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 9 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 11 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.

June 23, 2021 at 12:01am

#1012373

Dream a Little Dream

I've done entries in here before on dreams. It might even be too many; a search of "dream" in this blog yields five pages of results. But here's one more anyway.

Weird dreams train us for the unexpected, says new theory
AI inspires hypothesis that sleeping human brain might try to break its overfamiliarity with daily data

A reasonable hypothesis. Let's see if there's anything to back it up, or if it remains speculation.

It’s a common enough scenario: you walk into your local supermarket to buy some milk, but by the time you get to the till, the milk bottle has turned into a talking fish.

Wait, is this about dreams or LSD?

Dreams can be bafflingly bizarre, but according to a new theory of why we dream, that’s the whole point. By injecting some random weirdness into our humdrum existence, dreams leave us better equipped to cope with the unexpected.

That's also one reason why we have science fiction and fantasy.

Although Hoel’s hypothesis is still untested, an advantage is that it takes the phenomenology of dreams – particularly their sparse, hallucinatory, and narrative content – seriously, rather than viewing it as an unexplained byproduct of other background brain processes.

So, no, it remains speculation. But that doesn't mean it's wrong.

“However, as with so many theories that dreaming has a function, there is no evidence yet that dreaming is more than an epiphenomenon, a functionless byproduct of neural activity."

The idea of "function" can be a tricky one, biologically speaking. The way evolution works, something might start out having no apparent function, and then acquire one that's useful to the organism and promotes survival, which is then passed on. With dreams, well, it seems that dreaming is common to many animals. The question is, even if it is a byproduct of other processes, could it then be useful? I'd guess yes, but I'm hardly an expert.

The article finishes with a summation of some of the other possible functions of dreaming. I'm pretty sure I've mentioned some of them in here before. I'm not going to quote them here, but the article should be free to view if you're interested.

Of course, there's no reason why it couldn't be a combination of several of these functions (or even others). We have plenty of examples of body parts or processes serving more than one function. The mouth, for example, is used for both talking and chewing (hopefully not at the same time; were you born in a barn?) as well as... well... certain other things.

I'm not sure if they've ever been able to tell just when dreaming started along the evolutionary timeline, but as other animals do it, it might have been quite a long time ago -- in which case, it's had plenty of time to gain adaptive function. After all, at first glance, one might think that sleep itself isn't very conducive to individual survival, as it allows predators to sneak up on you. Dreams generally occur during sleep, though, and their benefit might very well outweigh any downsides to periodic unconsciousness.

But that's even more speculation, and in this case, coming from someone without any background or credentials. But that's okay -- as with a dream, the meaning can be very personal.

4 Like • Comment • View 3 comments

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