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 5 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 7 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.

April 15, 2024 at 8:25am

#1068809

That's Cole

Not to brag, but I've known this for a long time. Still, there's always new stuff to learn. This is a fairly old article from Vox:

Kale, Brussels sprouts, cauliflower, and cabbage are all varieties of a single magical plant species

And collard greens. Also kohlrabi, but every time I mention kohlrabi people be like wut?

In some circles, kale has become really, really popular.

Too popular. I think people may have finally calmed down on it a bit since this article came out in 2015. I don't know what the latest fad food is, but I can almost guarantee that behind the fad is someone with money trying to make more of it.

Once a little-known speciality crop, its meteoric rise is now the subject of national news segments. Some experts are predicting that kale salads will soon be on the menus at TGI Friday's and McDonald's.

I don't know about Friday's, but apparently McD's did come out with a kale salad that had more calories than a Big Mac.

Calories (if you haven't been following along, I talked about calories a few days ago) aren't the only indicator of healthiness in a food, but I do find that hilarious.

This makes it pretty interesting that kale and cabbage — along with broccoli, Brussels sprouts, cauliflower, collard greens, and kohlrabi, and several other vegetables — all come from the exact same plant species: Brassica oleracea.

Linguistic detour, as I often do: you might notice that most of those have some variant of "cole" in them. It's in the middle of broccoli, the beginning of cauliflower and collard and kohlrabi, and it's a slight vowel shift for kale. Not so much cabbage itself, but the syllable survives in that version's most perfect presentation, coleslaw. The odd one out is Brussels sprouts. One might be forgiven for thinking that this is where the species name brassica came from, but apparently not; brassica seems to have been Pliny the Elder's name for the cabbage group. No, Brussels sprouts were named for, believe it or not, Brussels, the one in Belgium, near which the plant was heavily cultivated.

Nor was Brussels

apparently named after brassica. The similarity of the first syllables there seems to be another linguistic near-coincidence.

How is this possible? About 2500 years ago, B. oleracea was solely a wild plant that grew along the coast of Britain, France, and countries in the Mediterranean.

Because France doesn't count as "Mediterranean?" I guess they meant the north and/or west coasts of the country.

As for "how is this possible," obviously, the article goes into detail. But it's the same sort of thing as saying "How is it possible that a chihuahua and a Great Dane can be the same species?"

Short version: artificial selection; that is, trait selection by humans. As we, too, are part of nature, the distinction between artificial and natural selection is, well, an artificial one.

For the long version, there's the article.

All this speaks to one thing: the remarkable power of human breeding and artificial selection.

Wow, that could have been worded better, couldn't it? Oh, well; I'm sure I've done worse.

5 Like • Comment • View 2 comments

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