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 11, 2019 at 12:26am

#964037

Inalienable

Oh, here we go again.

https://undark.org/article/pitfalls-of-searching-for-alien-life/

The Pitfalls of Searching for Alien Life
Scientists looking for evidence of extraterrestrials can draw media attention but also cynical, even hostile, reactions from their colleagues.

This just in: scientists are humans (as far as we know), and humans have opinions.

In October 2017, a telescope operated by the University of Hawaii picked up a strange cigar-shaped object (artist rendering in top image), which had slingshotted past the sun at a more-than-brisk top speed of 196,000 miles per hour. Scientists at the university dubbed it ‘Oumuamua, Hawaiian for scout, and at first labeled it an asteroid, then a comet, but agreed that it came from another solar system.

Let me get this out of the way, since it's in the lede: it's remarkably easy to eject a perfectly natural, orbiting thing from a solar system. All you need is the thing you're ejecting, and an enormous mass to overcome the gravity of the primary. Hell, Jupiter could do it - and we've found many, many objects more massive than Jupiter orbiting other stars. Eventually the objects in their orbits line up just right, and *sproing*.

Consequently,

...Avi Loeb, the chair of Harvard University’s astronomy department, published a paper in The Astrophysical Journal Letters the following year theorizing that the object could be artificial.

Hey, it's great to speculate, isn't it? "Could be artificial" translates to "can't rule it out," but as I noted above, it's unnecessary to postulate an artificial origin. Occam's Razor does the rest.

I've gone into my argument against a proliferation of intelligent alien life before. The short version: I like Star Trek as much as anyone and more than most, but... no.

The long version: It comes down to one term in the infamous Drake Equation.

Specifically, f_i, the fraction of planets with life that go on to develop intelligent life (the definition in the popnote above will suffice for this). Naturally, all we have for that is one data point: us. This tells us nothing about the probability of a planet producing intelligent life. As an analogy, think about if you won the lottery, but you have no idea what the odds were of winning the lottery - all you know is that no one you know has won the lottery. You don't even know how many of those people have played the lottery. The chance could have been one in a thousand or one in 10¹⁰⁰; it doesn't matter, because your chance of having won the lottery is exactly unity, once you've won it.

We do know that it didn't take long for life to develop on Earth, and that it took a comparatively very long time for "intelligent" life to emerge. And we've only been building spaceships for half a century, compared to the 4 billion + years life has existed, or even the 1 million + years that humans have existed. Meanwhile, other species are thriving just fine on our planet, having evolved for exactly the same amount of time that we have (they would probably be doing better if we weren't around, but that's another issue entirely), so there's zero evidence that evolution must produce that kind of intelligence. All we know for sure is that it can, and we know that because we define ourselves, as a species, as "intelligent."

The article I linked, though, seems to make the usual mistake I see in journalism on this subject: it conflates "life" and "intelligent life." Throw a dart at the timeline of Earth thus far, and you're almost certain to hit a time containing "life," and almost entirely unlikely to hit a time containing "intelligent life."

“I’m really surprised they get blowback,” said physicist Richard Bower of Durham University in England. He’s never gotten flak for his research in cosmology, which entails making computer simulations of possible parallel universes. He’s concluded that life elsewhere could be quite common, and others in his field back him up. “We used to say that life is incredibly rare and we’re lucky to live on a habitable planet,’’ he said. “But we’ve now observed so many planets that are plausible habitats. It seems, based on scientific evidence, there’s no reason to think that planets like the Earth are rare.”

I'm not arguing that. I'm arguing that Earth may well be a galactic outlier in terms of producing a species that has space travel capabilities (as rudimentary as they are right now). We also don't know how common any kind of life is in the rest of the universe - but as it's a big universe, it's probably safe to say it exists, somewhere.

It's also important to draw distinction between scientific inquiry and fantasy, though. The search for extraterrestrial life is important. Finding so much as a microbe, even just a strand of proto-RNA or its equivalent on another world, well, that would have a huge impact on our understanding. But the topic has been forever tarnished by fringe UFO conspiracies and popular imagination, to the point where even writing serious papers about it invites speculation about flying saucers, death rays, the JFK assassination, and whatnot.

Hence the blowback when serious scientists start speculating about alien intelligence. Such speculation is not without merit - it needs to be done. It's just that when being reported on, it gets all wrapped up in popular imagination and cultural zeitgeist.

Now, I'm not trying to change anyone's mind, here. You want to believe in little green men, or storm Area 51 next month, hey, that's your choice. And I could be wrong. I actually hope I am. All I'm doing is critiquing the way these things are presented in media. So when you see things like "this might be artificial in origin" or the ever-present "astronomers discovered an Earthlike exoplanet," just remember to take the news with a grain of salt. After all, to an astronomer, Mercury and Venus are Earthlike - and the last so-called "earthlike" planet I heard of was six times the size of our little world.

Imagine trying to escape that gravity well.

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