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:

tj in a winter wonderland

about 16 hours ago
In response to "Getting Cocker's"

I never bought into the "early to bed, early to rise" thing; it was a well documented fact that Ben liked to stay up until almost dawn recording YouTube videos.

1 Like

Wordsmitty ✍️

about 20 hours ago
In response to "Getting Cocker's"

I wonder why they never used or mentioned the book's full title ▼ ?

2 Like

tj in a winter wonderland

a day ago
In response to "Business or Pleasure?"

What instantly pops into my mind when I read "bleisure" and "frolleagues":

Frolleagues, an untreatable diseases that results in a severe rash, blisters, and bleisures.

2 Like

Wordsmitty ✍️

a day ago
In response to "Business or Pleasure?"

Bleisure with frolleagues. Sounds like kinky porn.

"Nichols, who travels frequently for work, said bleisure is a great way to get the most out of his business travel." and "He still uses all of his vacation days and views bleisure as a "supplement."" - So "b-word" really means taking advantage of your company for extra days and paid expenses, and hoping you don't get caught and fired. Just going out after your business is done each day is one thing. Extending days at the company's expense is another.

2 Like

tj in a winter wonderland

Jan 4, 2025 at 7:32pm
In response to "Giant Red Flag"

My thoughts were, purchase a bundle of the little red flags they stick in the ground for construction markers and just hand some out to everyone I meet.

2 Like

Leger~

Jan 4, 2025 at 12:00pm
In response to "Giant Red Flag"

Pic of the t-shirt on you or it didn't happen.

3 Like

🌖 HuntersMoon

Jan 4, 2025 at 9:44am
In response to "Giant Red Flag"

Elisa: Snowman Stik - Now, let's talk about procrastination...

2 Like

Elisa: Snowman Stik

Jan 4, 2025 at 8:19am
In response to "Giant Red Flag"

Don't worry (or worry). I'll be offering my perspective on this link this year, likely March or April.

2 Like

tj in a winter wonderland

Jan 3, 2025 at 3:09pm
In response to "The Year in the Sky"

It all sounds very interesting and hopefully I will be able to keep an eye on the sky and see some of these. It seems more often than not, it's overcast, or has been for past celestial events. I did, however, get a chance to view the Northern Lights around four this morning. The sky cleared sometime during the night and Max, my Husky woke me up to go out and we enjoyed a few minutes of a spectacular light show.

3 Like

Robert Waltz

Jan 3, 2025 at 11:00am
In response to "The Year in the Sky"

🌖 HuntersMoon

, depends on definition. Astronomically, it's a moon. Astrologically, it (along with the Sun) is a planet. Anyway, I read that to be "planetary parade plus Moon," but we can chalk it up to Lifehacker being misleading again.

2 Like

January 7, 2025 at 9:11am

#1082086

Getting Cocker's

A little less rant-worthy today—some history from Smithsonian.

After Failing Math Twice, a Young Benjamin Franklin Turned to This Popular 17th-Century Textbook
A 19th-century scholar claimed that “Cocker’s Arithmetick” had “probably made as much stir and noise in the English world as any [book]—next to the Bible”

But why didn't he just look it up on the internet?

Before Benjamin Franklin became a printer, newsman, author, inventor, philosopher, diplomat and founding father of the United States, he failed math twice.

They forgot "epic troll." Also, I'd never heard this "failed math twice" thing before, and couldn't find other support for the claim apart from things linked within the Smithsonian article. Those lead to Franklin's words, and, I reiterate, the man was an epic troll and in no way a reliable narrator. This doesn't mean it's false; I just can't confirm or refute it. We like to hear about great people's failings, and there was a similar rumor about Einstein that turned out to be pants-on-fire false.

Also that thing about Franklin's fellow Founding Father George Washington chopping down a cherry tree was false. So you'll have to deal with my skepticism.

It doesn't change what the article is mainly about, which is the math textbook.

Yet the story of the “book of arithmetic” that finally helped Franklin master the subject is little known today—another irony, because in his day, it was every bit as famous as he was.

Probably even more famous, at least when he was a brat.

Cocker’s Arithmetick was probably the most successful elementary math textbook published in English before the 19th century. It epitomized an age in which the expanding worlds of commerce and capitalism, education and Enlightenment, coalesced to make basic arithmetic the classroom staple it is today.

Oh, I thought they'd given up on teaching math. It sure seems that way.

Though perhaps unfamiliar to modern readers, in the 18th century, Cocker’s Arithmetick was as close to a household name as any math textbook is likely to ever be. Edited from the writings of London-based teacher Edward Cocker and published posthumously in 1678, the book included lessons on basic arithmetic with a commercial slant, posed as a set of rules to be memorized, as was typical of educational books of the day.

So it was kind of like the Ferengi Rules of Acquisition, but more math-oriented?

Cocker’s lessons covered addition, subtraction, multiplication and division, as well as calculation with pre-decimal British currency and a gentle introduction to 17th-century England’s bamboozling array of weights and measures.

I often found it ironic (or whatever) that, upon revolting, the US adopted a decimal money system while keeping the "bamboozling array of weights and measures." I think I did an entry about that a while back. Meanwhile, the UK kept the non-decimal money system and adopted, at least officially, SI units. If I remember right, the UK didn't switch from pounds/shillings/pence until around 1970.

Finally, the textbook taught arithmetic skills for business, such as dividing profits equitably between partners.

No wonder the book fell out of favor. Nowadays, we like books about how to screw over the other partners.

Whatever else a student might learn in Franklin’s grand scheme of general instruction, which included everything from oratory and ancient customs to drawing and geography, he maintained that arithmetic was “absolutely necessary.”

Nah, best to keep the populace ignorant of math so they don't notice when you short their paychecks or mess with their timeclocks.

Toward the end of his life, in a well-known letter of 1784, Franklin reflected in satirical fashion on the importance of waking early in order to save money that would otherwise be spent burning candles at night.

I'm just including this bit to emphasize that the "early to bed, early to rise" thing was satire.

Lots more history at the link, of course. Mostly, I just found it interesting that I'd never heard of the book before. Probably because no one can make money advertising it.

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