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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QPdoll

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I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 16 hours ago
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With all that walking, I hope you stayed hydrated.

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Annette in France

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Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
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a day ago
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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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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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