Complex Numbers (Book)

Blog Calendar

◄ March ► 2021
S	M	T	W	T	F	S
	1	2	3	4	5	6
7	8	9	10	11	12	13
14	15	16	17	18	19	20
21	22	23	24	25	26	27
28	29	30	31
Archive				RSS

About This Author

Contact Me

Come closer.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

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 9, 2021 at 12:01am

#1006051

Inside The Box

Warning: Math entry.

Well, actually a philosophy entry.

Philosophy of math.

Some Math Problems Seem Impossible. That Can Be a Good Thing.
Struggling with math problems that can’t be solved helps us better understand the ones we can.

Considering that simple arithmetic seems impossible to a lot of people, that's saying a lot.

Construct a convex octagon with four right angles.

It probably says a lot about me as a teacher that I assign problems like this. I watch as students try to arrange the right angles consecutively. When that doesn’t work, some try alternating the right angles. Failing again, they insert them randomly into the polygon. They scribble, erase and argue. The sound of productive struggle is music to a teacher’s ears.

Oh, that kind of teacher.

Finally someone asks the question they’ve been tiptoeing around, the question I’ve been waiting for: “Wait, is this even possible?”

This question has the power to shift mindsets in math. Those thinking narrowly about specific conditions must now think broadly about how those conditions fit together. Those working inside the system must now take a step back and examine the system itself.

I've known people who pride themselves on "outside-the-box" thinking. Without exception, every one of those people were utterly lousy at inside-the-box thinking. Otherwise known as "thinking."

It is true that breakthroughs happen when people think beyond the stated parameters of a problem. However, it is at least as important to acknowledge the constraints of the parameters.

Consider, for example, the nine-dots puzzle, which was my -- and probably a lot of peoples' -- first encounter with lateral thinking. I have only a very limited ability to post images here, but a full description of it, with illustrations (including solution), is provided at this Wikipedia page.

From that entry: "The puzzle proposed an intellectual challenge—to connect the dots by drawing four straight, continuous lines that pass through each of the nine dots, and never lifting the pencil from the paper. The conundrum is easily resolved, but only by drawing the lines outside the confines of the square area defined by the nine dots themselves."

The solution, as noted, relies on not seeing the outer boundary of the nine-dot square as a constraint; to draw the required lines beyond that imagined border. However, the constraints posed in the instructions are inviolable; that is, "connect," "four," "straight," "continuous," etc. One might very well wonder why such constraints exist, and what practical use such constraints might have. Violating these necessary parameters is what most people do when they smugly "think outside the box." It's pernicious.

Anyway, the article goes on to explain why the octagon problem posed in its first sentence is, in fact, impossible (assuming a flat Euclidean plane). There's a bit of math involved, but merely geometry. You can probably skim those parts and still get a good idea what the article's about.

I'm talking about this mainly because, at least at the time I plopped the article into my blog fodder stash, I'd been seeing an uptick in people going, "Nothing is impossible! Everything is possible if you try."

And that's utter codswallop.

As noted in the article:

To consider impossibility, we need to understand that just asserting that a thing exists doesn’t make it so.

A well-known example is: Consider an omnipotent being. Can such a being create a boulder so heavy that even the being itself cannot lift it?

If the answer is "Yes," because the being is, after all, omnipotent, then from the question itself, the being cannot lift the stone -- but that's impossible because the being can do anything (which is the plain definition of omnipotence).

On the other hand, if the answer is "No," then the being is not omnipotent after all.

Conclusion: Omnipotence is impossible.

And yet, we can construct the sentence: "An omnipotent being can create an object so heavy that even the being itself cannot lift it."

Many more things are possible in language than in practice. Is "everything" possible in language? I can't answer that, because I'd have to use language itself to do it.

Math is a kind of language, too -- one that is far more precise than plain English. You can say almost anything and make it look like math (for example, "2+2=143," but saying it doesn't make it true. Mathematics is based on rigorous proofs, including proofs of impossibility. Language isn't.

I do think language has its limitations; I have yet to encounter a satisfactory time-travel story, for example, and whenever we start talking about infinity, people can't grasp it. But that doesn't mean that either is impossible.

And yet, I will generalize from the article's assertions about the impossible: it's in thinking about the impossible, in any subject, that we can find solutions to problems that are merely difficult. I personally think that this is one of the main purposes of fiction writing.

For a long time, mathematicians thought that there were no numbers that, when squared, yield a negative number. But such numbers turn out to be useful, so they simply invented new ones: the so-called "imaginary" numbers that I ramble on about in this blog's intro that no one ever reads. So yes, we might find that some things thought to be impossible are actually trivial, in math or in philosophy or in other areas.

But other things will always remain imaginary, thus keeping fiction writers in business for the foreseeable future.

5 Like • Comment • View 2 comments

© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.