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!

about 8 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

about 10 hours 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.

January 10, 2024 at 9:25am

#1062150

Material World

More from that internationally renowned science reporting site, Cracked:

5 Alien Materials That Exist Naturally Here on Earth

Naturally, I need to nitpick the headline.

Planet formation is an active area of research, so they might not have the precise sequence or timeline nailed down. But whatever the actual mechanism, it's pretty clear that everything on Earth, and even the Earth itself, came from outer space. We still get literal tons

of material from outer space every day. Everything, including your body and the air you breathe, has an alien origin: it was not of Earth until it was, at which point some of it started getting recycled, mixed, and recombined.

Astronauts and space pirates are exploring the cosmos, in search of unobtanium, transformium and precious red matter.

That multi-source science fiction reference is doubly amusing to me, as I just downloaded the game Starfield yesterday (featuring astronauts and space pirates and valuable materials) and started playing it.

Their quest may be fruitless, since none of those materials really exist.

Well, philosophically, anything we can think of "really exists" in some manner. But no, like the Easter Bunny, unicorns, honest politicians, and (probably) Bigfoot, they're not part of consensus reality.

5. Flammable Ice

Yes, water is a mineral (just one that happens to be liquid at room temperature), and when methane is trapped in water crystals, that’s called flammable ice. It has a second, equally extreme name as well: “fire ice.”

While I generally don't bother fact-checking stuff from a comedy site, this sounded familiar, so I looked it up. Yep, they're talking about {xlink:https://en.wikipedia.org/wiki/Methane_clathratemethane clathrates. They even lifted the image in the article (and put "methane clathrate" in its caption) from that Wikipedia page. In their defense, the photo is credited to USGS, which I'm pretty sure means it's in public domain, or at least freely licensed.

What exactly a clathrate is, I leave to any curious readers to follow appropriate links. It's a more generalized thing, and fascinating, but irrelevant to the article or my commentary.

So, is this some promising new energy source, ready to power our kitchens and our factories? Maybe! Or, maybe the warming oceans will melt the ice and release gigatons of methane, setting off a chain reaction of further global warming to push us ever closer to catastrophe.

Burning methane produces carbon dioxide, so why not both?

4. Snail Teeth

What is the hardest substance in the world?

A congresscritter's head.

Many of you will answer “diamond.”

And many don't understand the difference between material hardness, toughness, and strength. And pure stubbornness, as with my answer above.

We’ve also found a natural composite material harder than diamond, in a most unlikely place: inside the mouths of snails.

Doubt they'll try to make jewelry out of that.

If you think you should now fear these snails, who will cut and smash through your body without mercy, put those worries aside. In reality, your body is so fragile that any animal’s teeth could make short work of it...

As some cats I've known were prone to proving.

3. Glass Bombs

We’re not totally sure where australites came from. Once we dispensed with the volcano theory, we dubbed them extraterrestrial, having come from meteorites, but we don’t think that’s true anymore. Instead, we think an asteroid struck the ground in the distant past, somewhere around 800,000 years ago. It hit so hard that it threw terrestrial material into space. This material then came back, now transformed by its journey and smoothed through reentry.

As I noted above (of course I had a point to my nitpicking), that terrestrial material was actually extraterrestrial material that had been here longer.

2. Spontaneously Combustible Coconuts

...is the name of my heavy metal Jimmy Buffett cover band.

You’ve run into coconuts many times when you’ve been stranded on desert islands or when you bought one from a roadside vendor. And yet the dried flesh of the coconut holds strange properties and is a class 4.2 hazardous substance.

No idea where the joke ends and facts begin here, and can't be arsed to look it up.

1. Fire-Forged Feces

Bird shit has always held fascinating properties, sending us rushing to mine the stuff to exploit it and sometimes sparking actual wars. When we weren’t using it as fertilizer, we were digging out compounds from it to use in creating gunpowder.

Pretty sure those are the same material: phosphorus.

Like everyone, the kestrel poops. Sometimes, it poops above a Russian coal mine, where the blasts of hot gasses transform the guano into something new. It becomes a mineral, which can be colorless, reddish or lilac.

But, sadly, doesn't spontaneously combust.

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