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').

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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.

December 11, 2022 at 12:01am

#1041676

Revisited: "Deep Dive"

When I pull the levers on the Wayback Machine, I exclude any entries from the last 12 months. Today's revisit is, however, from last year; specifically, July, after my cross-country trip was cut tragically short by a deer. Well, tragically for the deer; for me it was part minor inconvenience, part disappointment, and part unexpected adventure.

"Deep Dive"

This was a response to a then-current prompt from "JAFBG"

[XGC], one of the blogging activities around here that I sometimes do. It encourages rants, and I like to rant, so, big wins all around.

The entry my random number generator landed on was not, however, particularly ranty, just some discussion from a prompt about comfort zones. It's also recent enough that I don't think I've changed much as a person since then, and I mostly remember the context. But I still have some things to expand on.

Every time I travel, I step outside my comfort zone (which is about a three-mile radius centered on my house).

Oh, sure, there are times I can step back in, even when traveling. Most of the time, I feel at home in a brewery or brewpub.

Yesterday, I went to a beer festival. I always loved beer festivals, where a bunch of different brewers set up tables with, usually, 2-4 taps of their product. There weren't any of those for over 2 years. This particular one was kinda expensive, and only had ten vendors, including one vineyard, one cidery, and one sake brewery. Yes, we have a sake brewery here, one of the few in the US, and it's a good one.

The one advantage of having so few producers represented is that I was able to try one of everything. Of course, by the time I was done, I was pretty lit. I barely remember the Uber trip home. Point being, I was well within my comfort zone.

My near-visioin is still pretty clear, so often, what I'll do is take a picture of the taps or the bottles with my phone, and hold the phone up close to my face to read the words.

I find it somewhat amusing that I typo'd "vision" in that sentence. At the time, I was using prescription reading glasses to type, ones that also accounted for the cataracts I had at the time. They weren't perfect. Pretty sure that's not the only typo I made in that period. I don't think I misspell things very often, so when I do, it's somewhat embarrassing.

Speaking of comfort zones, I've lived most of my life with near-perfect vision, with only a slight astigmatism to deal with -- this was only a real impediment to stargazing, and I never even noticed it in my daily life. Later in life, I learned to deal with needing reading glasses for computer work and books. Now, in the space of two years, I've gone from near-perfect distance vision to not being able to read beer taps from across the bar. That's uncomfortable, and I don't like it.

I don't remember if I noted this at the time: when I went in for a cataract surgery consultation last year, the ophthalmologist went over some different options for replacement lenses. The cheap ones, the ones insurance covered, would be the kind that only worked for distance vision; I'd need glasses for anything closer. There was also an option not covered by insurance that worked for both near and far. But, as he explained, the side effect would be a halo/starburst effect around light sources at night. "Usually that's only a problem if your hobby is astronomy," he said.

Well, you know me. Astronomy is one of my hobbies.

Still, balancing that with its benefits, I opted (pun intended) for the fancy ones. The surgery was almost a year ago, now. Indeed, lights (and stars and planets and the moon) have a halo/starburst effect. Worth it for not needing glasses. As it was in my younger days, I can switch from reading a screen to watching a far-away cloud, both with roughly equal clarity.

Sorcery.

My vision with the upscale lenses isn't perfect, but it's good enough, and definitely better than having cataracts.

So, would I dive into a dive again? Sure. But I think the real question is, "Would you step out of your comfort zone again?" And the answer to that is still yes.

At least, yes, once I buy a car again, which I might just put off until after cataract surgery. Which itself is so far out of my comfort zone that it might as well be on a planet orbiting Proxima Centauri, and an experience that, once I have it, I doubt I'll be in any hurry to repeat.

So, as I noted in here, I did indeed buy another car—though not until this past October. Not because of vision, but because availability was low and prices were high.

I feel another road trip coming on, too. Difficulty: possibility of winter weather. We'll see. It'd be another affront to my comfort zone, but at least the car is comfortable.

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