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 22, 2018 at 12:26am

#947959

Bread

As you know, I really like science. It's not that I'm actually trained as a scientist or anything; it's just that I know it's the best tool we have for understanding ourselves and the universe. As a non-scientist, it's more like I'm checking out its butt as it goes by - I don't understand much of it, and I'll never be very close to it, but it sure does look good.

But, as I've noted before, I have serious issues with nutritional science - or, at least, how it's reported on. It's difficult to do it right, and even more difficult to get science reporters to really understand the findings. That's why you get the sense that they keep going back and forth on what's good for you and what isn't.

Some things are known to a high degree of certainty - for instance, the bad effects of trans fats on cholesterol. But other things keep changing: eggs have gone from bad to good to bad to good in my memory, and diet advice keeps changing with every study - biased or not - that seems to produce some result or another.

Worst of all, a lot of studies are biased because they're funded by people who want a certain result, and that can creep into the reports.

That's why I never did buy into the low-carb nonsense. Part of that was that I simply didn't want to: bread is the greatest of all foods, and if eating it makes me die sooner, then so be it; a life without bread is a life not worth living.

As it turns out, though, that may not be a problem.

https://qz.com/quartzy/1487485/the-scientific-case-for-eating-bread/

Bread has long been a foundational part of the human diet, but a revolt against it has been building for years—and seems to be reaching a crescendo. Today, many regard bread as a dietary archvillain—the cause of bigger waistlines and the possible origin of more insidious health concerns. Popular books and health gurus claim that bread and the proteins it harbors can cause or contribute to foggy thinking, fatigue, depression, and diseases ranging from Alzheimer’s to cancer.

But go digging through the published, peer-reviewed evidence on bread and human health, and most of what you’ll find suggests that bread is either benign or, in the case of whole-grain types, quite beneficial.

So there you have it. Time to make bread great again.

Now if only someone would admit that donuts are a health food...

3 Like • Comment • View 3 comments

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