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 20 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 22 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.

February 28, 2020 at 12:48am

#976520

I Think Therefore I Think

Running a little late today thanks to my good buddy Al Q. Hall. So this one's gonna be short. It's also very obviously an ad for the author's book, but hey, that's okay; they're up front about it this time, and I don't begrudge another writer the opportunity to promote their book, as long as it's not a stealth ad.

https://www.scotthyoung.com/blog/2018/12/14/think-better/

How to Think Better

Periodically, someone comes along and says, proudly, something like "I specialize in thinking outside the box." Here's the thing: First learn to think, period.

Imagine if you could take a pill that would double your intelligence. What would that feel like?

LSD, very likely. Not that I have any experience with that.

Writing your thoughts is the key to better thinking.

Duh.

Cognitive scientists believe that working memory is one of the major components of intelligence. Working memory is like the RAM for your mind. It consists of all the things you’re keeping in mind simultaneously.

If working memory is key to intelligence, than I'm dumb as a rock.

People who can hold more numbers in their heads, have higher working memories and are often more intelligent generally.

Correlation? Causation? Who cares?

Are you struggling with an important problem in life or work? Your first instinct should be to get a piece of paper and start writing it down. Jot down all the elements of the problem, including all your different ideas for a solution.

Who uses ancient technology anymore?

One of my most popular studying tactics was the Feynman Technique. This technique boils down to using writing to make it easier to understand hard problems in math, science and other subjects.

No snark here; Feynman was a legit genius. Still, that doesn't mean we'll be geniuses if we emulate him.

Without writing, it isn’t simply that I would have tons of unrecorded ideas, bumping around in my skull, but that those ideas wouldn’t exist. They are created by the act of writing, much more so than they are being recorded.

I feel like this is dead obvious, but maybe not to someone who's not a writer.

I could go into another discussion of how language is a compression algorithm, but I'm going to need to drink a lot of water, take some Advil, and veg out by bingeing Season 2 of Altered Carbon on Netflix. In other words, the opposite of thinking.

2 Like • Comment • View 3 comments

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