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 5, 2019 at 12:02am

#963736

Kelpless

https://www.atlasobscura.com/articles/what-is-nori

Whether you like sushi or not (I do), this is a fascinating read.

Though she lived halfway across the world, Kathleen Drew-Baker played a monumental role in saving Japan’s multi-billion dollar nori industry.

The article goes on to describe nori and its history in Japan, which is pretty cool by itself.

But the main part of the story is what caught my attention.

In 1948, shortly after the end of World War II, the nori disappeared. A series of fierce typhoons—coupled with the effects of pollution and industrialization—had ravaged the coast. Unlike past, unlucky years, this time the nori didn’t bounce back the next year.

So first they lose a war, then famine strikes. Any other horsemen want to weigh in?

But then it gets interesting.

But the research needed to revive Japan’s nori had begun 20 years earlier, across the globe in Manchester, England, when Kathleen Drew-Baker was fired. She had been a lecturer in cryptogamic botany at the University of Manchester, but the college did not employ married women.

So, enter our protagonist, a victim of institutionalized sexism... and a subject of the Crown with which Japan had been enemies up until a short time before.

At the time, botany was one of the only sciences considered appropriate for women to enter. “Botany was a safe science for women,” Kassinger says. “It was the one that didn’t involve mathematics or cutting things up.” As a cryptogamic botanist, Drew-Baker studied plants that reproduced by spores, such as ferns. “Ferns were a good idea for women to study because they didn’t have flowers and therefore they didn’t seem to involve sex,” Kassinger says.

What the eukaryotic fuck, sexism?

She goes on to make an important discovery about the life cycle of algae. This discovery turns out to be crucial to Japan re-establishing its seaweed trade.

Despite the financial setback she faced as a married female scientist, she managed to raise two children while conducting her own research and training younger botanists. In 1952, she was elected the first president of the British Phycological Society.

In short, this chick was a badass.

So if you’re in Wales, try some laverbread, the pulpy seaweed paste kneaded from wild laver. And if you’re ever in Uto, stop by Drew-Baker’s memorial, smell the salty, weedy scent of the Ariake, and pay your respects to Japan’s Mother of the Sea.

So the Japanese kelp farmers put up a statue of this badass scientist from a formerly enemy country, and they pay their respects to her every year.

There's still institutional sexism out there, in the US as well as in the UK... and in Japan. It's not as bad as it used to be, though there's a ways to go still. But science doesn't care about your plumbing or where you stick it; nor does it matter to the people whose lives and livelihood you save.

And it's not just all about research. Science is still the best means we have of figuring out how the universe operates, and as Randall Munroe, the artist behind the webcomic xkcd, said, "it works, bitches."

1 Like • Comment

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