when you're cold, dry, and high up. I can handle dry and elevated. Cold repels me.

There was a night many years ago when most of the planets were visible in just one quadrant of the sky. Just after sunset, one could view (if I recall correctly) every classical planet except Mars, between the horizon and the zenith. We went up to the Blue Ridge to get a nice view. And we froze our asses off, but got to see Jupiter's moons and Saturn's rings through a telescope.

You’ll also want to figure out what’s going on with the lunar cycle, because the moon can either improve or obscure your view, depending on what you want to see. If you’re looking for the Milky Way, for instance, or faint, distant objects, your best bet is to wait for a moonless night. If you’re planning to look for something on the moon itself, though, you probably want to do so when it’s full.

I'm going to quibble a bit about this one. When the moon is full, some of the central features can get washed out, because they're not casting shadows. It's like high noon here on Earth -- short shadows, which can make it difficult to see contours. There's a lot to see when the moon is in an intermediate phase, especially around the terminator. Without a telescope, the lunar terminator looks like a smooth curve (or line if it happens to be a half-moon). Thorough even binoculars, though, you start to make out details: some crater floors darken before the rims, and mountain tops can be visible even on the dark side.

This page

explains it better, and with helpful images.

I don't really need to quote more of the article; it really is a good beginner's guide if you're even slightly interested in astronomy. Click through to the links for more information; there are apps there that can really help in locating night sky objects.

I once spent a week up in the Rockies (cold, dry, high) with a professional astronomer, and we saw some really cool stuff. That was expensive, but as the article notes, you don't have to travel to get some nice views.

One more personal note: when the eye doctor was talking about new lenses for my eyes (actual eye lenses, not glasses), he said something like, "...and with this kind, you can change focus to near or far, but the downside is that small lights can get halos around them. This isn't usually a problem unless your hobby is astronomy."

Me: "Well, actually..."

Still, the advantages of these lenses outweigh the disadvantages. I'll still probably be able to see stars and planets better than I do right now (I see blurry fireworks instead of lights). But it's likely my stargazing days are behind me. That's okay; I've seen what I can see. The only thing left on my list is viewing the planet Mercury. I have never seen it and known it for sure. It's possible that I mistook it for a star, as it tends to be near the horizon and small. The other visible planets are really obvious, once you know what you're looking for.

So with this, it's my hope that more people will look up at night sometimes. There really is nothing else like it.

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