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!

about 3 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

about 5 hours 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.

April 16, 2021 at 12:03am

#1008600

Healthy, Wealthy, and Wise

Today's article talks about something I've harped on before -- the unreliability of food science studies and reporting.

Here’s Why It’s So Impossible to Get Reliable Diet Advice From the News
What’s good for you seems to change every week. Maybe we should stop blaming the media and look at the studies underneath the stories, too.

And maybe we should stop blaming ourselves and being so damn neurotic about these things.

It is easy, especially as someone who is on the research side of things most of the time, to fault the media for sensational coverage of individual studies that fails to consider the broader context. And certainly there is a healthy dose of that all around us (for example, why write a headline like “Do Tomatoes Cause Heart Attacks?” when the answer is “no”?).

Uh, I know that was a rhetorical question, but the answer is "for clicks." It's best to assume that the answer to any headline in the form of a question is "No."

I would argue the main problem is that the studies that underlie this reporting are themselves subject to significant bias.

I've been saying this for years. If you've been following along, you're probably sick of hearing me go on about it.

Take coffee as an example. On the one hand, coffee drinkers are more educated on average, and they exercise more.

How do you know that? A study? But didn't you just say that studies are biased? And of course coffee drinkers want to think they're smarter, but if they're so smart, how did they get addicted to caffeine? I should commission a study that shows how guys with long hair who wear Birkenstocks and Hawaiian shirts are far, far sexier than average.

I set out to analyze—using a top-notch data set and standard analysis techniques—a simple question: “Which foods make you thin?”

None. None foods make you thin. Lack of food makes you thin. Duh.

In the data, eating quinoa is associated with a whopping 4.2 point reduction in BMI, whereas eating breakfast sandwiches is associated with more than a 2 point BMI increase.

So if you eat a quinoa breakfast sandwich, it balances out.

There are other issues with quinoa, cultural ones. But this isn't a post about American cultural imperialism.

The article goes on to describe kind of a study of studies, with lots of bar graphs and explanation. It's worth at least glancing through if you're interested in this sort of thing.

But if not, here's the bottom line: Educated rich people are healthier than ignorant poor ones. (Educated people who wear Birkenstocks and Hawaiian shirts are also way more attractive.)

Or something like that. Honestly, she loses me about halfway through, but that's probably because I have a bit of a mind fog right now. No, I haven't been drinking; I got my second Trump Mumps shot yesterday and, while I'm generally feeling pretty good, my brain's kinda wonky.

In any case, it's pretty well established that eating vegetables is better for your health than eating candy bars. I don't think any of us need a study to know that, but since candy bars taste way better than any vegetable, it helps to be reminded from time to time.

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