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 20, 2019 at 12:43am

#954627

Balance

Today is the equinox.

You might have heard stuff about the moon tonight too. "Super moon" they call it. There's some science behind this, but as always, popular descriptions of it show up highly exaggerated. Here's a link to a pretty good explanation.

https://earthsky.org/?p=301190

Just to clear some things up:

No, you can't balance an egg today and only today. That's idiotic.

You won't even notice the slightly larger moon. But that shouldn't stop you from taking a few minutes to look at it. Anything to get people looking at the sky is a good thing.

I'm not going to go into my usual rant about Blue Moons (of which there is one coming up in a couple of months). From the above article:

Last but hardly least, this March 2019 full moon gives us the first of four full moons in one season (between the March equinox and June solstice). Most of the time, a season – the time period between an equinox and a solstice, or vice versa – only harbors three full moons. But since this March full moon comes very early in the season, that allows for a fourth full moon to take place before the season’s end.

Some people call the third of four full moons in one season a Blue Moon. So our next Blue Moon (by the seasonal definition of the term) will fall on May 18, 2019.

By which they mean that correct people call this the Blue Moon. The other definition:

The next Blue Moon by the monthly definition – second of two full moons in one calendar month – will come on October 31, 2020.

...is, simply, incorrect; and that's all I'm going to say about it here. I've ranted enough about it elsewhere. Still, it's something to look forward to next year: a full moon on Halloween. No meaning to it, unless you've read a certain book by Roger Zelazny, but it's cool anyway.

Again, if it makes you go look at the Moon - fine.

Bottom line: Here in the Northern Hemisphere, soon it will be spring. Finally. Naturally, I saw a weather report for freezing temperatures here next week, so my satisfaction at this change of season is tempered by that.

On a personal note, it was on the solstice that I committed to eating less and exercising more in an effort to lose weight. All I'll say about that is that I've been successful - so far. Eventually, I will fail. And yeah, I had a couple of overindulgences over the past three months, but I've always managed to get back on track.

It won't last, but it's something of an accomplishment for me.

Edited to add:

At this moment, the link above isn't working properly. It's not my fault. But this is pretty cool too. https://apod.nasa.gov/apod/ap190320.html

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