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 12 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 14 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.

May 9, 2020 at 12:05am

#983119

Lies, Damned Lies, and Memories

All memories are inaccurate.

PROMPT May 9th

Choose an event in your life that someone else remembers differently. Describe both memories and debate the differences. Who do you think is right? Why do you think you remember it differently?

When I was a kid, I loved astronomy. Okay, I still do. But when I was young, my father would take me out on cold winter nights and point out stars and planets, and I was fascinated. For a few minutes, until I got cold.

The only thing that kept me from being an astronomer is that I absolutely despise the cold, and in my voracious reading on the subject, I discovered that mountaintop telescopes are generally not heated. Fuck that. Next idea. Hell, I turned down admission to MIT because I didn't think I'd survive a Boston winter. Cambridge. Whatever.

Point is, I remember seeing Jupiter's moons without optical aids.

Now, this isn't completely unreasonable. People have reported seeing the Big Four, or a subset thereof, for centuries, and some probably saw them before Galileo resolved them in his telescope, but they didn't know what they were seeing. It's rare, but it happens.

Thing is, I also remember wearing glasses to correct astigmatism as a kid. It's unlikely that both of these memories are accurate. I'm going to go with the "glasses" one because I also remember seeing a class photo of Kid Me with a goofy haircut wearing goofy glasses.

So why do I remember seeing Io, Callisto, Ganymede, and/or Europa? Probably because I wanted it to be true that I was one of the few who could resolve them with the naked eye. I'm far more certain that, at some point, I saw them through a telescope or binoculars, and later conflated that memory with observing the Jovian system without lenses.

I don't have a direct response to the prompt. That would require me talking to someone I've known for a long time, probably at length, and that's not going to happen. The thing with Jupiter's moons is as close as I'm going to get to it.

But it's not just me. Science has known for a long time now that memories are usually distorted, often confused, and sometimes flat-out false. There was a whole scare back in the 80s and 90s involving Satanic ritual abuse of children. It was a Big Deal at the time; in those pre-internet days, the "news" still spread around the country like wildfire. The problem is, as I recall (which could mean nothing), not a single one of those allegations turned out to be true and, moreover, most of them were connected with a single psychologist who was found to be encouraging false memories in older "survivors" of this "epidemic." It would have been mildly amusing if the allegations hadn't resulted in the ruination of several innocent caretakers.

Or, again, so I recall. The details are fuzzy and I can't be arsed to look them up, because the whole thing was just so stupid and unbelievable... and yet otherwise rational people believed it. Some say the Salem Witch Trials were the result of ergot poisoning or some such, but I maintain that it was just simple human herd mentality, the same mentality that caused people to actually believe that there was systematic abuse of children in the name of The Dark Lord, or the herd mentality that insists that covid-19 is a hoax.

What I can be arsed to look up are citations for my assertion that memory is unreliable.

https://www.healthline.com/health-news/mental-memory-is-unreliable-and-it-could-...

https://www.psychologytoday.com/us/blog/hidden-motives/201203/unreliable-memory

An excerpt from this second one:

We tend to think that memories are stored in our brains just as they are in computers. Once registered, the data are put away for safe-keeping and eventual recall. The facts don’t change.

But neuroscientists have shown that each time we remember something, we are reconstructing the event, reassembling it from traces throughout the brain. Psychologists have pointed out that we also suppress memories that are painful or damaging to self-esteem. We could say that, as a result, memory is unreliable. We could also say it is adaptive, reshaping itself to accommodate the new situations we find ourselves facing. Either way, we have to face the fact that it is “flexible.”

So, it doesn't matter what the memory is. Someone else is always going to remember it differently. Their recollection will be wrong. So will yours. Even if it's on video, you're not going to remember accurately the things outside the frame, or your emotions, or what you were thinking, or any number of thousands of other details. You may be absolutely certain that events unfolded in a certain way, were caused by this, led to that, whatever, but you are wrong.

I probably didn't see Jupiter's moons without aid. I'm pretty sure my father wrapped me in a blanket and mispronounced "Betelgeuse." I know for damn sure it was cold, because I remember freezing my little tuchis off and it usually is cold on nights when you can see Betelgeuse, and besides, the blanket. It was a red and white plaid thing with a weird, stiff, brown plastic fringe, and I have no idea what became of it. But do I really know for damn sure, or am I mixing different memories and filling in the blanks?

You can never know for sure, but there are some things that you want to know for sure; in my case, that my father encouraged me to love science and learning, so whatever memories reinforce that, I'll keep.

"Who do you think is right?" Both. And neither. Life is about reconciling what you think are your memories with what you wanted to happen, what you want to happen, and how you want to feel in the present.

And I'm okay with that.

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