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:

Turkey DrumStik

about 5 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

December 27, 2018 at 12:22am

#948275

Trust Your Gut

Like many people, I struggle with weight issues.

It is a source of endless frustration to me, knowing that the things that I want to eat are generally unhealthy. We all know this, of course, and yet some people are better at eating healthy foods. I am not one of them.

I waver. Sometimes, this is because I am basically lazy. It is easier to eat premade food than it is to make my own. Sometimes, even something as simple as cooking up a serving or two of broccoli (a vegetable that I actually like) seems like a monumental task, especially when factoring in having to clean up afterward.

The ideal "healthy" food, to me, is carrots. Minimal preparation, great when eaten raw (in my opinion anyway), and little to no cleanup. But when I'm hungry, do I think of grabbing a handful of "baby" carrots, a conveniently prepackaged serving of bite-sized root vegetables? No, of course not. I end up going to Taco Hell.

So yes, I'm one of those people who would rather have a pill to take to get and/or keep my weight down, something that would enable me to eat whatever the hell I want. I desire this even more than flying cars, jetpacks, or warp drive. Maybe not more than holodecks though. A fully programmable holodeck would be ideal.

But I digress.

Almost every article I've seen about Western obesity makes some version of the following observation: "We live sedentary lifestyles and eat junk food."

This observation is usually unsupported by evidence - at least, when comparing the people of today to the people of, say, the early 70s. It's an assertion that implies that the reader will agree with the writer. And, usually, because so many such articles have been written, it tends to be taken as fact.

But I remember the early 70s. One thing I remember clearly is the news reporting on this new "jogging" trend. People actually went outside and ran! This was unheard of! It's preposterous, and look what they're wearing!

As a group, we are more aware of the benefits of exercise than ever before. Office jobs have existed for over half a century, and even back then, people would come home and plop themselves in front of the TV. And yet, as a group, we're far heavier than we used to be.

Even accounting for the proliferation of "junk food" (which, I can assure you, existed in vast quantities 50 years ago), I've always felt that something else must be at work. That something else, I hypothesized, was that, at about the same time as the obesity "epidemic" began, there had been great advances made in the creation of antibiotics. And these antibiotics (again, according to my amateur hypothesis) might have attacked certain beneficial gut microbes, ones that assist in digestion and help us keep from becoming too fat.

I'm not a scientist, of course, and as I said, I'm lazy. So I never really followed up on it. Something at the back of my mind, that part of me I rarely pay attention to, also said, "Come on, Waltz. That's just an excuse. Eat more carrots and fewer cheeseburgers, and exercise." And, indeed, I'd never seen any legitimate scientists explore that area of research.

Not that they would. There's a pernicious streak of puritanism lurking in American society. The idea that there might be an easy answer is anathema. Rather than working on ways to reduce the bad effects of alcohol, for example, we urge people not to drink. Instead of concentrating on a cure for lung cancer, we tell people not to smoke. Anything perceived of as a "vice" has to be rooted out and destroyed, rather than mitigated by the awesome power of science - because we just have to ensure that anything remotely enjoyable has to have consequences.

And if we're fat, this puritanical streak insists that it's our fault. A moral failing.

Now, look, I admit to many moral failings, including laziness and apathy. Maybe even gluttony sometimes. But that's no excuse not to pursue avenues of research that could lead to some solution other than "if you don't follow our dietary and exercise advice, you're worthless." Especially when, as I've noted repeatedly, nutritional science is rife with bias and bad data - and extremely questionable conclusions.

Anyway, enough. Turns out I might have been onto something.

https://getpocket.com/explore/item/how-the-western-diet-has-derailed-our-evoluti...

I'm not taking this as the absolute truth either, of course. But it's worth studying further. That is, by people who aren't lazy.

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