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 2 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 4 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.

April 2, 2021 at 12:02am

#1007556

Scout Chip

I know I've talked about this before, but here's Cracked's take on it.

5 Facts About That Enigmatic Object That Invaded Our Solar System

A couple years ago, a big hunk of something flew into our solar system. It took a bit longer for the world to really stand up and pay attention to it, and we're still keeping an eye on it, even though it's now billions of miles away now.

You said "now" twice. So it's twice as now?

Hey, look, I'm not claiming I never make mistakes. I also don't have an editor.

5. First, The Most Important Part: What's With That Name?

Just kidding, the name isn't the most important part, but let us set this thing up for you.

In October 2017, astronomers were looking up at the sky, which is their favorite thing to do. They thought they spotted a comet, so they named it, using the standard comet designation protocol. Then they looked closer, saw no hint of a tail, and figured it had to be an asteroid, which meant they instead needed the asteroid designation protocol. But as they got out their notepads and tried to trace its path, they realized it had to have come from outside the solar system. It was the first ever identified interstellar object, which was very exciting, because it meant they had to create a NEW designation protocol.

You know, the way this is generally reported, it makes it sound like nothing ever passes through this solar system from outside it, that this is a once-in-a-billion year happening, that it's unique and precious.

Of course, we don't know how often it happens because we've only recently had the tools to detect that it's a) an object that b) comes from outside the solar system. This was the first detected, not the first ever. Probably. Or the last. Probably.

Cracked, unlike any other article I've seen on the subject, goes into detail about the origin, prononciation, and meaning of its name: 'Oumuamua. So, way to go, dick joke site?

4. We Don't Actually Know What This Thing Looks Like

At the top of the page, and copied again right below these words for your convenience, is an artist's impression of 'Oumuamua:

For which I'm afraid you're just going to have to click on the link. We only get a limited number of image files here, and I'd rather not use up all of mine on rocks.

Scientists, and the media, unanimously refer to this as "cigar-shaped," which is the term they use when they want you to think about dongs but are too afraid to just come out and say "dong."

Way to go, dick joke site.

All this is to say -- and this will become important in a minute -- there's a whole lot of very basic info about 'Oumuamua we don't know. But we have been able to monitor its movement, and that's what's stumped so many scientists.

Surprised they didn't joke about it moving in and out and in and out and in and out...

3. 'Oumuamua Is Moving Very Weirdly

As those astronomers were tracking 'Oumuamua (during the few breaks allowed to them by their droves of oversexed groupies), they realized their interstellar visitor could not have come from the fairly close star system Vega just 600,000 years away, nor could it have come from any of the first several other close alternatives they checked out. The paths just didn't line up. Instead, 'Oumuamua might have spent billions upon billions of years traversing the cosmos before it landed in our solar system.

Which is really kinda cool. By the way, they're not saying Vega is 600,000 light years away; I believe they're implying that at the speed this thing was traveling, it would have taken it that long to get here from the vicinity of Vega. If you want to know for sure, ask a Vegan. (You have NO IDEA how long I've been wanting to make that joke. NO IDEA.)

But that's not the weird part about its motion. The weird part: 'Oumuamua was accelerating. Not just accelerating due to the Sun's gravity, as expected, but also accelerating due to ... something else.

And this, as I mentioned before, inspired a lot of people, including a supposedly respectable dude from Harvard, to start shrieking "ALIENS! ALIENS! AAAAALLLLIIIIIEEENNNNS!" But, as Cracked loquaciously puts it:

2. So, Is This An Alien Spaceship?

No.

2. C'mon. Couldn't It Be Aliens?

Okay, we can't rule out that aliens made it. But if you've heard the theory that 'Oumuamua is an alien spaceship -- and if you heard of 'Oumuamua before today, you probably heard this theory -- you should know that despite all the coverage it got, this idea comes from just one dude. Well, two dudes, counting his paper's co-author, who doesn't seem to chase the spotlight as much. Well, three dudes, if you count the ancient aliens guy who has endorsed the theory.

And since I harped on this before, I'm just leaving this bit here.

1. Let's Go And Find Out For Sure

There's a reason we're still talking about 'Oumuamua today, and that reason is Project Lyra. This group of scientists is considering sending an interstellar ship to catch up with 'Oumuamua, even though it's already billions of miles away and going 500 million miles farther every year.

Are they nuts? No, no they are not. We absolutely do want to catch up with interstellar objects eventually, and 'Oumuamua's a good candidate because at least we know where to find it. The other plan for reaching an object like that has been approved by the European Space Agency and involves sending a vessel into space by the end of the decade and then just chilling for a while till we figure out where to direct it. Compared to that, Project Lyra seems downright reasonable.

The idea that we can chase this sucker with a probe is *almost* as farfetched as the "alien ship" hypothesis.

Before they launch, Project Lyra also needs funds, and personnel, and manufacturing facilities, and lots more. The group behind it (the "Initiative for Interstellar Studies"), they're just some scientists writing a proposal, not a space agency. But we're absolutely rooting for them to come up with a plan that makes sense and to put it in the hands of those who can make it happen. And if they do discover that 'Oumuamua is a functioning spacecraft, we will admit we were wrong, and then we'll commandeer the vessel and fly it back to where it came from.

But we're clever (except for the guys shrieking about aliens); we can find a way.

Because that's what we do. When we're not making dick jokes, anyway.

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