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 8 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 10 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.

March 9, 2024 at 9:27am

#1065919

Daydream Believer

Like to daydream? Well, consider this article from The Conversation. Author is British, so there are some weird-to-Americans spellings:

Daydreaming’s dark side: the compulsive, complex fantasy disorder that dominates some people’s daily lives

If daydreaming has a dark side, wouldn't that be a daymare?

Despite what we’re often taught to believe, daydreaming can be immensely useful.

Especially when you're approaching a deadline.

Daydreaming, when defined as thoughts that aren’t tied to what you’re currently doing, occupies a good chunk of our waking lives – an average of around 30% of the time if you randomly probe people.

Hopefully not while you're at work. Because that would be time theft. You'd be stealing from the Company. You're not stealing from the Company, are you, Janice?

But it’s estimated 2.5% of adults experience a type of excessive daydreaming which is defined as the disorder “maladaptive daydreaming”. So-called maladaptive daydreamers compulsively engage in vivid fantasies and daydreaming plots so excessively that it interferes with their ability to function in daily life.

I suppose there is no activity (or, in this case, usually inactivity) that someone, somewhere, can't find a downside to.

Unlike typical daydreams which can be fleeting (lasting seconds), maladaptive daydreamers can spend several hours at a time in a single daydream.

Okay, fine. That does seem a bit excessive.

These invented worlds are often rich and fantastical, with complex plots and intricate storylines that evolve over many years.

You've just defined fiction writers. Except for the, you know... writing it all down part.

With maladaptive daydreaming, there’s a strong urge to daydream and annoyance when this is not possible or interrupted. Most also find it difficult to stop or even reduce the amount of time they spend daydreaming.

I'm no expert, but isn't that pretty much the classic definition of addiction?

It’s important to note that immersive daydreaming and vivid fantasy activity isn’t by default maladaptive.

I'm just leaving that quote here in case you can't be arsed to read the article (perhaps because you're too busy daydreaming), and are wondering if you should be concerned about yourself. Chances are, no. I think the article is about making the rest of us aware that people like this exist, not, for once, trying to tell us that we're doing something wrong.

There seems to be a strong relationship between OCD and maladaptive daydreaming. One study found that over half of participants with maladaptive daydreaming also exhibited signs of OCD. This may suggest possible shared mechanisms between the two disorders, including intrusive thoughts, dissociation and a lack of cognitive control.

Again, far from expert here, but I'm pretty sure that by OCD, they mean diagnosed by a professional, not just someone being coy about their cleaning and/or organizational habits.

The fact that maladaptive daydreaming is not recognised as a psychiatric condition also means we know little about treatment options. There is one documented case study published in a peer-reviewed journal showing a 25-year-old man was able to cut the time he spent daydreaming in half – from nearly three hours daily to under an hour and a half. This was done over the course of six months using a combination of psychological treatments such as cognitive behavioural therapy and mindfulness.

At last, a practical use for cognitive behavioral therapy and mindfulness. Maybe. As always, one data point does not a scientific study make. Somehow, "directing the patient to become a writer" doesn't seem to appear in the list of treatments.

Finally, I can't do a title like that without posting the song I cribbed it from, which was written by John Stewart, but not that John Stewart:

4 Like • Comment • View 6 comments

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