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:

Kåre เลียม Enga

a day ago
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

a day ago
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.

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

1 Like

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.

2 Like

JayNaNoOhNo

Nov 5, 2024 at 1:16pm
In response to "Anomalies"

*Toronto, Canada submits a sternly-worded petition and disputes the center of the universe claim.*

2 Like

July 24, 2024 at 12:00pm

#1074345

It's Out There

From aeon, an article on a subject I've discussed before:

Alien life is no joke
Not long ago the search for extraterrestrials was considered laughable nonsense. Today, it’s serious and scientific

Though I'm repeating myself here, I feel the need to note that "life" doesn't imply "technologically-capable life."

Suddenly, everyone is talking about aliens.

For those of us paying attention, people have been talking about it all my life, and even before.

The next big multibillion-dollar space telescope (the successor to the James Webb) will be tuned to search for signatures of alien life on alien planets and NASA has a robust, well-funded programme in astrobiology.

Good. I hope they find some. Not Klingons, of course. Even just evidence of microbes or their equivalent would be a Big Fucking Deal.

Meanwhile, from breathless newspaper articles about unexplained navy pilot sightings to United States congressional testimony with wild claims of government programmes hiding crashed saucers, UFOs and UAPs (unidentified anomalous phenomena) seem to be making their own journey from the fringes.

On that front, the science is far less secure.

For decades, scientists wanting to think seriously about life in the Universe faced what’s been called the ‘giggle factor’, which was directly related to UFOs and their culture.

I once tried to start a conspiracy theory that went like: We have been trained to mock UFO enthusiasts because the extraterrestrials among us wanted to ensure that anyone who called them out for what they were got scorn heaped upon them.

I suck at starting conspiracy theories.

But for me, as a researcher in the field of technosignatures (signs of advanced alien tech) – the new face of SETI – getting past the giggle factor poses an existential challenge.

Once again, there's an implied conflation between "life" and "technolife." I call it that to avoid the inevitable tired jokes about "intelligence." You humans can be considered "technolife," but there's nothing that we know of about evolution that requires its development. There's probably exolife in our near stellar vicinity (perhaps even in our solar system), but I'd be surprised if technolife was anything but rare.

It is, however, in our best interests to search for both. It's probably not in our best interests to keep beaming signals to Proxima Centauri with messages like "Visit scenic Earth!"

There follows a bit of history about seeing UFOs (more recently renamed to UAPs, likely in an attempt to avoid mockery). And also the history of people looking for alien life.

I have no doubt that people have seen weird shit in the sky. What I have an issue with is immediately jumping to the "must be aliens" conclusion. It's like seeing ghosts. Perhaps you did see something; that doesn't mean it's a dead person's disembodied spirit.

Remarkably, when it comes to exoplanets, we are now also able to see exactly which planets are in their star’s habitable zone, where liquid water (the key, we believe, for life) can exist. That means we know exactly where to look in our search for life (something Drake could only dream of).

The "habitable zone" thing is real, but again, some people seem to equate "habitable zone" with "definitely must harbor life." No, it's a probability thing. If you're looking for a bear, you look in the woods, not a desert. If you're looking for a fish, you look in the ocean, not a mountain. Yes, sometimes you get fish in mountain lakes or streams or whatever, but your best bet is ocean.

Also, keep in mind there's a real chance that Jupiter's moon Europa harbors (simple) life under its ice cover, and it's not in the Sun's "habitable zone."

We’ve also demonstrated that there is no reason to suppose that biosignatures will be more common than technosignatures.

Yeah, that sounds like wishful thinking to me.

Since the exact same techniques are required to search for both bio- and technosignatures, there’s every reason to carry out both kinds of search at the same time.

But please, keep looking.

But if my colleagues and I claimed we’d found life on another world, we’d be required to provide evidence that meets the highest scientific standards. While we should let future studies lead us where they may, there is simply no such evidence surrounding UFOs and UAPs that meets these standards today.

And that bit is really what I care about: evidence. Not this leap to conclusions that definitely wasn't instigated by the aliens hiding among us.

The next big space telescope NASA is planning will be called the Habitable Worlds Observatory.

I'm betting that's a working title and it'll end up being named after someone, like the others were. One hopes they choose someone not associated with bigotry, this time.

Just remember, when they find biosignatures, don't immediately leap to "little green men" or their pop-culture alien equivalent. We're trying to keep our existence a secret, dammit.

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