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.

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.

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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.

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February 12, 2023 at 12:03am

#1044647

Revisited: "Crossing My Path"

Time for another trip to the olden days. This one's from way back in May of 2020: "Crossing My Path"

A 30DBC entry, this was written to a rather open-ended prompt: Start your entry today with the words: “I used to believe...”

I could have gone pretty much anywhere with this one. I chose what I consider to be a safe route; some of my former beliefs are profoundly embarrassing to me now. For instance, for a while there, when I was much younger (but old enough to vote), I thought that libertarians had the right ideas. It pains me to even admit that, but there it is.

As I noted to open the entry, though:

I used to believe that we stayed basically the same person all our lives. Then I read something recently, but didn't bother to save a link, about how some scientists did a study on young people, then studied them again when they were older, and what they found that Old You doesn't really have anything in common with Young You. I mean, we've known this from a physical standpoint - cells get replaced and all that - but it kind of calls into question the idea of continuity of consciousness, and identity itself.

I notice this even more with Drunk Me. He gets me the best presents from Amazon, though.

So I banged on, in that entry that feels so long ago but really isn't, about an article I'd found about superstition.

But what struck me today, revisiting this, was that I didn't really talk about "belief" itself.

It's one of those words that's a bit slippery. Sure, one can quote a dictionary definition ("an acceptance that a statement is true or that something exists" or "trust, faith, or confidence in someone or something"), but dictionaries don't capture all the nuances of a word. It can also mean a conviction, which has got to be confusing to those not used to English, because "conviction" also means that the legal system has determined that you've done a crime. You can have a conviction that you didn't do the crime, but you've still been convicted.

To me, there are three major kinds of belief:

1) a trust in someone (possibly yourself), as in "I believe in you!" This isn't the same thing as saying "I believe in aliens," because the existence of the "you" in that sentence isn't in question. Which leads us to my second kind of belief:

2) Certainty that something exists, despite a paucity of evidence, no evidence whatsoever, or evidence against. Belief in space aliens, God, Bigfoot, fairies, Russel's Teapot, etc. is in this category. And then there's:

3) Certainty that something exists, with solid evidence to back it up. Gravity, the (mostly)spherical shape of the Earth, evolution by natural selection, or cats, for example.

Wait, okay, there's another one: 4) to understand or remember something in a certain way, as in "I believe they said they would do that." I'm not really counting that one, though; it's got nothing to do with certainty.

So what we need is a different word for the third kind, in my opinion (or belief, if you will). Because saying "I believe in evolution" isn't equivalent to "I believe in leprechauns" in any way except linguistically. Assuming that it does leads to all sorts of confusion; for instance, one might say "I believe in climate change" and someone who doesn't accept the science might retort with "Sure, and I believe in Mothman."

The difference is substantial. It's why I bang on about science in here and try not to use the word "belief" in those contexts.

But I used to believe it wasn't that big a deal.

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