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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 23 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

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.

December 31, 2019 at 12:15am

#972288

Tribe

Well, here we are at the end of December, the end of 2019, and (in popular imagination anyway) the end of a decade.

If I'm being pedantic, the decade ends a year and a day from now (2020 is a leap year). But let's not be pedantic.

No one seems to know what to call this past decade. I came up with a name for the first decade of the 21st century: the noughties. It didn't catch on, though I did see the BBC use it once. It is, after all, rather British. I don't have a similarly awesome name for 2010-2019.

Next month, I'm once again participating in "30-Day Blogging Challenge ON HIATUS"

[13+], so I'll switch back to that format tomorrow. But for today, I have an older article from 2013. I selected it at random, but it's appropriate enough.

http://nautil.us/blog/we-are-all-princes-paupers-and-part-of-the-human-family

We Are All Princes, Paupers, and Part of the Human Family

Chances are, if you have a famous ancestor far enough back that finding out about them is a surprise, you share them with a small city of other people. And the farther back you go, the truer that is.

Chances are, you have a famous ancestor, period. Ghengis Khan if no one else. Or at least a significant one.

This line of thought led to the revelation that everyone of European heritage alive today is a descendant of Charlemagne, who ruled over much of Europe as the first Holy Roman Emperor.

That gives me an excuse to post one of my favorite songs.

It doesn’t get any less weird when you look at it from the other angle: While you more than likely have four distinct grandparents and eight distinct great-grandparents, past a certain number of generations back, your number of ancestors stops growing exponentially, because they start being the same people.

A good point.

Stretch this back a few thousand years and you can see how you wind up being related to every other member of your species.

Stretch this back a lot farther than that and you can see how you wind up being related to your cat.

Geneticist Luke Jostins did a nice mathematical analysis and estimated that you have only about a 12 percent chance of being genetically related to an ancestor 10 generations ago; by the time you get to a 14-generation ancestor, the probability is nearly zero.

And that's why having kids just to pass along your genes doesn't make any sense. Never bought that argument for it, and I have plenty of other reasons not to have reproduced, but people keep talking about it. Yeah, biological drive and all that, but we also evolved to understand these things and have the persistent illusion of choice. (There are, of course, plenty of other reasons to have kids, and for most people, these seem to overwhelm the reasons not to. That's okay. I'm just talking about my own choices here.)

And the illustrious folks we might like to lay claim to have so many descendants that being descended from them is kind of a moot point.

I have a friend whose brother traced their family line back to Pocahontas. I don't know how accurate the genealogy was, but it was a point of pride for him. While this is cool, if true, I didn't tell him he's probably not one in a million, but one of a million.

The article ends with:

We’re all part of this enormous human fabric, full of fascinating tendencies and bizarre biochemistry. And research is revealing more and more about humanity as a whole and our incredibly beautiful, incredibly unlikely perch in the universe. That’s a tradition to be proud of.

There is no "us" and "them." If I had one wish for the twenties, it would be that we continue to work toward ending tribalism, or at least its more negative effects.

It's not going to happen, but the end of the year is traditionally the time for hopes, right?

4 Like • Comment • View 3 comments

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