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:

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.

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.

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

February 2, 2023 at 12:01am

#1044036

A Star is Born

Not a lot I can say about this one; I just thought it was too cool not to share.

Before I get into it, though, a quick note: I'll be participating in "Journalistic Intentions"

[18+] this month (though not today). Check it out and join in—you have nothing to lose and everything to gain. Well, a couple of awards to gain, potentially, but that's not nothing.

And now, let there be light.

Astronomers Say They Have Spotted the Universe’s First Stars
Theory has it that “Population III” stars brought light to the cosmos. The James Webb Space Telescope may have just glimpsed them.

A group of astronomers poring over data from the James Webb Space Telescope (JWST) has glimpsed light from ionized helium in a distant galaxy, which could indicate the presence of the universe’s very first generation of stars.

Don't worry; the article later explains this better later on.

These long-sought, inaptly named “Population III” stars would have been ginormous balls of hydrogen and helium sculpted from the universe’s primordial gas.

I guess "ginormous" is a scientific term now, probably along with "bajillion" and "metric shit-ton." As in, there were a bajillion ginormous stars in the early Universe, and each one weighed a shit-ton.

Theorists started imagining these first fireballs in the 1970s, hypothesizing that, after short lifetimes, they exploded as supernovas, forging heavier elements and spewing them into the cosmos.

"Fireballs" is misleading, but "exploded" is apt.

That star stuff later gave rise to Population II stars more abundant in heavy elements, then even richer Population I stars like our sun, as well as planets, asteroids, comets and eventually life itself.

Got that? Summary: Pop III came before Pop II which came before Pop I. No idea what they're going to name the fourth generation of stars. Population X, I hope. If this seems backwards, it's because the classification system was developed before people figured out the order of things. Again, the article explains this later.

The early Universe was mostly hydrogen, some helium, a tiny bit (relatively speaking) of lithium, and not much else. It takes nuclear fusion or other processes to create the rest of the elements. This is why we say we're made of star material.

Confirmation is still needed; the team’s paper, posted on the preprint server arxiv.org on December 8, is awaiting peer review at Nature.

Important disclaimer, so you can be That Person at the party when someone gushes about how astronomers found the first star or something.

Because they are so far away and existed so briefly, finding evidence for them has been a challenge.

Remember, we're at the center of an inside-out universe; what's furthest away is oldest. Well. We appear to be at the center, anyway; so does every other point in the Universe.

The rest of the article goes into more depth, and while it's not nearly as paradigm-shattering as, say, detecting life on another world (a subject I've tackled in here before), getting confirmation, or even negation, of this model of the history of star formation would be a step forward in astronomy and cosmology. Thus, the cool factor.

4 Like • Comment • View 3 comments

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