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 24 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

a day 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.

December 29, 2019 at 12:02am

#972174

Too Much Like Work

https://www.theguardian.com/news/2019/jan/03/why-exercise-alone-wont-save-us

Article's a year old now, but it refers to New Years resolutions, so it's cycled back again.

Why exercise alone won’t save us

Our relationship with exercise is complicated. Reports from the UK and the US show it is something we persistently struggle with. As the new year rolls around, we anticipate having the drive to behave differently and become regular exercisers, even in the knowledge that we will probably fail to do so. Why do we want to exercise? What do we expect it to do for us? We all know we are supposed to be exercising, but hundreds of millions of us can’t face actually doing it. It is just possible the problem lies at the heart of the idea of exercise itself.

Well, duh. It's a pain in the ass and always takes longer than you think, what with having to change clothes twice, probably shower, and maybe get to the gym or back. Like, I always think, "30 minutes on the treadmill and then some weight-pumping" and I end up being gone for nearly 2 hours.

Evidence about bone strength and density gleaned from fossils of early humans suggests that, for hundreds of thousands of years, normal levels of movement were much higher than ours today.

You know how I'm always bitching about evolutionary arguments? Specifically, there, I'm bitching about wild speculation about how psychological, not physical, traits evolved. The former is guesswork; the latter is revealed by some physical evidence. Whether they're interpreting the evidence reasonably is another issue.

After two centuries of trying, we should accept that exercise is not working as a global fitness strategy while it remains an addition to the working day. In the long view, it is starting to look a lot like a fad. Government guidelines in the UK and other countries that encourage sport and exercise are failing. These strategies struggle because we are trying to get people to give up what little leisure time they have to pursue activities that require substantial additional effort.

I have a different take on it.

First of all, technology got us to this point because it kept inventing things to make our lives easier, as the article points out. Consequently, the obvious solution is technology that keeps our bodies from falling apart if we don't exercise. I think I've posted stuff before decrying the common idea that failing to keep oneself fit is considered a moral failure, a failure of will, whatever - basically, YOUR FAULT. Instead, if we really cared about this sort of thing, we could put all of our efforts to developing a Leisure Pill - something that allows us to sit around all day with no ill effects. You may dismiss this idea as the "easy way out," but the "easy way out" is apparently what got us here; it works for us, and we're really, really good at it, so why not extend the concept?

Second, maybe we're looking at all of this the wrong way. The hidden assumption underlying all of these health articles is that the goal is to live longer. Of course, few people want to die, but after seeing two parents live into their 80s and beyond, both dying of mental degeneration, I'm not sure that a longer life is necessarily better.

There's something to be said for living life the way you want to. Sure, you could stay active, eat no more than 1100 calories a day, don't drink, don't smoke, breathe filtered air, never climb a mountain, avoid driving, etc., and science says you could live a lot longer. And hey, if that's how you want to live your life, great. But if it's not, if you're more of a hedonist like me, does it really make sense to give up the pleasures that make life worth living, for the possibility - not the guarantee, but the possibility - of a few more years of elderly frailty?

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