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 an hour 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 3 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.

November 19, 2023 at 8:56am

#1059766

Revisited: "On Merit"

It's time travel time, and the random entry I'll feature is from August of 2020: "On Merit"

The article I discuss therein

is still up, from Aeon, apparently published in early 2019—the Before Time. It discusses how belief in meritocracy is not only false, but dangerously false.

And I don't think my views on the topic have changed in the past three years. What has happened in the intervening time is that I found, read, linked and commented on other articles that touched on the role of luck (as opposed to merit) in an individual's level of success. Though, as the article points out, "merit" is itself a product of luck: someone was born at the right time, with appropriate talent, and was raised in an environment that promoted use of that talent.

So what I want to say today comes from an experience I had yesterday evening.

Yesterday was quite pleasant for mid-November around here. Relatively warm temperatures continued on into the early evening, so I took my laptop outside to enjoy what's surely the season's last gasp of habitability outdoors. Thing is, I live about a mile and a half from UVA's football stadium, and it was Saturday, and there was a home game.

I know this because I heard pretty much every word the announcer said, every cheer, every drumbeat.

Now, if I made sustained noise that could be heard a mile and a half or more away, I'd be visited by cops. Understandable? Sure. Fair? Not so much.

Honestly, the noise didn't bother me that much. It's the principle of the thing that got me wondering. We put up with a lot of things to support Sacred Sports that simply wouldn't fly in other circumstances.

College athletes, from what I understand, don't get paid directly, though some go on to the NFL or whatever and pull down salaries that most of us can only dream of... all because their talent, experience, and physical parameters fit the needs of the sport. Work and practice are involved, sure, but almost every occupation requires work and practice.

But a belief in meritocracy might cause someone to believe that, by virtue of these elements, the athletes are more deserving of this measure of financial success than, say, math teachers. And the proof seems to be right there: athletes get paid more than math teachers, so their contribution to society must be proportionately greater. This is a circular argument.

"But, Waltz, a football player only has a decade or two to make bank, while math teachers can do their thing well into middle age and often beyond." Okay, fine, forget athletes; consider, instead, movie actors, rock stars, or some other profession of limited use to society compared to, say, teachers, civil engineers, or firefighters. I'm not knocking the individuals, here; you have the talent and experience, I don't blame you for squeezing every last dime out of leveraging your abilities. I don't follow sports, but I do get entertained by movie actors and musicians, and I appreciate their work. No, I'm questioning the idea that people get rewarded in accordance to their contribution to society.

Because they obviously don't.

And that's not even getting into systemic barriers to entry in many fields, with more opportunity still going to those groups who are historically privileged. I'll say this for sports: it's one way for someone without many other opportunities to achieve some measure of success.

There are, of course, other ways of measuring success besides money. But it's pretty damn obvious (from this and many other examples) that, in the world we live in, financial success isn't a measure of one's value.

Or, to quote from the original article one more time,

Although widely held, the belief that merit rather than luck determines success or failure in the world is demonstrably false. This is not least because merit itself is, in large part, the result of luck. Talent and the capacity for determined effort, sometimes called ‘grit’, depend a great deal on one’s genetic endowments and upbringing.

Another way to get rich is to win a lottery jackpot. We tend to view such people as "lucky," but it's just as lucky to be born into a place and time when your abilities are in high demand, and thus rewarded more.

In short, no, the world isn't fair. If it were, though, I'd be worried. As it is, we can work to make it less unfair, but I agree with the point made in the article: belief in meritocracy only makes it less fair.

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