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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 14 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 16 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 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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