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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Elycia Lee ☮

about 3 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 16 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 19 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

about 21 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

April 29, 2020 at 12:02am

#982296

Vitamin?

Time once again to kill some cherished ideas.

https://www.bbc.com/future/article/20161208-why-vitamin-supplements-could-kill-y...

Why vitamin pills don't work, and may be bad for you
We dose up on antioxidants as if they are the elixir of life. At best, they are probably ineffective. At worse, they may just send you to an early grave.

"We," my ass. I haven't taken supplements in years. Yeah, there was a time I thought they might be helpful, but then I, you know, found out about science.

For Linus Pauling, it all started to go wrong when he changed his breakfast routine.

If nothing else, the article is worth clicking on for the photo of Pauling with a cat on his shoulders.

But it's also worth reading because, in relatively plain language, it goes into why people thought supplements were a good thing to begin with (hint: no, it's not entirely because supplement makers funded the studies).

Just seven studies reported that supplementation led to some sort of health benefit from antioxidant supplements, including reduced risk of coronary heart disease and pancreatic cancer. Ten studies didn’t see any benefit at all – it was as if all patients were given the sugar pill also (but, of course, they weren’t). That left another 10 studies that found many patients to be in a measurably worse state after being administered antioxidants than before, including an increased incidence of diseases such as lung and breast cancer.

True believers: "Well, just look at these seven studies!"

We now know that free radicals are often used as molecular messengers that send signals from one region of the cell to another. In this role, they have been shown to modulate when a cell grows, when it divides in two, and when it dies. At every stage of a cell’s life, free radicals are vital.

Without them, cells would continue to grow and divide uncontrollably. There’s a word for this: cancer.

So the things people take out of fear in order to prevent cancer might actually be causing it.

We would also be more prone to infections from outside. When under stress from an unwanted bacterium or virus, free radicals are naturally produced in higher numbers, acting as silent klaxons to our immune system. In response, those cells at the vanguard of our immune defense – macrophages and lymphocytes – start to divide and scout out the problem. If it is a bacterium, they will engulf it like Pac-Man eating a blue ghost.

It is trapped, but it is not yet dead. To change that, free radicals are once again called into action. Inside the immune cell, they are used for what they are infamous for: to damage and to kill. The intruder is torn apart.

Loving this imagery.

Put another way, freeing ourselves of free radicals with antioxidants is not a good idea. “You would leave the body helpless against some infections,” says Enriquez.

I mean, who ever thought "antioxidant" could be a good thing?

No one would deny that vitamin C is vital to a healthy lifestyle, as are all antioxidants, but unless you are following doctor's orders, these supplements are rarely going to be the answer for a longer life when a healthy diet is also an option. “Administration of antioxidants is justified only when it is evident that there is a real deficiency of a specific antioxidant,” says Villanueva. “The best option is to get antioxidants from food because it contains a mixture of antioxidants that work together.”

But then how can I sell my magic pilllllllls?

It doesn't matter anyway. People are so stuck in their beliefs, so emotionally attached to what they've been doing, that they're not going to change, any more than I'll quit drinking. At least with drinking I know the risks, but I'm willing to bet the pill-poppers think they're doing something to benefit themselves. And not even getting drunk in the process.

2 Like • Comment

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