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 8 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 10 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.

March 31, 2024 at 8:32am

#1067220

Revisited: "Heartbreak"

You know what today is, right? Oh, sure, it's April Fools' Eve, of course, but being Sunday, it's time to revisit an older entry. This one was from five years ago, and follows my usual comment-on-an-article format: "Heartbreak"

The BBC article

is still up, as of today. Remarkable, considering how many episodes of early Doctor Who the BBC destroyed to make room for newer things. Maybe they learned their lesson.

To summarize, there was some evidence that a particular heart condition is at least partially caused by stress or strong emotion.

Me: The more I learn, the more I think that the commonly stated dichotomy between mind and body is bogus.

I would probably phrase this differently, now. Also, throughout my entry, I seem to have used "mind" and "brain" interchangeably, which I think I'd be more careful to avoid, these days.

My understanding of the internal workings of a person or other animal is rudimentary, but from what I've heard, the brain is a physical, identifiable organ located mostly in one's head. It does lots of stuff, including stuff we don't understand, but it seems to be responsible for, among other things, telling your feet to walk and your chest to occasionally fill with air. The "mind," however, is a far more nebulous concept. I seem to recall doing another entry about that at some point in the not-so-distant past; it involved René Descartes, who asserted that because you can't put a set of spatial coordinates (Descartes invented coordinates) on the mind, but you can on the brain, that there's no way that one can act upon the other. Might as well link it, too, for reference: "Mind, the Gap"

In addition to walking and breathing, the brain is sometimes fairly good at thinking. More in some people than in others, but "thinking" is a legitimate activity, even if it doesn't look like much of an activity to other people. But many people think (pun intended) that it's the mind doing the thinking. Once you're done thinking, often, you do some physical activity. For example, writing what you just thought. We draw that separation between "thinking" and "doing" because we love categorizing things, like cats and dogs being different species, or Pluto not being a full-fledged planet after all.

Point being, a lot of the seeming contradictions of mind/body dualism, such as in the last entry I linked, go right out the window when you consider that the mind is a product of the body. It's probably more complicated than just being conjured up by your brain, but the brain is a body part too. Descartes' assertion that the mind cannot influence the body is, therefore, wrong. Brilliant guy, but in this case, he was working from a shaky premise.

Anyway. The original entry, from up top, ends with the literal video version of Total Eclipse of the Heart, and I'm pleased (state of mind and body) to see that the video is still in existence. I have a whole entry on that song (the actual song, not the parody) coming up at some point; it's in my queue and will pop up at random.

3 Like • Comment

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