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 11 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 13 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.

May 14, 2022 at 12:01am

#1032374

Any Storm in a Port

And now for another fragment of history...

On This Day
14-May-1915
May 14 Revolt

As with most of my oeuvre, I select these things semi-randomly, and in this case, I find myself way out of my comfort zone. That's okay. That's one reason I'm doing this.

Portugal is not a country I think about often. I mean, sure, I have a vague desire to visit there, but I have a vague desire to visit lots of places. I'm aware that it was once a major naval power -- for one thing, they achieved what Columbus failed to do, find a sea route to Asia -- and that it once controlled Brazil. Other than that, the true importance of Portugal to me has always been the existence of port wine, which is delicious and can get really expensive.

So, apparently, Portugal was a monarchy up until 1910, at which point the peasants revolted. This was, of course, long after France, a few years before Russia, and don't ask me about Spain because that history is a real mess.

This Wiki page is particularly questionable, and when that happens I usually look at their listed sources for clarification. But in this case, all the sources are books that I can't be arsed to buy, and web pages that are in -- believe it or not -- Portuguese. Half of them are 404 anyway.

But of course that's not stopping me from riffing on it.

Anyway, from what I can gather, the May 14 Revolt was a revolt of the revolt. That is, the supposedly representational government that got installed after the 1910 overthrow of the monarchy. According to this Wiki page

, "The Republican Party presented itself as the only one that had a programme that was capable of returning to the country its lost status and place Portugal on the way of progress."

Wow, history really does echo, doesn't it?

Point is, as is often the case in these situations, the government they replaced the government with was dysfunctional, so another revolt was called for in order to restore the values that the first revolt was supposed to have installed but didn't. At least that's my take on it; like I said, there's not a lot of detail on that page.

The insurrection was, from what I can gather, swift, bloody, and successful.

At least, at first. A few years later, there would be yet another bloody revolution, this one getting the country back to a dictatorship. It wouldn't be until 1975 that the country returned to some semblance of democracy.

From this, I have learned several things, including but not limited to:

Sometimes, revolution is necessary and ultimately beneficial, if bloody

I would not have wanted to live in Portugal before 1975

I'd venture to guess that most people don't know this bit of history, from which it follows that they are doomed to repeat it.

Democracy is fragile

Those bullets are meant to be bottles of port. Preferably 40 year vintage port, which at this point could have been produced after Portugal achieved a semi-stable democracy. Port gets its name from the city of Porto, where, historically, the wine was shipped out to the great benefit of other countries (and the financial benefit of the vineyards and Portugal).

Oh, and one other interesting bit. The name "Portugal" is of somewhat disputed origin, but is most recently derived from the Roman name for it, Portus Cale. You can guess what Portus means. Cale is the disputed part

. It might have been a Celtic word for "port," which would make the name of the country translate to "Port port."

I like that.

As a final disclaimer, I might have gotten some of this shit wrong; like I said, it's not something I've looked into before. Don't take my words (or Wikipedia's) as the gospel truth. Just consider this another port in the storm.

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