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 2 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 4 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.

December 3, 2021 at 12:04am

#1022713

Know One

I'm more than a little hung over right now, so I'm not going to have much commentary on this article. But read it anyway. It's a Cracked link, and I think it says a lot about the state of internet media right now that a dick joke site that evolved out of a juvenile magazine gets shit more right than more "serious" publications.

Stop Telling People To Believe The Science

"But Waltz, you're all about science! How can you post such a thing?"

Well, like I said, it's worth reading.

No one disbelieves science, but we’re all selective in our embrace of it. I “believe” in science, but I still consume alcohol and junk food, which science says are terrible for me.

Doing something that's objectively bad for you isn't the same thing as doing things that are objectively bad for other people. For example, I don't care if you smoke. But lots of people care if you smoke in a crowded restaurant.

Some “believers” in science believe in rather unscientific propositions, from the imminent triumph of transhumanism to Elon Musk not being full of shit.

Transhumanism (feel free to look it up; I can't be arsed right now) is fantasy couched in scientific terms. Kind of like scientology.

Here's the money shot, though:

Science—actual science—isn’t something you believe in. Inertia and friction don’t need your belief to keep existing. A scientific theory, tested by an experiment, will either produce or fail to produce evidence, regardless of what you hope or believe will happen. If you actually believe in science, you should welcome reasonable questions and criticism and skepticism, because that’s how science moves forward.

As I've noted before, science is often wrong. That's part of the deal. It's a feature, not a bug. Unlike with religion, you don't get "infallible" diktats intoned from on high; all you get is support or negation of your hypothesis.

I'm okay with that. Everyone should be okay with that, but I get that some people aren't, and they want certainty.

But there's no such thing as certainty.

In the early days of COVID we were told not to wear masks.

I gotta call this one out because a lot of the moron brigade loves to trot this out whenever the subject comes up. Yes, they told us not to wear masks. This wasn't because masks are ineffective, though (they are not). It was not because "science" said it wasn't transmissible by aerosol (it did not). It was because there weren't enough masks. It wasn't a scientific decision but a pragmatic one. Remember how people hoarded toilet paper back then? Imagine how much worse it would be with a run on something that people generally didn't use at the time. You have to take human stupidity and greed into account, or things get even worse.

There’s a tendency to let “belief” in science slide into scientism, the idea that the scientific lens is always the best way of looking at the world and that scientists could, in theory, solve all of our political and ethical problems if we just got those dumb politicians and philosophers out of the way.

I'm pretty sure I've said this before, but philosophy and science are entirely separate ways to approach knowledge. And yet, philosophy has to be grounded in science... and science must be guided by philosophy. As an example of the latter, think of animal testing. We consider many forms of research on animals to be immoral. But that's a philosophical matter. Science itself is not moral, immoral, or amoral. It's also not a collection of facts; rather, it's an approach.

I watched a thoroughly mediocre science fiction show on Amazon a while back. As part of the plot, a journalist and a scientist walk into a bar (yes, that could be a setup for a joke). The bar is having Trivia Night. The scientist is portrayed as absolutely confident that he will win the trivia contest, and he and the journalist bet each other.

Thing is, scientists are 1) not automatically more intelligent than anyone else; 2) generally focused on one or two narrow fields of study and 3) entirely too immersed in said study to know, or care, who played Steve on Soaps of our Lives or who was the second-string wide receiver for the Raiders in 1997. The show's writers fell into a common trap: conflating intelligence with knowledge.

All else being equal, I'd put all the money on the journalist in a trivia contest against a scientist.

At best, that leads to bizarre assumptions about how knowledge works, like how the media loved to have Stephen Hawking—a theoretical physicist and cosmologist—issue ominous warnings about the threat of AI, aliens, and imminent human extinction.

Hawking has been an ex-Hawking for a while now, and yet I still see articles about his prognostications. Yes, he was brilliant. No, none of his "warnings" should carry any more weight than that of your average science fiction writer. Even the smartest people in the world have their biases.

People have to be engaged on an emotional level, regardless of how unpleasant that task can be, because otherwise we’re ignoring how they reached this point.

To me, that's not just unpleasant, but absolutely disgusting. Still, on a basic level, I understand that it's impossible to reason someone out of a position they didn't reason themselves into.

Okay, I guess I had more commentary on the article than I expected, hangover or not. Believe it or not, there's a lot more I'd like to say. I'm not going to, though. In case it's not obvious, however, I'm not saying "ignore science." Far from it. Neither is the writer. But I think we all need to watch the movie on a wider screen.

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