Complex Numbers (Book)

Blog Calendar

◄ July ► 2022
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:

Write_Mikey_Write!

about 5 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 7 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.

July 12, 2022 at 12:02am

#1035065

Imagine

It's not very often that I talk about the plain meaning of this blog's title. I generally let the various puns on it rule here. "Complex" has several meanings, as does "numbers."

But today is one of those days. I know a lot of readers don't "get" math, but the video I'm linking today is more about history than it is about equations and such—though there are certainly equations used to illustrate the points. Periodically (pun intended), I go down a YouTube math/science video rabbit hole (though I have enough sense not to link a lot of them here), but I can't remember if this link was the result of one such journey, or if someone brought it to my attention.

Since it's a YouTube video, I'm linking it here as usual, but also embedding the video if you can't be arsed to click through.

How Imaginary Numbers Were Invented

Since it's a video, I can't do my usual quote-mining here, but to paraphrase what I feel are the important takeaways: Mathematics started out as a way to quantify the world. As such, things like negative numbers were anathema for a very long time (you can have three oranges, but you can't have negative three oranges). It was only when mathematicians let numbers be the abstractions that they actually are that math became a powerful tool for science, paradoxically allowing us to gain a much deeper understanding of reality.

Such abstractions shouldn't be so difficult to understand. As writers, we work in abstractions too—we call them metaphors, and as with math, they often reflect a deeper reality than the one in front of our noses.

This, by the way—the ability to think in metaphor—is, in my estimation, the thing that makes humans different from all other animals on Earth. It's why we're able to send rockets into space, some of them containing powerful space telescopes

that search deep into space and time, giving us an even greater understanding of the universe.

(The link in the previous paragraph is to the BBC article talking about the first image released from the Webb telescope, just yesterday. I have some issues with the way the article describes things, but this entry isn't about that, and it's good enough to get a sense of what's going on. Also the picture is really damn awesome.)

So it turns out that so-called imaginary numbers and, by extension, complex numbers, really do describe reality at its most fundamental (so far as we've been able to determine) level.

A far cry from counting grains of wheat, for sure. But it all comes from the same mindspace.

The video quotes Freeman Dyson (yes, the guy who conceptualized the Dyson sphere): "...nature works with complex numbers and not with real numbers."

The phrase "real numbers" has a specific definition in mathematics, but the rest of us have a different conception of the meaning of "real." Using the layman's understanding of "real," well, all numbers are real, even the imaginary ones that don't seem, at first glance, to correspond to anything we can see or touch.

Or, alternatively, all numbers are metaphors, describing an abstraction of what we can see or touch.

Either way, math is just crazy good at describing the world we know of as "real." In many cases, such as the one in the video above, the math existed first, only to later find a use in science.

It's all real. Even the stuff we don't understand. Especially the stuff we don't understand.

4 Like • Comment • View 5 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.