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:

Write_Mikey_Write!

a day 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.

October 2, 2019 at 12:28am

#967109

Fat Chance

https://www.theatlantic.com/health/archive/2015/09/why-it-was-easier-to-be-skinn...

Why It Was Easier to Be Skinny in the 1980s

A new study finds that people today who eat and exercise the same amount as people 20 years ago are still fatter.

I haven't been talking about my weight loss much lately because I suspect no one wants to hear about it, but it's continuing, albeit more slowly now.

So articles like this are of interest to me. Let's see if it passes the manure test...

A study published recently in the journal Obesity Research & Clinical Practice found that it’s harder for adults today to maintain the same weight as those 20 to 30 years ago did, even at the same levels of food intake and exercise.

For the record "a study" doesn't mean diddly-shit. Who funded it? What biases might the scientists have had? How big are the sample...

The authors examined the dietary data of 36,400 Americans between 1971 and 2008 and the physical activity data of 14,419 people between 1988 and 2006.

Oh. That's... a pretty significant sample size, I think.

They found a very surprising correlation: A given person, in 2006, eating the same amount of calories, taking in the same quantities of macronutrients like protein and fat, and exercising the same amount as a person of the same age did in 1988 would have a BMI that was about 2.3 points higher. In other words, people today are about 10 percent heavier than people were in the 1980s, even if they follow the exact same diet and exercise plans.

2.3 points, or 10%, doesn't sound like a lot. And that certainly doesn't explain why I looked like a marshmallow. That was my own fault.

The fact that the body weights of Americans today are influenced by factors beyond their control is a sign, Kuk says, that society should be kinder to people of all body types.

Okay, being nice to people is a good idea in general, but that conclusion doesn't follow from the science. If factors beyond our control account for 2 to 3 points of BMI, then any additional points on top of that may be within our control. As someone who struggles with this sort of thing, I know it's easy to make excuses, to lie to ourselves. "Oh, I weigh 300 pounds but it's not like I can do anything about it." (I never hit that number, myself, but I'm just saying.) Seeing a study like this, it's reasonable to assume that someone out there feels vindicated by the numbers, but, again, the numbers just aren't that compelling past a certain point.

Just to be clear: Definitely be nice to fat people (and everyone else while you're at it), but this is a study about the mechanics of weight gain, not psychology.

“There's a huge weight bias against people with obesity,” she said. “They're judged as lazy and self-indulgent. That's really not the case. If our research is correct, you need to eat even less and exercise even more” just to be same weight as your parents were at your age.

As a lazy, self-indulgent person, I can say with some certainty that you can lose weight and still be lazy and self-indulgent - just not to the extremes you may be used to.

Full disclosure, though: the three hypotheses mentioned in this article - industrial chemicals , prescription drugs, and microbiome issues - I've had my suspicions about for quite some time. So there might be some confirmation bias here on my own part. It's important to remember that a "hypothesis" is somewhat higher than a "guess" and lower than a scientific "theory." So more research is warranted.

Conclusion: passes the manure test, within limits. Now I want to see if anyone in this puritanical, "everything is your own fault and you can do something about it if you just try" society takes notice.

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