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 24 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

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.

December 15, 2019 at 12:30am

#971477

Occam's Nail Clippers

Today I'm linking an article about Occam's toothbrush. Wait, no... razor. That's it.

https://fs.blog/2019/10/occams-razor/

How to Use Occam’s Razor Without Getting Cut

Occam’s razor (also known as the “law of parsimony”) is a problem-solving principle which serves as a useful mental model. A philosophical razor is a tool used to eliminate improbable options in a given situation. Occam’s is the best-known example.

Before you go running away, this isn't limited to profound scientific inquiries. It's a useful tool for ordinary life. For example, if you come home to a messy house after a short vacation, you can make up all kinds of scenarios: an opossum got in, burglars trashed the place, the cops executed a search warrant, aliens teleported in and were looking for their missing zap gun, etc.; but if you have a teenager, well, Occam's Razor says it was probably the teenager.

Among competing hypotheses, the one with the fewest assumptions should be selected.

A "hypothesis" is basically a possible explanation. In the above example, all of the possible scenarios are possible. Well, except maybe the aliens one. Maybe.

In simpler language, Occam’s razor states that the simplest explanation is preferable to one that is more complex. Simple theories are easier to verify. Simple solutions are easier to execute.

More importantly, simpler theories are easier to falsify. Your car is rattling? Check the easy stuff first. If it's not that, move on to the more complex.

For example, the principle of minimum energy supports Occam’s razor. This facet of the second law of thermodynamics states that wherever possible, the use of energy is minimized. Physicists use Occam’s razor in the knowledge that they can rely on everything to use the minimum energy necessary to function.

That's certainly true for me. Why expend more energy than necessary?

The article goes on to expand upon the use of this philosophical tool in several disciplines, but, like I said, it's also useful for everyday life.

It is important to note that, like any mental model, Occam’s razor is not foolproof. Use it with care, lest you cut yourself. This is especially crucial when it comes to important or risky decisions. There are exceptions to any rule, and we should never blindly follow the results of applying a mental model which logic, experience, or empirical evidence contradict. When you hear hoofbeats behind you, in most cases you should think horses, not zebras—unless you are out on the African savannah.

When you hear hoofbeats behind you, best not to think too hard about it - just run. To the side. Don't do what they do in the movies and when there's something bearing down on you, just keep running in a straight line. I've heard this called "The Prometheus School Of Running Away From Things," but nobody remembers that awful movie now.

This is why it’s important to remember that opting for simpler explanations still requires work. They may be easier to falsify, but still require effort. And that the simpler explanation, although having a higher chance of being correct, is not always true.

And that bit is why I bothered to link the article. Too many people stop thinking once they've worked out the simplest explanation. To go back to the example I used above, when you come home to a messy house and blame your teenagers, what if it turns out they weren't home, either, and there are also scorch marks in your front yard? Well, then, the space alien explanation is looking more probable, isn't it?

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