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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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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').

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

March 17, 2021 at 12:03am

#1006536

Learning

Check out this One Weird Trick.

How I Rewired My Brain to Become Fluent in Math
The building blocks of understanding are memorization and repetition.

You might have noticed that I have a wide range of interests. I could never choose between math things and language things. As a result, I never got really good at any of them.

But here I am spending my time attempting to improve my writing and learn new languages. For me, learning won't stop until I'm brain-dead.

I was a wayward kid who grew up on the literary side of life, treating math and science as if they were pustules from the plague.

This article is from 2016. It turns out most of us don't actually avoid the plague.

So it’s a little strange how I’ve ended up now—someone who dances daily with triple integrals, Fourier transforms, and that crown jewel of mathematics, Euler’s equation. It’s hard to believe I’ve flipped from a virtually congenital math-phobe to a professor of engineering.

And already the author is beyond my paltry knowledge of mathematics.

But these hard-won, adult-age changes in my brain have also given me an insider’s perspective on the neuroplasticity that underlies adult learning.

The point being that if a person wants to learn something, is motivated to do it, they usually can.

If you can explain what you’ve learned to others, perhaps drawing them a picture, the thinking goes, you must understand it.

Pretty sure Richard Feynman, the late physicist, was an advocate of that technique.

The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition.

I'd say this is the case for anything. Since my memory isn't great, I have to repeatedly review earlier language lessons, doing exercises over and over. I still don't always get it right, but I get better.

There is an interesting connection between learning math and science, and learning a sport. When you learn how to swing a golf club, you perfect that swing from lots of repetition over a period of years. Your body knows what to do from a single thought—one chunk—instead of having to recall all the complex steps involved in hitting a ball.

Again, that's probably the case everywhere. A while back, I came to the conclusion that learning is largely about consolidation. Simple concepts build up to more complex ones. You see this a lot in math, where they often consolidate complex equations into one symbol for ease of manipulation, and then unpack it later. It's kind of like a compressed file on a computer.

But in my case, from my experience becoming fluent in Russian as an adult, I suspected—or maybe I just hoped—that there might be aspects to language learning that I might apply to learning in math and science.

This, too. Math is really just another language, or at least that's one way to look at it.

I think this is the money quote, though:

Understanding doesn’t build fluency; instead, fluency builds understanding. In fact, I believe that true understanding of a complex subject comes only from fluency.

Her path is opposite mine - she learned languages first and then mathematics, whereas my journey is reversed from that - but I think her insights are helpful in either direction. If I have one quibble with what she says here, and there's a whole lot that I'm leaving out so definitely check out the article, it's a focus on learning for a purpose rather than learning for the sake of learning. However, if having a purpose for what you're learning keeps you motivated, I can't fault that.

But in my view, the biggest obstacle to learning anything is when you tell yourself you can't. Sometimes this comes from outside, as when a parent or a teacher might have discouraged you from pursuing a particular field. But sometimes it comes from having tried, and failed. If you think you can't, then it becomes a self-fulfilling prophecy. But once you open your mind to the idea that you can learn something, then you will, at least to some extent.

And I say this as someone who has been convinced, from time to time, that he can't. But I want to change that. And I think I can.

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