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 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.

January 20, 2023 at 12:01am

#1043396

'Sup, Y'all?

I thought y'all would want to see this.

‘Y’all,’ that most Southern of Southernisms, is going mainstream – and it’s about time

I use "y'all" as a second person plural, a part of speech that English otherwise lacks. And it can't always be inferred from context.

Southern Living magazine once described “y’all” as “the quintessential Southern pronoun.” It’s as iconically Southern as sweet tea and grits.

I like grits, but sweet tea can kiss my... ass.

“Y’all” fills that second person plural slot – as does “you guys,” “youse,” “you-uns” and a few others.

"You guys" is considered sexist these days, "youse" is still pretty much limited to a small area in the Northeast, and I'm not sure about "you-uns." I think Pittsburgh uses "yinz."

I’m interested in “y’all” because I was born in North Carolina and grew up saying it. I still do, probably a couple dozen times a day, usually without intention or even awareness.

I use it too, but more intentionally. I thought I used it more, but a quick search of this blog of over 2,000 entries only yielded 64 entries with "y'all." This would be #65.

Back in 1886, The New York Times ran a piece titled “Odd Southernisms” that described “y’all” as “one of the most ridiculous of all the Southernisms.”

Damyankees.

Like the Southern dialect in general, the use of “y’all” has often been seen as vulgar, low-class, uncultured and uneducated. As someone noted in Urban Dictionary, “Whoever uses [y’all] sounds like a hillbilly redneck.”

The only way to change this perception is to use it with intention.

The etymology of “y’all” is murky.

So is the etymology of a lot of other words.

My examples push “y’all” back 225 years before the citation in the “Oxford English Dictionary,” and they show that the word appeared first in England rather than the United States.

I think it’s important to point out that it originated in a more formal context than what’s commonly assumed. There are none of the class or cultural connotations of the later American examples.

Now, I can't be arsed to research this right now, but I think older versions of English made a distinction between second person singular and plural. That's how we got "thee" and "thou" and other constructions that are now associated with the KJV and maybe Shakespeare. Or something like that; like I said, not looking it up now.

Still, there it is, in an English poem written in 1631.

Not long after Shakespeare, really. Y'all Brits invented the language; we just perfected it.

“Y’all means all” – that’s a wonderful phrase that seems to be popping up everywhere, from T-shirts and book titles to memes and music.

Sounds good to me.

Now, how about we come up with a first-person plural that distinguishes between "us, including you," and "us, not including you?" Like if I said, "We're going to a party," does that mean you're invited? No. No, it does not, and now I'm embarrassed because you inferred that it did.

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