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:

Brandiwyn🎶

about 22 hours 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

November 5, 2024 at 10:17am

#1079535

Anomalies

Ah, yes, November 5, and Election Day in the US. The UK will be celebrating an attempted terrorist attack, while over here, we're trying to avoid a terrorist attack.

The Random Number Gods have chosen to bless us with another Cracked link today.

5 Spots on Earth Where the Laws of Physics Fall Apart
There’s a gravity hole in the ocean

Okay, but, no, there are no spots on Earth where the laws of physics actually fall apart. Well, maybe, sometimes, at CERN, but they do it on purpose.

Still, these are rather interesting.

There are some places on the planet where things get weird. For instance, you ever heard of the Bermuda Triangle? Well, it turns out there’s nothing weird about that bit of the ocean at all — it sees a lot of traffic, but vessels that travel there are no more at risk than those anywhere else.

Like Bigfoot, that's not going to stop humans from making shit up about it.

5 Gravity Drops Near Sri Lanka

Once you learn that the force of gravity is slightly variable across the planet, it stands to reason that there are some spots where it's less and others where it's more. Finding out where it's less, though, that's what science does.

Gravity varies from place to place — and in some places, it varies a lot. In the ocean near Sri Lanka, gravity is so much weaker than in the rest of the world that the sea level is more than 300 feet lower than it would otherwise be.

Hey, I just came up with a fix for rising sea levels! Just increase the gravity of the Indian Ocean, and presto!

To know why gravity’s so low there, we’d have to burrow deep into the planet, and possibly cut it in two, which is inadvisable.

Awww.

Now, look, what that article's not telling you is that it's a minuscule effect. The variation from average, above or below, is about 0.5%. We'd never feel it. Sure, it has a profound effect on sea level, but look at what the comparatively really very tiny effect of the Moon's gravity does to that on a daily basis.

4 The Toasty Bit of Norway

In the Norwegian Sea, we have one disturbing bit called the Lofoten Vortex, where the water stores an unusual level of heat.

Lofoten Vortex can be the name of my Bjork cover band.

Yes, yes, I know Bjork is from Iceland, but come on, look at a globe. (A flat map inevitably distorts the distance between Iceland and Norway.)

3 On Top of Paraguay, Magnetism Disappears

I recently saw an argument for why the Earth's magnetic field isn't as important as we thought for maintaining our atmosphere. But this isn't about that.

Without looking at the map, if you had to guess one spot where the magnetic field gets weird, maybe you’d point your finger at one of the poles. But the planet’s rotational axis, which defines where we put the north and south poles, isn’t the same as its magnetic axis (which creates the magnetic field). As a result, we have this belt of radiation around the globe that dips down and comes close to us at this unlikely spot above Paraguay.

So, it doesn't "disappear." It just gets weaker there. How much weaker, I can't be arsed to look up. I can forgive a comedy site for hyperbolic headers, but they made it sound like, I don't know, compasses won't work in Paraguay or something.

2 The Tulsa Center of the Universe

You might not have known that the center of the universe is in Tulsa, but that’s what this spot is named, and Oklahoma wouldn’t lie to us.

Can't be the center of the universe; I don't live in Tulsa.

We’d go investigate ourselves, but that would mean having to spend time in Tulsa.

Good reason.

1 The Cave Where Energy Comes from Rock

I mean, technically, coal is rock. Uranium ore is rock.

But then you have Movile Cave in Romania. The interior is totally cut off from the outside world, and the creatures in there get no energy through photosynthesis, either directly or indirectly. Instead, the producers of this food web are bacteria that get their energy through chemosynthesis.

Here, though, they're talking about an entire cave ecosystem that doesn't rely, at its base, on solar energy input (apart from, you know, it not being frozen solid and all). Which is outside our normal experience, and definitely doesn't break the laws of physics any more than the other examples do, but is really interesting. Because now we know for sure that life can exist without photosynthesis.

Exist, yes, but imagine crawling out of that cave to go vote. Who are the candidates, now?

3 Like • Comment • View 2 comments

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