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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10 Most Recent Blog Comments:

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.

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.

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

1 Like

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

February 20, 2023 at 8:50am

#1045128

Remember What Now?

I don't remember when I saved this article, but it's from January, so it couldn't have been that long ago.

Want to improve your memory? Try these unexpected tips.
Your memory probably isn’t as bad as you think.

What? I can't even remember what I had for breakfast this morning, and it's 8:30 am as I write this.

The brain is an extraordinary organ, with many wonderful qualities, including the ability to forget — which may actually be a good thing. “If we remembered everything that we experienced, our brains would be hoarders, clogged with all sorts of useless crap that gets in the way of what we really need,” says Charan Ranganath, a professor of psychology and the director of the Dynamic Memory Lab at the University of California Davis.

Yeah, except my brain decides that I really need jokes, song lyrics, and something embarrassing I said when I was 9, rather than where I left that damn screwdriver or someone's name five seconds after they tell me.

In today’s constantly plugged-in, always-on world, people are faced with a barrage of information — emails, news, pointless meetings, traffic updates, chitchat from family members — far more than anyone can process, Ranganath explains.

I don't know about you, but I'm sick and tired of people blaming tech for everything.

These issues with remembering often rear their heads at the least convenient times: when you’re in a rush and can’t find your keys, when you enter a room and don’t know what you came for, when you’re talking with an acquaintance whose name escapes you, when a friend refers to a nice moment you shared and you have no recollection.

The room-entering thing at least has a plausible explanation: you're in, say, the living room, and you notice that a panel of your awesome vintage 70s wood paneling is loose. So you decide to get a hammer and some finishing nails from the garage. You step out of the gloriously golden light of the wood-paneled room into the harsh LED glare of the shop, completely different surroundings. And your memory is tied to the visuals from the living room. You shake your head, go to the kitchen, make a sandwich, return to the living room... where you suddenly remember, oh, yeah, hammer and nails.

Another method to help you pay closer attention to the tasks at hand is what Braun calls the PLR technique: pause, link, and rehearse. This can help you both remember someone’s name and recall the reason you walked into a room. If you’re hiding a birthday present for your kid but fear you won’t remember where you put it, take five seconds to pause and focus on where you’re putting the gift, as “opposed to just putting it down and looking away and doing something else,” Braun says. Then look at the surroundings — this is the “link” step — and contextualize where you hid the present with its environment: in your closet, next to a bin of shoes.

That supports my surroundings hypothesis.

Use technology to your advantage, Ranganath and Schacter agree: Put meetings in your phone’s calendar (be detailed about who you’re meeting, where, and why) and make sure alerts are turned on, set reminders, and take photos of events to refer to later.

See? Tech is also a solution. No, I don't always remember who played the weird cousin in that obscure buddy movie from somewhere in the 70s or 80s, but Google does. I refer to it as my auxiliary memory.

Events that occur during heightened emotional states — fear, joy, anxiety, excitement, sadness — are more memorable. It’s why you remember your wedding day and perhaps not your 10th date. In order to remember more mundane things — where you’re storing dress shoes you wear once a year, a name, an item you need to pick up at the store — make these things extraordinary, says five-time USA Memory Champion and memory coach Nelson Dellis.

So, hire someone to scare the living shit out of you when you have something you'd like to remember. Got it.

Dellis recommends spending five minutes before bed recalling what happened that day. Did you see a beautiful sunset? Did your kid have a funny retort to a simple question? Did you eat something delicious? Replay small but lovely occurrences you’d like to savor.

That sounds nice in theory, but in practice, you're going to spend those five minutes berating yourself for neglecting to tip the delivery driver, or coming up—finally—with the perfect retort for that zinger your friend shot your way at lunch.

Okay, look, I'm not saying this article doesn't have some value. On the contrary. I just like to make snark. But as someone with a memory like a steel sieve, I'll probably forget to implement any of its suggestions.

2 Like • Comment • View 3 comments

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