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:

tj-Merry Mischief Maker

about 15 hours ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

about 18 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

about 21 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

a day ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

May 8, 2024 at 10:33am

#1070719

Be Positive

One reason conspiracy "theories" flourish is that, sometimes, an actual conspiracy is brought to light. While today's article, from MPR, doesn't quite rise to the level of major international conspiracy, it's enough to make you wonder what other similar things are going on.

Online dietitians backed by General Mills, big food companies using ‘anti-diet’ language to promote junk food

This is like finding out that PETA, the pseudo-animal-rights organization whose idiotic performance art ensured that I will never become a vegetarian, is actually funded by KFC. And yet, this is both hilarious and entirely predictable.

Cupcakes. Frozen Pizza. Ice Cream. Cereal. These are some of the sugary foods that a slew of social media influencers are encouraging folks to enjoy guilt free.

Honestly, I'm not opposed to that message. One reason we're so damn neurotic as a society is that lots of us have been convinced to feel guilty about everything pleasurable, instead of simply enjoying it. It's Puritanism all over again, and I won't put up with it.

But I say these things without any benefit other than my own enjoyment at turning out to be right. Certainly, no one's paying me. Not even attention.

However, a joint Washington Post and The Examination investigation found that major food companies, including Minnesota-based food giant General Mills, are reportedly capitalizing on the campaign by paying online dietitians to get people to consume more of their highly processed foods.

Calling the people involved "online dieticians" is questionable, at best. Having watched a couple of videos about diet advice doesn't make you a dietician, but that, combined with a helping of chutzpah and earnest confidence and maybe a nice paycheck from Little Debbie, can convince people that you are. It's like how people spread the misinformation about ivermectin and Covid.

Recently, General Mills, one of the biggest food companies in America, has started adopting the anti-diet philosophy as a way to promote their products. So, they’re doing this through a multi-pronged campaign that involves funding influencers online to promote this message, funding research and also hiring lobbyists to fight back against federal regulation.

Just to be clear: I think that all of these things are acceptable in a society that values freedom of speech and association. What's not acceptable is lying about it, or failing to disclose it.

We reached out to General Mills, and they say that they comply with federal regulations, and are working closely with scientific health nutrition experts to make sure that the information that they’re putting out there is accurate and based in evidence.

I can hear its nose growing from all the way over here.

I think it’s best to, when we’re approaching social media, to have a skeptical mindset. And, probably I would recommend people turn to their doctors for advice, as opposed to influencers online.

And sure, doctors are human too, and susceptible to outside influence. The shady practices of pharmaceutical companies are one way they can be led astray. So they're not always right. But they're far more likely to have the knowledge and experience to consult on things like this than some rando on the internet (myself included).

Skepticism (that's scepticism if you're British) is a good thing. Better that than naîve acceptance of everything you see online (or in ads or wherever). But that includes skepticism of conspiracy theories. Or conspiracy-adjacent theories.

7 Like • Comment • View 2 comments

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