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:

Write_Mikey_Write!

about 24 hours 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.

December 4, 2019 at 12:27am

#970931

Science Is Almost Always Wrong, And That's A Good Thing

Today, I'm going to snark on physics. Don't worry; it's an article for the uninitiated.

https://www.quantamagazine.org/einstein-symmetry-and-the-future-of-physics-20190...

The Simple Idea Behind Einstein’s Greatest Discoveries

Lurking behind Einstein’s theory of gravity and our modern understanding of particle physics is the deceptively simple idea of symmetry. But physicists are beginning to question whether focusing on symmetry is still as productive as it once was.

The flashier fruits of Albert Einstein’s century-old insights are by now deeply embedded in the popular imagination: Black holes, time warps and wormholes show up regularly as plot points in movies, books, TV shows.

Yes, and most of them get it wrong. I mean, seriously, how hard is it to get the damn science right? I personally know a guy who works as a science consultant to certain TV shows, and the problem is they don't listen to him.

Perhaps ironically, though, what is arguably the most revolutionary part of Einstein’s legacy rarely gets attention. It has none of the splash of gravitational waves, the pull of black holes or even the charm of quarks.

You know, probably about 5% of your readers got that last pun. I hope it was worth it.

The key insight came to Einstein in one of his famous thought experiments. He imagined a man falling off a building. The man would be floating as happily as an astronaut in space, until the ground got in his way.

1. There weren't any such things as "astronauts" then; 2. Newton figured shit out with falling apples, not humans. WTF, Albert?

It took a while for him to pin down the mathematical details of general relativity, but the enigma of gravity was solved once he showed that gravity is the curvature of space-time itself, created by massive objects like the Earth.

For the record, this is a little bit misleading. Yes, he solved one "enigma" of gravity (the article specifically mentions the one; it's how heavy and light objects fall at the same speed). But physicists are still ultimately puzzled by gravity. Maybe if they find (or fail to find) the hypothetical "graviton," some of that will clear up, but science is science, meaning that'll just lead to more puzzles.

So the one fundamental force that we all feel all the time - with the notable exception of a select few astronauts, for brief periods - is the one we have the least understanding about. Life is funny that way.

Nearby “falling” objects like Einstein’s imaginary man or Galileo’s balls simply follow the space-time path carved out for them.

That's going to be my new curse. "Hey, Waltz, someone keyed your car." "Galileo's balls!"

At the same time, symmetry-based reasoning predicted a slew of things that haven’t shown up in any experiments, including the “supersymmetric” particles that could have served as the cosmos’s missing dark matter and explained why gravity is so weak compared to electromagnetism and all the other forces.

1. Dark matter isn't "missing;" they're all but certain it's there; it's just that they have no idea what it is. 2) The relative weakness of gravity starts to make sense if you hypothesize invisible dimensions. ... which I suppose doesn't help much.

Periodically - and fortunately, this particular article doesn't do this - you see someone going "Einstein was WRONG!" This plays to the masses, who yearn to see some genius or authority taken down a peg or two. But it displays an ignorance about the way science works. Was Einstein wrong, ever? Most definitely. For all his vaunted intelligence and insight, he was human, and humans are wrong from time to time (some more than others, granted).

Einstein was no more "wrong" than Newton was wrong; he just didn't have all the data. Newton's theories about gravity work to a really high degree of accuracy in both our everyday lives and the things we observe. Einstein extended them. It's analogous to how we think of the Earth itself: it's basically a sphere, and can be modeled as such for all kinds of purposes, such as charting airline flight paths or drawing maps. But to get greater precision you have to take into account the difference between the equatorial diameter and the polar diameter, which is (roughly) 40 miles, for most purposes a rounding error in the average diameter. Discovering the equatorial bulge doesn't make "the Earth is round" wrong; it just adds precision. And of course it's also lumpy, but not 40 miles worth of lumpy; Everest, for example, tops out at only about 5 or 6 miles above sea level.

Point is, though, the theories Einstein came up with have withstood numerous experimental tests. So no, he wasn't "wrong" in that sense. But we keep searching for more refinements to the theories. And I find that sort of thing fascinating.

3 Like • Comment

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