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!

a day 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

a day 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.

August 7, 2019 at 12:08am

#963836

No Kidding

https://www.outsideonline.com/2258581/mountaineers-choice-be-sterilized-climbing...

*sigh* I guess I'll link to this site again, even though it has the damned word "outside" in the title.

A Mountaineer’s Choice to Never Have Kids
The accomplished alpinist Lydia Bradey looks back at her life, 31 years after making the decision to get sterilized

I decided long ago that I never wanted kids. After having been active in several childfree forums on the internet, I've come to the conclusion that the decision is different for men and women, but social expectations are similar.

Lydia Bradey always knew she didn’t want kids of her own. She knew that when she was 18, and had already summited both of New Zealand’s most famous peaks, Mount Aspiring and Mount Cook. She didn't change her mind at 19, when she attempted the south face of Alaska’s Denali, nor at 26, when she became the first woman to climb an 8,000-metre Himalayan peak, Gasherbrum II in the Karakorum range, alpine-style without oxygen.

I also made the decision to never climb mountains, and yet, for some reason, that doesn't come with the same cultural baggage.

The topic of motherhood in mountaineering wouldn’t ignite the public's attention until 1995, when British climber Alison Hargreaves died on K2 in a violent storm. Hargreaves was one of the best mountaineers of her generation, also summiting Everest without supplemental oxygen and soloing all six of the classic north faces of the Alps in a single season. But rather than being remembered for her skill, she was attacked after her death as irresponsible and selfish for being a mother of two who indulged in such a risky career.

Damned if you do. Damned if you don't. Can't please anyone if you're a chick, I suppose.

Still, I'd like to point out that anyone can die of anything at any time, so if you have kids, you're always risking orphaning them. It's a huge responsibility, mountaineer or not.

It's not that Bradey made the decision lightly, and she admits that over the years, her thoughts about having children began to evolve. “You begin to meet young people and you start to think about how having children would be rewarding,” she says. “As you get older, you see that parenthood is a compromise.”

There are occasions when speaking in second-person works. This is not one of them. I have never, not once, considered that having children might be rewarding. Or rather, perhaps I have, but the potential rewards were always outweighed by the potential downsides.

Other people, I know, felt the converse. Most people seem to breed without much thought, though. To be fair, "thought" isn't something required for that; even gerbils do it.

In her early 40s, Bradey began the process to adopt a child. “When I was very young, the concept of adoption came up a lot,” she explains. “I’d decided when I was little that if I were to have children, I wanted to adopt.”

Well, I certainly can't say she hadn't given it thought. For the most part, I respect the people who want to adopt - there are way more people who have kids and can't take care of them than there are people who want kids and can take care of them. As an adoptee myself, I only wish more people would do that rather than demand their own genetics be passed on.

Spoiler alert if you don't read the article: She didn't adopt.

“I reckon if I could afford to have a permanent nanny and I could do full-time adventures, I would definitely have had children, and I’d have had no problem being a Himalayan guide.”

I don't detect much regret in this statement, just a plain stating of fact that if things had been different, things would be different. A tautology, really. I mention this because "regret" is one of the things the childfree - especially women - get asked about most often. When it comes up, I usually point out that if you're going to have regrets, it's better to regret not having 'em than it is to regret having 'em. The former affects no one but yourself. The latter? Kids can pick up on that shit.

I could write an entire treatise on this, but no one would read it so... meh. All I'll say is that it's an individual choice. A lot of ethical questions can be resolved by asking "would it still be the right thing to do if everyone did it?" Well, if "everyone" decided to have kids, we'd be overpopulated that much more quickly. If "everyone" decided not to, we'd die out pretty fast (whether that's a good or bad thing is outside the scope of the question, but inside of 100 years there'd be no one left to debate it). So my conclusion is that it has to be an individual choice, made for whatever reasons you find compelling. Examples of non-compelling reasons: "My parents demand grandchildren," "All my friends are doing it."

So I'm not writing this to rag on the people who choose one way or the other, just to throw out the idea that maybe it would be good if we could stop the social pressure - and maybe accept that, sometimes, fulfillment and accomplishment comes from other sources. Biology, after all, isn't the same thing as destiny.

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