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 16 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 18 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.

March 28, 2020 at 12:01am

#979382

Groundhog Year

This one's been hanging out in my queue for three months. If I thought I'd still be around in 9.75 years, I'd drag it out then. As it is, I'm leaving it here so y'all can laugh about it at the end of 2029.

https://www.vanityfair.com/news/2019/12/new-technology-2020-will-change-life-as-...

“It Won’t Be Pretty”: How the Next Decade’s Technological Tsunami Will Change Life as We Know It
Self-driving cars, iPads, Uber: The technological advances of this decade have happened so quickly that their breadth is difficult to comprehend. The next decade? That, times ten.

The great thing about prognostications is how terribly wrong they all are in hindsight. Even, sometimes, three months later.

This time a decade ago, there was no such thing as an iPad.

As far as I'm concerned, there still isn't.

There were no food delivery meal kits.

Or these.

You didn’t speak to a machine called Alexa or Siri

Still don't.

or get laid with an app called Tinder.

hahahahahahaha

You stayed in hotels, not Airbnbs.

I have a prediction about Airbnb. It's short-term. You can guess what it is. Anyway, I've never used it. Too much potential for scamification.

You telephoned a cab company, rather than pressing a button and waiting for an Uber or a Lyft.

Okay, I do the ride-share thing. Or I did, back when there was anywhere to go.

You didn’t waste hours of your day on Instagram, scrolling from one box to the next like a gerbil running on a wheel as an algorithm watches and takes notes.

Still don't.

Jobs that are now performed by hundreds of thousands of people—Uber driver, gif-maker, social media influencer—didn’t exist.

Out of those three, only one is an actual job.

You likely read the newspaper in the morning, watched the news at night, and consumed a trickle of information in between by going to Yahoo News or through the RSS feed you’d painstakingly constructed, rather than drinking from the fire hose that is Twitter.

What's a "newspaper?" And I've never had cable TV, so no nightly news. I still have an RSS feed I comb through daily, because I fucking hate Twitter with the fiery power of a thousand supernovae.

Things felt slower.

Did not.

Now, this whirlwind of a past decade could be just a taste of what’s to come.

Or it could be the last gasp of a dying civilization.

We’re just now seeing deep fakes used to make even more believable fake news for political gain.

Thus leading to the inevitable conclusion that everything you see is fake, so you start to disbelieve everything.

And yet a decade from now, on the eve of 2030, we’ll look back at today in astonishment at how primitive life was in 2019.

While the last dregs of humanity are stalking bison on the prairie.

By then, it’s likely that cars will drive themselves.

Right, because we'll have convinced people that 10,000 deaths a year from driverless cars is objectively better than 30,000 from human drivers.

They won’t even look like cars, more like traveling gyms or gaming cars or mobile beds to nap on during your commute.

Mine would look like a drunk tank. What's a commute?

Some will fly. (Maybe most of them will.)

Uh huh. And we'll have unlimited clean energy from fusion, and warp drive, and teleportation, and a post-capitalist utopia.

Your phone could be replaced with a contact lens, or some glasses that (finally) look like glasses.

A little bit of actual thought would tell anyone why contact lenses won't work for communication, at least not until we have a better brain/machine interface, at which point, what's the use of lenses?

On the flip side, health care will go through the most substantial changes since the invention of antibiotics.

Snort.

And you might get to live until you’re 150.

In this world? Dear gods, please, no, just end it already.

My guess is that a decade from now, society will look nothing like it does today—and it won’t be pretty.

And finally we get a prediction we can actually bank on.

That's right. I poked my head out of my hole and saw my shadow, and I'm predicting six more decades of winter.

3 Like • Comment • View 5 comments

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