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!

a day 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

a day 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 7, 2019 at 12:23am

#953874

Jersey Devils

Reality is weird.

I mean, the universe is overwhelmingly, mind-bogglingly, surrealistically weird.

Peek outside your ordinary, day-to-day, Newtonian experience, and things get really bonkers.

And yet...

And yet, for some reason I have experienced but do not comprehend, people still have to invent weirdness, as if reality weren't strange enough.

Case in point:

https://gizmodo.com/ongs-hat-the-early-internet-conspiracy-game-that-got-t-18322...

Ong's Hat: The Early Internet Conspiracy Game That Got Too Real

Ong’s Hat is one of the internet’s earliest conspiracy theories, but before that, it was a place, a ruin almost 3,000 miles away from Santa Cruz, deep in the woods of New Jersey’s Pine Barrens. Rumors swirled for years that something profound had once happened there, a confluence of mad science and the paranormal that had warped reality itself, opening a door into strange, unfathomable worlds.

It's rare that one can trace the actual origins of a legend. Things get lost to time and entropy, the people involved die, and ideas morph and take on traits of similar ideas. If it lasts long enough, it becomes part of popular consciousness: Bigfoot, the Roswell Incident, various religions, etc.

But this is one of those rare cases, its origins clearly set out as deliberate fiction. And in much the same way as Orson Welles' radio broadcast of War of the Worlds was reported to cause people to think it was reality (reports which, incidentally, were themselves overblown, thus resulting in a kind of mythception as many still believe many people were fooled), this had the effect of actually warping reality.

By which I mean this: At some point, it doesn't matter whether something has a concrete reality or not, because all it takes is a belief in the thing to evoke thoughts and actions exactly as if it were real. To avoid offending too many people, I'll bring up Santa Claus as an example. We're all adults here, and we know that there's no physical being possessing all the traits attributed to Santa Claus. And yet, every Christmas, many of us like to pretend there is - and in pretending, give him a kind of reality.

Humans have a tenuous grasp, in general, on the line between fact and fiction. Some things we know are fiction - Harry Potter, e.g. Some things we know are fact, such as the JFK assassination (even if the details surrounding it are subject to myth-making). And then, some things... well... depending on the individual, some things blur the distinction.

This may be a uniquely human trait. I don't know. I suspect it's related to our propensity for coming up with new ideas. After all, in order to create an invention, we first have to imagine it - to make it a reality in our heads. Some imaginings can be made to become physical things, and thus we go from chipping a wheel out of stone to sending robots to Mars.

Clearly, though, this trait has a dark side, in that imaginings become obsessions, as with the Ong's Hat thing linked here.

I'm just not sure we can have the one without the other.

2 Like • Comment • View 2 comments

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