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.

July 14, 2019 at 12:16am

#962565

Word World

In the age of the internet, most everything we say and do online will be preserved forever. Even though our opinions, beliefs, and actions change over time, should we still be held accountable for our words, even words spoken or written years ago?

Oh, boy. This can of worms.

Short answer: No. And yes.

Long answer:

One of the founding principles of the USA is freedom of speech. Much has been written on the subject - as one would expect, given that same freedom of speech - so I won't belabor the point, but the upshot of it is that, within certain circumscribed boundaries such as issuing death threats or inciting to riot, the government is not supposed to be able to imprison us for mere words. Hence, you can say "The President is a fuckwad," and a lot of people will disagree with you, or object to your use of profanity, but you can be assured that you won't be sanctioned or imprisoned for those words alone. In theory, anyway.

This was a big deal 250 years ago or so, when saying something mean about the King could get you hung in the public square.

Still, "the government can't arrest you for those words" is a pretty low bar to clear. It says nothing about how friends, family, potential employers, landlords, or your dog should treat you.

I vaguely remember a situation a few years ago when some chick said something on Twatter like, "Going to Africa. Hope I don't get AIDS" before she got on a plane to somewhere in Africa. I can only imagine how her head must have exploded when she landed, checked Twatter, and discovered a) a shitstorm of criticism, to put it mildly, and b) she no longer had a job.

Just because you have the right to say something, that doesn't mean you don't have to take responsibility for it.

However, I think sometimes we go too far with the unofficial, private sanctions. On the one hand, it's a useful tool for weeding out, say, a pool of candidates for employment: if Candidate A is on record saying the "n" word, and Candidate B is not, and they're otherwise similarly qualified, you go with Candidate B (assuming your business isn't run by racists). On the other hand, people do change, and at some point it becomes grossly unfair to restrict the opportunities of someone who said something stupid when they were 20 - especially if they've also shown regret or apologized.

Even most actual crimes have statutes of limitations, and incarceration limits.

Where to draw the line is a matter for debate - and in a culture that is founded on the principle of freedom of speech, there can be a lot of debate - but I'm asserting that there should be a line, based on each individual circumstance.

As for me, I just hope no one ever finds the "poetry" I wrote when I was 14. Shudder.

2 Like • Comment • View 5 comments

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