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 10 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 12 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 25, 2022 at 12:01am

#1032826

Later

Today's article is five years old. Given the subject matter, it seems appropriate.

Why some people are always running late
It’s not always rudeness or scatter-brained behaviour – it can be something much deeper, writes Laura Clarke.

I thought I might have linked this one before, but I couldn't find it in a search. I've been at this for a long time and I have a shit memory.

We all know that person: there’s the child minder who is always late, the colleague who misses every deadline, even if just by a few hours, the friend you must tell to arrive 30 minutes earlier than she needs to for your lunch reservation.

30 minutes? Amateur. I dated a woman who had to be asked to show up two hours early. Even then, she was often late for the actual time. Hell, one time we left for a trip 28 hours after we were supposed to. She called to change plane reservations (this was effectively pre-internet) three times.

We didn't date very long. Bad enough my first wife was two hours late to our wedding. She had an excuse, at least (illness).

A look into the psychology of lateness offers a glimpse into a mind that that may be malfunctioning.

Seems to be a lot of that going around these days.

Perceptions of unpunctual people are almost always negative — even if misguided.

They should be negative. However, I'll tentatively accept the article's assertion that it's not always the reasons we think.

Some excuses, particularly for acute lateness, are fairly universally accepted —an accident or illness, for example. But others aren’t so easy to swallow. Some late people will pass it off as a symptom of being big-thinking and concerned with loftier matters than time-keeping, as an endearing quirk, a mark of doing one’s best work under pressure, or having the body clock of a night owl rather than a lark.

Okay, go right to hell with all those, especially that last bit. Maybe it's why you're late for work -- I almost always was, and for that reason -- but it's definitely not an excuse to miss your 8pm dinner reservation.

Being consistently late might not be your fault. It could be your type. The punctually-challenged often share personality characteristics such as optimism, low levels of self-control, anxiety, or a penchant for thrill-seeking, experts say. Personality differences could also dictate how we experience the passing of time.

Well, we can get all philosophical about whether those things are your fault or not, delving into questions of free will, but either way, it's still your responsibility. It's like... maybe you have the kind of personality that makes you love driving fast. I can understand that. But it's still your responsibility to pay the inevitable speeding ticket.

In 2001, Jeff Conte, a psychology professor at San Diego State University ran a study in which he separated participants into Type A people (ambitious, competitive) and Type B (creative, reflective, explorative). He asked them to judge, without clocks, how long it took for one minute to elapse. Type A people felt a minute had gone by when roughly 58 seconds had passed. Type B participants felt a minute had gone by after 77 seconds.

Leaving aside for the moment my skepticism about Type A and Type B, how many people did he study? Were they all undergrads? Did they control for people on the autism spectrum? What was the spread (I highly doubt every single participant had exactly 58 or 77 seconds; there had to have been some overlap)? Was the study replicated? The link from the article isn't much help.

Late people often have a “bizarre compulsion to defeat themselves,” wrote self-proclaimed late person and TED speaker Tim Urban in 2015.

Hey, I have one of those, but I'm not a late person. Sure, I'll procrastinate stuff, but when it comes to meeting up with other people, I'm on time. Usually.

He gave these poor souls a name: CLIPs, Chronically Late Insane People.

You go to hell with that acronym, too. It's demeaning to people with actual mental illness.

And depression often comes with low energy, making mustering the motivation to get a move on all the harder.

I can relate to that, as well, but when I'm depressed, I'm more inclined to just blow something off rather than be late for it.

For those left waiting, there is hope. You, too, can dictate what you’re willing to put up with.

“Instead of getting angry or upset, you can take a stand and set boundaries,” she says. “Talk about what you will do if the other person isn’t on time.” For instance, tell your late friend you’ll go into the movie without them if they’re more than ten minutes late.

That's way too generous. I'm not going to miss one second of the previews. You get there on time or early, or I watch the movie without you.

In summary, the article might have some good points. I don't know. But perhaps it will help someone. I especially liked the part about the rest of us not putting up with other peoples' lateness.

2 Like • Comment • View 3 comments

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