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:

Turkey DrumStik

about 8 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

November 18, 2018 at 1:07am

#945780

Thanks

With Thanksgiving just a few days away, there are three things I can be sure of:

1) A plethora of "helpful" articles on the internet about traveling over the Thanksgiving holiday;
2) Nonstop ads and news "stories" about Christmas shopping;
3) Endless talk about being grateful.

I've heard people wondering why atheists observe Thanksgiving. After all, who or what are we thanking? Well, fortunately for you, I'm here to atheixplain.

There are two distinct forms of gratitude. The first is what I talked about in my last blog entry: simply expressing gratitude to someone who has done you a service (whether or not you paid for it). Religious, or spiritual, people use God, or gods, or the Goddess, or the Universe, or whatever, as the target of their gratitude, because they feel the deity (or whatever) has done them a service, so for them, it's all this first type. I have some experience with this, because I used to be religious. Of course, there's a great deal of variation among religious/spiritual people, so I'm sure one can find counterexamples, but I'm writing about generalities.

The second form of gratitude doesn't have a target. Or, perhaps, the target can be one's younger self - which certainly doesn't qualify as a deity, but what Young You did might have set the stage for today's bounty. Either way, this kind of gratitude, I think, is more about expressing a feeling of appreciation for what one has. It's an internal thing, a kind of acknowledgement that, okay, things could always be worse, but I'm going to focus on the positive.

In a more logical language than English (should one actually exist), perhaps these two expressions might be described by different words, much as the Greeks had more than one term for "love," or (and this is probably apocryphal) some Inuit supposedly have a multitude of words for "snow."

Because of this second definition, atheists can be perfectly happy to celebrate Thanksgiving in the US. I'd argue that, either way, you're acknowledging that things could be worse - and that has certain psychological benefits. Rather than dwelling on regrets, drama, or missed opportunities, you're reflecting on all that is good and helpful in your life: family, friends, french fries, fuckbuddies, finances... whatever.

Also, there's stuffing ("dressing" for my fellow Southerners). That in itself is sufficient reason to have a traditional Thanksgiving, regardless of one's religious or political point of view.

Unfortunately for me, I'm a bit of a contrarian. Okay, I'm a lot of a contrarian. In the past, I've purposely made lists on or around Thanksgiving of the things for which I am most decidedly not grateful. Telemarketers. Ice storms. Coors Light. Autoplay videos on websites. The cancellation of Sense8. Those sorts of things.

But I... I don't think I'm that person anymore. Near-death experiences can do that, I've heard. And I've had at least four of those, so sometimes the Universe or karma or fate or whatever has to kick me in the balls a few times before I'll pay attention. I just hope that doesn't mean I'm losing my edge. When snark becomes part of your personality, you come to depend upon it.

So... I'm grateful that I almost died at least four times? Maybe? It's probably disappointing to other people, though. Just goes to show that you can't please everyone, so you might as well drink.

Thank you, scotch!

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