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

March 16, 2023 at 9:22am

#1046446

Eight Isn't Enough

Today's "You're Doing It Wrong" article is from, not Lifehacker for once, but Quartz.

Why eight hours a night isn’t enough, according to a leading sleep scientist

Hahaha. Yeah. Right. Eight hours a night isn't enough, whatever income you're making isn't enough, sex twice a day isn't enough, and one slice of pizza definitely isn't enough.

I suspect this article was written by a cat.

For something that we spend a third of our lives doing (if we’re lucky), sleep is something that we know relatively little about. “Sleep is actually a relatively recent discovery,” says Daniel Gartenberg, a sleep scientist who is currently an assistant adjunct professor in biobehavioral health at Penn State.

"Discovery?"

As anyone who has lay awake at night contemplating the complexities of the universe can attest, sleep is a slippery beast.

Reminds me of the old joke about the dyslexic agnostic insomniac, who stayed up all night wondering if there is a dog.

There are also many, many misconceptions about sleep: that you can “catch up” on the weekend for lost hours of shuteye.

Oh? See "Sleeper Agent"

That you can get by on four hours’ sleep a night.

You can, for a little while. Source: experience.

That a nip of whiskey before bed helps you sleep better.

Even I, with my alcohol-positive lifestyle, knew that. I just don't use it as an excuse NOT to drink.

Even that eating cheese before snoozing causes nightmares.

I rather enjoy eating just before bedtime. It usually results in the most vivid and memorable dreams. That could be a kind of placebo effect, but who cares, if it works?

To set the record straight about being horizontal, Quartz spoke to one of the world’s most-talked-about sleep scientists.

Yawn.

He’s also an entrepreneur who has launched several cognitive-behavioral-therapy apps, including the Sonic Sleep Coach alarm clock.

So this whole thing is an ad. A crummy commercial.

Oh well, at least it's free, unlike most interesting stuff on the internet these days.

Some topics we cover:

why 8.5 hours of sleep is the new eight hours

Spoiler: It's not, really. They're including time trying to get to sleep and wake up and coming up with "total bedtime."

the genes that dictate if you’re a morning person or a night owl

Like that's going to change anything, except that now you get to blame genetics for something.

why you should take a nap instead of meditating

Finally, an idea I can wholeheartedly support.

how sleep deprivation can be a tool to fight depression

Another spoiler: this is, to say the least, disputable.

why sleep should be the new worker’s rights

Right, like anyone has time to fight for worker's rights while holding down three jobs and a side hustle.

and tips on how to get a better night’s rest (hint: it’s not your Fitbit)

I'm going to go out on a limb here and say that the answer is the Sonic Sleep Coach Alarm Clock.

The rest of the articleadvertisement is in interview form, and I won't reproduce much else from it. It's actually pretty interesting, and covers more than just the topics I just listed and snarked on. So reading it shouldn't put you to sleep.

3 Like • Comment • View 4 comments

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