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 3 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 5 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.

September 17, 2020 at 12:02am

#993535

Lessons

Sometimes I wonder if I've learned any life lessons at all.

PROMPT September 17th

What life lessons have you had to learn the hard way?

Wait... there's an easy way? Dammit, I've been doing it wrong!

I have a bit of a stubborn streak. While I'm flexible enough to change when new data comes in, I want to discover the information for myself.

That is to say, hypothetically here, if I'd spent my childhood in an underground bunker, and people kept telling me the sky was blue, I'd only believe them provisionally -- better if I could escape and see it for myself.

This has benefits as well as downsides. As a benefit, it keeps me curious, always wanting to know more. For instance, you might ask, as kids do, why said sky is blue. Turns out it's from preferential scattering of light in the atmosphere, the same effect that gives a rising or setting sun a more reddish hue. The details are important, but no need to get into them; the point is, I had to figure this out for myself by reading and learning about physics.

But what's especially bad for me is that this means that some of life's unwritten rules, those social forces that guide behavior, don't really sit well with me. I prefer my rules to be written. To take a simple example, it turns out that when you're visiting someone, you bring a gift. There, now it's written down, and I can follow it. But it took me a while to figure that one out.

All my life, I've wanted to know "why." When it comes to science, like the blue sky thing, that's actually pretty easy to figure out (and it was even B.I., before internet). But when it comes to social situations, the "why" almost always escapes me.

This is probably why I don't get invited to parties. Well, this, and the puns. But mostly this.

Like, in the Before Times, a handshake was considered a proper greeting. I thought I knew the rules: firm grip, eye contact, whatever. Not that eye contact is easy for me, but I fake it by looking at the person's nose. But then I went to shake someone's hand (as a teenager), and got corrected, "No, you wait for the female to invite the handshake." Okay, so it's different with men and women. Why? I don't get it.

I try to follow the rules anyway, most of the time, to try to make a good impression. I just don't know all of them.

There are science textbooks. There are no "being part of polite society" textbooks. Oh, sure, Miss Manners and all that, but most of what's in those texts makes no sense to me.

Don't get me wrong; I try to be considerate of others' feelings. But subtle cues always seem to escape me.

Anyway, some minor life lessons I've picked up the hard way:

Never go grocery shopping hungry.

Money can't buy happiness, but it can buy just about everything else.

Any internet argument that goes on long enough devolves into a discussion of epistemology.

It's okay to suck. You get better.

Making one's bed is a gigantic waste of time.

Never choose lodging solely on the basis of proximity to the brewpub.

Always carry a backup of essential items. Especially maps/GPS.

When a cat displays the soft fur of its belly? It's a trap.

I'm sure there are dozens more, if not hundreds, but that sampling should at least give you an idea of my mindset. I'm sure some people know some of these things intuitively, but like I said, I have to work on it.

And I'll probably be working on it until the day I croak.

6 Like • Comment • View 13 comments

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