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 2 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 3 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.

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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').

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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.

May 15, 2023 at 9:21am

#1049626

Extinct

Nothing is forever.

11 Extinct Foods From History

I didn't fact-check these, so beware. If true, there are some on this list I'd never heard of.

As per normal, I'm not copying all of them here.

1. Ansault pear

Unlike other items on this list, the Ansault pear appeared relatively recently. First cultivated in Angers, France, in 1863, the fruit was prized for its delectable flesh.

Angers, France? That's not Nice.

Irregular trees and the rise of commercial farming contributed to the fruit's demise.

Seems to me that, if we really wanted to, we could recreate this one. Fruit varieties are generally made by some sort of cloning or hybridization.

3. Auroch

You may have heard aurochs mentioned in Game of Thrones, but this creature doesn’t belong in the same category as dragons. The real cattle species was domesticated 10,000 years ago in the early days of agriculture. They were big (“little below the elephant in size,” according to Julius Casear) and leaner than modern cows.

Apparently these lasted longer than I thought, all the way to the 17th century.

Here, the article leaves out an interesting bit about the aurochs: it was so important, so integral to developing civilizations, that the pictogram for it became a letter. Phoenicians called it aleph. The Hebrew script still does. In Greek, alpha. We know it as the letter A.

5. Dodo

Dutch sailors first visited the island chain of Mauritius in 1598, and less than two centuries later the archipelago's native dodo went extinct. Sailors relied on the birds as sustenance during long voyages at sea, but that isn't the primary reason they died out; habitat and the introduction of invasive species like rats and pigs ultimately wiped out the animal.

Pretty sure they mean "habitat loss," not "habitat."

It's my understanding that it was fairly common, at the time, for people of the European variety to believe that God put all the other animals (and plants, etc.) on Earth for our benefit, and would never allow one to become extinct.

That turns out not to be the case.

6. Steller’s sea cow

German naturalist Georg Wilhelm Steller identified the Steller's sea cow around the Commander Islands in the Bering Sea in 1741. Growing up to 30 feet long, it was significantly larger than the sea cows alive today.

Cue Hindenburg disaster narrator: "Oh, the huge manatee!"

7. Mammoth

Wooly mammoth meat was an important component of the diets of our earliest human ancestors. We ate so much of them that hunting may have contributed to their extinction around 2000 BCE (though climate change was likely a bigger factor).

So, apparently, there were mammoths wandering around at the same time as there were pyramids in Egypt.

Not in the same place, though.

8. Taliaferro apple

Thomas Jefferson cultivated Taliaferro apples at Monticello. In an 1814 letter to his granddaughter, Jefferson said the small fruit produced "unquestionably the finest cyder we have ever known, and more like wine than any liquor I have ever tasted which was not wine."

Including this one in my commentary for literal local flavor. But also because many people might not be aware that the Virginia pronunciation of Taliaferro is, inexplicably, Tolliver.

9. Great auk

Modern humans primarily killed great auks for their down, leading to the species’s extinction in the mid-19th century, but prior to that they were hunted for dinner.

I knew about this one because I had an English teacher in high school who loved to point out awkward sentences in his students' compositions by writing a big red AWK and circling the offending phrase. He called it the "Great Awk."

It should surprise no one that I had a truly stupendous Great Awk collection.

Anyway, there's obviously more at the link, and they're all interesting, even if, as is the case with the passenger pigeon entry, some of them are already well known.

4 Like • Comment

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