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 12, 2018 at 12:50am

#945385

Ships in the Night

A ship is safe in harbor, but that is not what a ship was built for. - William H. Shedd

I like bars.

This should come as no surprise to anyone.

One of the reasons I like bars - besides the obvious - is that I sometimes get to meet interesting people. I was sitting at a bar in my town once when I overheard the woman two seats down from me say something to the bartender that indicated that she - the customer, not the bartender - was in the Navy.

So of course I butted in; the seat between us was empty. "You're at the JAG school?" (UVA is home to the Judge Advocate General school, and there's little other reason for Navy officers to be around.)

"That's right. I teach there."

The officer was a slim woman, perhaps in her forties, with short-cropped dark brown hair and the kind of steely eyes you only get if you're in the military.

"That's cool," I said. "My dad was Coast Guard."

Now, I'm not the type to pick up sailors in bars, but I do like chatting with fellow drinkers, so we got to talking about ships and boats. "Something I've always wondered," I told her. "And my dad never really explained it. What's the difference between a ship and a boat?"

She laughed. "That's a big thing in my classes," she said. "Every once in a while I'll show a picture and ask, 'Ship? Or boat?'"

"Not always easy to identify, then?"

"Well..." she went on after ordering another beer. Navy people can drink, let me tell you. "Sometimes it is. Anything that stays on a river? That's always a boat."

The bar was starting to get busy, so I shifted to the seat between us so I could hear her better. I don't regret a single rock concert, but they haven't done wonders for my hearing.

"Huh." I thought about this. "Yeah, I've never heard of a ship on a river."

"Unless it's a ship that's sailing to or from a harbor on a river."

"...Oh. Yeah. What about on the Great Lakes, like the Edmund Fitzgerald?"

"That was a ship."

"Because it had cargo?"

"Yep, but not all ships have cargo."

"I think I need another beer." For the record, the connections between many of the Great Lakes are known as rivers, and yet the Fitzgerald was a ship. That's why I needed another beer.

"Also, a ship can carry a boat, but a boat can't carry a ship."

"It can push one, though," I noted, thinking of tugboats.

She nodded.

"I've been thinking about it like: A ship travels between ports, but a boat always returns to its own port."

"That's not a bad distinction," she acknowledged. "Oh, and a submarine is always a boat."

"My head hurts."

"You could always just Google it."

And so, later, I did.

Now, it's been a few months, so the conversation didn't go exactly like that, but I think I covered the gist of it. I keep going back to that bar, but I haven't seen her again. I guess you could say that ship has sailed.

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