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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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.

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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.

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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.

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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.

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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.

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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

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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.

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September 12, 2023 at 9:35am

#1055633

Morning Star

We're back to the solar system articles today. This is a hot one.

The Romantic Venus We Never Knew
Venus used to be as fit for life as Earth.

The article's from 2016, but that's a bit over 10 days on Venus (...yes, I did the math; the article came out 10.1893 Venusian days ago).

On the day that I was born—winter solstice, 1959—a headline in Life magazine proclaimed “Target Venus: There May be Life There!”

In that era, we were trying to find the slightest hint of how life may exist on any other planet.

It told of how scientists rode a balloon to an altitude of 80,000 feet to make telescope observations of Venus’s atmosphere, and how their discovery of water raised hopes that there could be living things there.

For context, this was after Sputnik 1 but before humans put other humans into orbit.

The very first thing that scientists discovered with a mission to another planet was that Venus was not at all the Earthly paradise that fiction and speculative science had portrayed.

From what I can gather, the surface of Venus will kill you even quicker than vacuum will.

As a possible home for alien life, it has been voted the planet least likely to succeed.

You probably heard the breathless hype over the detection of phosphene in that planet's atmosphere a few years back, touted as a possible sign of some sort of life there. What you probably didn't hear was that the data didn't withstand scrutiny, and we're back to square one.

Russian and American spacecraft also found hints that the primordial climate might have been wetter, cooler, and possibly even friendly to life.

Which is a far cry from claiming that life still exists there today.

For most of Earth’s history, Venus may have been the nearest habitable planet and possibly even home to a thriving biosphere. For billions of years, our solar system may have had two neighboring wet, geologically active, habitable rocky worlds. They may even have very occasionally exchanged life when meteors struck and catapulted shrapnel from one planet to the other.

I just want to emphasize, again, that this doesn't imply sentient Venusians, now or in the past.

In any case, the article is necessarily full of speculation such as this, because we just don't have enough data. Which is a shame, because it's often the closest planet to us. The Moon isn't technically a planet, Mars is sometimes closer to Earth than Venus, and, on average, according to some calculations, the nearest planet on average is Mercury,

which makes sense when you think about relative orbital speeds. What's not in dispute is that it's the planet with the closest orbit to ours.

While the author bangs on about climate change and the greenhouse effect, I can excuse that in this case because he's actually a planetary scientist. Well, an astrobiologist, anyway.

Close or not, Venus presents challenges Mars simply doesn't. The latter doesn't have much atmosphere, for instance, while the former has, arguably, too much of one. We can handle near-vacuum, but not corrosive acid at immense pressure. Not well, anyway.

But we're trying, and we're clever and curious. We'll probably have more data in a few days. A few Venus days, anyway./size}

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