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:

Writer_Mike

19 minutes ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

QPdoll

about 16 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

about 19 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

July 8, 2023 at 7:27am

#1052318

Unsung

Despite what airy nonsense your life coach may have spouted, it is not possible for one person to change the world.

5 People Who Changed the World and Got Jack Squat for It

And even if it were possible (which it's not), there's always the chance that no one would notice.

Link is from Cracked, so don't try to use this in any world-changing scholarly papers.

5. Rosalind Franklin

I agree she got dicked over (pun intended), but I suspect more people recognize her contribution now.

Before 1953, though, the structure of DNA was unknown, until the now ubiquitous double-helix structure was modeled by James Watson and Francis Crick. If you know any names related to the study of DNA, it’s most likely theirs.

To the best of my knowledge, Watson was unrelated to A.G. Bell's Watson, or Sherlock's.

Watson and Crick did admit that solving DNA’s structure without the Franklin data would have been “unlikely, if not impossible,” but apparently putting her name at the top of the paper was a step too far.

In case you're wondering, this is one reason I say it's impossible for one person to change the world: everyone has been influenced by others, and, in many cases, others have to let themselves be influenced by that person.

4. Ignaz Semmelweis

I didn't recognize the name at first, but the thing he did was something I'd heard of.

So what was Semmelweis’ incredible, world-changing medical discovery? It was that doctors should maybe start washing their hands.

This really was tantamount to heresy at one point.

3. Michael Delligatti

When you think of a central figure related to McDonald’s, you might think of the founder, Ray Kroc, or more likely, a fictional and mildly unsettling clown.

Kroc founded McDonald's like Musk founded Tesla Motors: in mythology only.

You may have assumed the Big Mac was cooked up in some secret burger laboratory deep beneath McDonald’s HQ. In reality, it was Delligatti who suggested a two-patty burger and developed the special sauce, an idea that McDonald’s hated at first.

This may be heresy, too, but I've never really liked the Big Mac. I do miss the McDLT, so it's all for the best that it's no longer available.

So what did Delligatti get for forever changing the cholesterol levels of the human race? The ultimate bare minimum: a plaque.

We all get plaque from eating at McDonald's.

2. Philo Farnsworth

If you’re born with a name like Philo Farnsworth, you basically have two viable career paths: some sort of strange preacher, or a brilliant inventor.

You forgot "supervillain."

I don't feel like copying much of this section; basically, he invented the television. Which, come to think of it, may have caused as much suffering as the Big Mac. (The part that was surprising, to me, was that said invention was nearly 100 years ago.)

1. Henrietta Lacks

As with Franklin above, she's more widely known, these days. Unlike Rosalind Franklin, though, it wasn't what she did that was significant (which in no way should diminish her significance).

For whatever reason, her cells reproduced at an unbelievable rate, and “HeLa” cells have been used in research ever since. She did finally get a book and a movie in which she was played by Oprah, but she and her relatives got a whole lot of absolutely nothing for years after her death.

Played by Oprah? Talk about adding insult to injury.

Moral of these stories: sometimes it just doesn't pay to be influential.

3 Like • Comment • View 3 comments

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