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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Write_Mikey_Write!

a day 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.

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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.

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').

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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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

May 28, 2019 at 12:47am

#959762

On Being Wrong

On this last Talk Tuesday of May, let’s talk about bias. How do your own biases influence your life? Is it possible to ever be truly objective?

To answer that last question first: no.

Everyone has bias. It's as inevitable as death, taxes, and construction on the Capital Beltway. The best any of us can do is try to identify those biases and account for them.

Albert Einstein is generally considered the greatest mind of the 20th century (I think it was actually Richard Feynman, but I may be biased.) Whether that's the case or not, there is no denying that he had a towering intellect. Einstein is one of those rare people whose image is practically a dictionary definition; in his case, of the word "genius." Nearly single-handedly, he turned physics on his head and revealed truths about the universe that caused a tectonic shift in our understanding thereof. The theories of relativity are so distant from our everyday perception that, in order to formulate them, one has to throw out a lifetime of experience and personal observation - basically, overcoming one's bias.

Science is like that. It acknowledges that we humans have bias, and provides a working system by which, in time, those biases are minimized, through observation and testing. As unbelievable as relativity seemed at the time, it was independently tested and verified, and refined; and yet, in our everyday lives, it's nearly irrelevant. Sure, you have to take it into account when, say, designing a GPS device, but not when you use one.

Shortly after that particular cosmic brainstorm, however, other scientists were developing what would become known as quantum mechanics - the study of the very small. Einstein made his own contributions to that (notably, it's what he was awarded the Nobel Prize for), and yet, when the full ramifications of this theory began to make themselves apparent - and they're far, far weirder than relativity - Einstein balked.

See, quantum theory deals heavily with probabilities. Instead of "this object exists at this location," it's more like "This thing is both a particle and a wave, and the more we know about its location, the less we know about its motion." and "We can't know what this electron is doing, but there's a 90% probability that it's doing this one thing and a 10% probability that it's doing this other thing" (yes, that's an absurd oversimplification, but that's not the point (pun intended)). When physicist Niels Bohr pointed this out, Einstein objected, based on his own perception of reality. "God," he supposedly said, "does not play dice with the Universe."

To which Bohr, so the story goes, replied: "Al, stop telling God what to do."

Whether that particular conversation ever took place or not, it does sum up the attitudes of the scientists under discussion.

Point is, even Einstein had his biases. And in this case, he was wrong. Since that probably-mythical exchange might have occurred, the probabilistic nature of quantum theory has been shown to be correct through observation, experimentation, mathematics, and all the other things that science does to try to disprove a theory. (That's what it does, you know - tries to disprove, not tries to prove.) Einstein tried to disprove it, and hit a dead end. The Universe, it turns out, is, at its most basic level, non-deterministic.

Whether there's a literal God or not is still under debate and probably always will be. Einstein couldn't prove or disprove that, either. No one can. "God" was his shorthand for something like "the basic forces of the universe." We can all have our own opinions about it, but they amount to biases, one way or the other.

Me? I try to keep an open mind, but not so open that it collects dust. I know that no one knows everything. I know that, for any given topic, whatever I think may be wrong. My biases do show up from time to time, which I'm sure you realize if you've been following this blog. All I can do is be willing to change my mind if the right kind of evidence presents itself, and to try to be a big enough person to admit when I'm wrong.

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