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 23 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

a day 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.

December 5, 2019 at 12:03am

#970979

Observability Bias

Today's link comes to us from the UK, so be prepared for British spelling. In contrast to yesterday, it's a short one and probably more relevant to our daily lives.

https://www.theguardian.com/lifeandstyle/2017/oct/27/lonely-short-of-friends-loo...

Lonely? Short of friends? Try looking at it differently

You never see your friends at home alone in their pyjamas, eating pickled onion Monster Munch...

Okay, so I was prepared for "pyjamas," but what in the culinary hell is "pickled onion Monster Munch?" Jeez, England...

Psychologists are regularly berated for spending their workdays reaching blindingly obvious conclusions about the world... At first glance, it’s tempting to respond that way to a new study from the University of British Columbia, explaining why people tend to assume that their friends have more friends, and lead less solitary lives, than they do. Can you guess? That’s right: because every single time we see our friends, they’re socialising.

That's like every time my friends see me, I'm drinking, so they probably assume I'm an alcoholic. But I rarely drink when there aren't other people around. Don't get me wrong; I'm not averse to drinking alone - it's just that alcohol helps me socialize, but what it does not help me do is play video games, read, or follow TV shows. Perhaps if I actually wrote more, I'd drink more.

You’re never there when [your friends] wake in the dark at 3am, wondering where their lives are headed.

I'm not immune to existential crises, but this never happens to me. Is it common? Maybe it doesn't happen to me because I like to stay up until at least 3 am. Okay, that's a technicality. But in general, the only times I wake up when I ought to be sleeping are a) a cat has knocked something off of something else or b) I just had a nightmare, in which case I think, "Gah, another nightmare. Oh well. Just a dream" and go back to sleep.

So, yes, the fact that we only ever experience loneliness when it’s happening to us is blindingly obvious, I suppose. But blindingly obvious in an almost literal sense: it’s so self-evident, we barely ever see it.

I'm pretty solitary by nature, so I'm rarely lonely. I'm not even sure that I've ever experienced it in the way that others do, kind of like I'm never sure if my "yellow" is your "yellow." I like people, in general; I just can only deal with so much socializing before I have to retreat.

Anyway, so add "observability bias" to those things we have to watch out for.

Speaking of which, I'm about to travel again, so posts aren't guaranteed for the next week or so. I'm sure you'll all find a way to cope.

1 Like • Comment • View 1 comment

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