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

June 3, 2024 at 8:51am

#1072084

One for Sorrow, Two for Joy...

It's fairly well known that corvids are hardly birdbrains, but this recent article from SciAm delves into further evidence, while also punning on one of my favorite bands.

Crows Rival Human Toddlers in Counting Skills
Counting crows proclaim “caw, caw, caw, caw” when staring at the number four

The rock group Counting Crows were onto something when they chose their band name. Crows can indeed count, according to research published this week in Science.

That's not why they picked the name, but by Waltz's Law, never let facts get in the way of a good joke. Or a bad one. Especially a bad one.

But, you know, that's just, like, a pinion, man.

Okay, okay, puns out of the way for now, there's some interesting stuff here.

The results show that crows have counting capacities near those of human toddlers who are beginning to develop a knack for numbers, says lead study author Diana Liao, a postdoctoral researcher in neurobiology at the University of Tübingen in Germany.

So, no, they're not about to take over the world or build rockets to Mars.

“We think this is the first time this has been shown for any animal species,” she adds.

I know there's been research done in this area, but I can't be arsed to search for examples countering what the researcher says here. Yes, there have been claims that horses, e.g., can count, but those usually turned out to be hoaxes or misunderstandings.

They presented the birds with randomly ordered cues, four of which were visual—colored Arabic numbers that appeared on a touch screen—and four of which were auditory, including a short guitar chord and a drumroll.

Please, please tell me they sampled the latter from Counting Crows albums. Well, in my headcanon, they absolutely did.

Onur Güntürkün, a biopsychologist at Ruhr University Bochum in Germany, who was not involved in the research, says the new paper is “excellent”—even if the findings are “not unexpected” given all that scientists already know about crows and many other species’ intelligence.

Like I said: we know they're smart. They also apparently have a long memory. There are plenty of stories about people who pissed crows off, only to find themselves harassed by black-winged birds for the rest of their lives, indicating that not only does the pissed-off crow take revenge, but gets all their corvid buddies and offspring to punish the offender. People associate elephants with never forgetting, but elephants ain't got nothin' on crows.

On the flip side, do something nice for a crow, and they could become your army of darkness. I've thought about doing this by leaving shiny things out for them to collect, but I'm entirely too lazy, and what would I do with bird minions, anyway? Get them to attack surveillance drones? No, they might get hurt. Besides, my cats might get jealous.

Obviously, I can't leave this entry without an appropriate video:

If you think you need to go
If you wanted to be free
There's one thing you need to know
And that's that you can't count on me

2 Like • Comment • View 2 comments

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