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 7 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 17, 2018 at 1:01am

#945717

Sounds of Silence

I'm introverted and I'm trying to eat more healthy food, so I have groceries delivered. That way, I don't have to deal with people at the grocery store or get tempted by Oreos. One of the regular delivery drivers is deaf.

I've never learned ASL (I don't think the New Jersey State Bird counts as ASL), but I looked up how to sign "thank you," on the theory that one should always be courteous to someone who is handling one's food.

He's not the first Deaf person I've encountered, of course, but the situation did get me thinking about how life would be different without hearing. I'd imagine that, as with blindness, it would be different if one is born without the sense than if one lost it later in life.

A fair bit of the English language is built around onomatopoeia - such a fun word to say and spell. There are words describing animal noises, such as bark, meow, or moo; every child, I think, loves to learn these words and associate them with their respective sounds. You also get words describing the sounds of nature, such as the whisper of wind, gurgling of a stream, roar of the ocean, crunch of leaves or snap of twigs underfoot. Even more artificial sounds have associated onomatopoetic words - the ringing of an alarm, the beeping of a clock, the squeak of a shoe, the gong of a gong, and so on.

But what of those who lack the referent for these sound-words? I don't think it's the same thing as trying to describe color to a person blind since birth; it seems to me that, perhaps, the formation of these sounds themselves, in speech, might help with understanding their source. When I say, for example, "quack," I'm reminded of the sound a duck makes - but what do those who have never heard a duck make of the word "quack?"

It's not like I'm going to ask the delivery driver these things. That would be rude; besides, it's far more likely I'll encounter Deaf people to correspond with online.

I've sometimes wondered if onomatopoeia might have been the ultimate origin of language. I don't think we can know this for sure, but it would make sense that humans started out by mimicking the sounds around them. Like, maybe, the hiss of a snake might have become a name-word for the reptile, for example.

Most of our words are pretty arbitrary in their representation, but onomatopoeia connects language to what it represents - symbol to referent. This has been used to great effect in comics. Comics, and comic books, are primarily a visual medium, but when you see a "Bang!" or "Pow!" or "Snikt!" your mind translates that to sound. And I think that's pretty cool. I still wonder how Deaf people process that, though.

4 Like • Comment • View 7 comments

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