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

June 19, 2023 at 8:22am

#1051306

Unplanned Obsolescence

When it comes to invention, some of us can be pretty clever. Well, not me. But some of us.

5 Fascinating Technologies the World Forgot Once They Were Replaced

From Cracked, so the usual countdown list and jokes.

An icon of a floppy disk represents saving in Word documents. That’s even though floppies have been almost nonexistent for years, and haven’t been the primary place for saving docs in decades.

And? Some "hang on a sec" computer icons display an hourglass, which is even more obsolete.

“Kids today,” more than one commentator has speculated, “probably don’t even know what a floppy disk is. They just think that’s the ‘save symbol,’ and it doesn’t stand for anything in real life!”

And those ignorant bastards also think # (technically called an octothorpe, which I mention because that's an awesome name) means something other than "number" or "pound."

5. Before Trains, Horses Pulled Goods Through Water

Okay, I can kind of excuse being ignorant about obsolete technology. What I can't excuse is being ignorant about music, and if you weren't exposed to "Low Bridge, Everybody Down" as a kid, your teachers failed you.

In England, people were using these horse barges as far back as Roman times, and the practice came back during the Industrial Revolution, once we finally had a lot to transport. In the 1790s, one single canal eased transport so much that the price of coal halved. The country approved dozens more canals in the years immediately thereafter, and by the middle of the next century, they measured 5,000 miles in length.

The canal system in England is fascinating, and we don't hear much about it here in the US.

4. Before Plastic, We Mixed Sawdust With Blood

Okay, some older technologies won't be missed. Though with current whingeing about plastics, perhaps this one will stage a comeback tour.

Hemacite sounds like some sort of mineral.

Yeah, that's because there is a mineral called hematite.

But as all you vampires and fans of etymology know, hema- means blood.

As does hemo- but not hemi- or homo-.

People manufactured hemacite to deal with a pressing problem: a surplus of blood. Slaughterhouses produced blood in huge quantities, and people had limited success using the sticky stuff in fertilizer.

Apparently, that's not a big an issue anymore. So if there's a resurgence in hemacite, maybe we can use blood from organ donors. Blood's an organ, right?

3. Pre-Satellites, We Used to Bounce Signals Off the Moon

Scientists first used a satellite to relay human communications in 1954. This milestone will raise some eyebrows from most students of space history, who know that Sputnik didn’t launch until three years later.

Saves me from quibbling about how the moon is a satellite.

The military stopped using the “Moon bounce” in the 1960s, now that they had communications satellites. But amateur radio operators can still totally try EME themselves today. It’s not easy, but you can do it. May as well get to practicing now, so you’ll be prepared once a catastrophe knocks out all satellite communications and you need to send a message of hope to all survivors worldwide.

Except the moon needs to be in the sky for this to work, so at best, you'd only catch about half the globe. Unless, of course, the flat-earthers are right [spoiler: they are not].

2. Letterlocking: Envelopes Before Envelopes

This bit is actually pretty cool, and much less gross than hemacite. I'd been seeing various articles about it recently. I don't know why; maybe because it's more secure than Twatter?

The most secure way to send a message is physically. Seal it in an envelope, and the spies who pursue you will not easily be able to open it without leaving some evidence behind.

Because we're all being spied upon, even outside the internet.

1. People Had to Listen Really Closely for Planes

And I can't really do this one justice; you have to see the illustrative photographs.

So, not really a lot of deep meaning in today's article. Just some cool history stuff.

2 Like • Comment • View 1 comment

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