Complex Numbers (Book)

Blog Calendar

◄ January ►
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:

🌖 HuntersMoon

28 minutes ago
In response to "Cell Mates"

Sum1's Home! / Jayne - You make me feel like a stranger in a strange land.

1 Like

Jayne

43 minutes ago
In response to "Cell Mates"

I'm charging you 85 gps for making me think of Heinlen. Feel free to ignore the invoice, my molecular makeup is too unorganized to develop a collections department

2 Like

Sum1's Home!

about an hour ago
In response to "Cell Mates"

One word in your post made me smile, and ask this question to myself. How many people will understand the word 'Grok', and where it originally came from? I must have read that book 2-3 times before I really grasped its meaning.

2 Like

tj in a winter wonderland

about an hour ago
In response to "Cell Mates"

I think we all know folks who are sacks of loose molecular parts; thankfully none of them are organized.

2 Like

🌖 HuntersMoon

about 3 hours ago
In response to "Cell Mates"

We've kept it a secret this long... Don't go blabbing it now!

2 Like

Robert Waltz

a day ago
In response to "Samurai Wine"

Annette

, if my understanding of the article is correct, it was one stop on his journey, and the one where he met the cult leader.

1 Like

Annette

a day ago
In response to "Samurai Wine"

How does Scotland figure in this?

2 Like

Elisa: Snowman Stik

a day ago
In response to "Samurai Wine"

"And that's why Hollywood won't touch it: there's no happy ending."

I don't know. Hollywood has given Guillermo del Toro money over the years, and happy endings aren't his strong suit. Also, Amazon Studios has toed the waters of unhappy endings (hello, Manchester by the Sea). Nothing's impossible, really.

2 Like

tj in a winter wonderland

a day ago
In response to "No Parking"

This reminds me of a past visit to The Badlands National Park. There are walkways (boardwalks) to prevent walking in the grass where rattlesnakes may be slithering about as well as signs everywhere warning of rattlesnakes. On this particular visit, a woman told her preteen children (three of them) to step off the walk and go over and stand by the sign for a picture. Luckily no one was snake bit; that time.

1 Like

🐕GeminiGem🎁

Jan 8, 2025 at 2:29pm
In response to "No Parking"

Flash flooding is a problem even when you aren't at a National Park. Around here, it is typically caused by "burn scars" on the mountains, where the land cannot handle rainfall after having a wild fire. It is super scary.

2 Like

January 10, 2025 at 11:01am

#1082204

Cell Mates

Sometimes, I just find something I think explains something pretty well. This is one of those times. From Quanta:

Meet the Eukaryote, the First Cell to Get Organized
All modern multicellular life — all life that any of us regularly see — is made of cells with a knack for compartmentalization. Recent discoveries are revealing how the first eukaryote got its start.

While it's amusing to think of cell organization as a bunch of oppressed worker cells getting together and striking for more pay and better benefits, that's not quite what happened.

Three billion years ago, life on Earth was simple.

Ah, yes, the good old days.

Single-celled organisms ruled, and there wasn’t much to them. They were what we now call prokaryotic cells, which include modern-day bacteria and archaea, essentially sacks of loose molecular parts.

In fairness, I know a few people who are little more than sacks of loose molecular parts.

Then, one day, that wilderness of simple cells cooked up something more complex: the ancestor of all plants, animals and fungi alive today, a cell type known to us as the eukaryote.

Think of the eukaryotic cell as like those old Reese's commercials where "You got chocolate in my peanut butter!" "You got peanut butter in my chocolate!"

“Eukaryotes are this bananas chimera of bacteria and archaea,” said Leigh Anne Riedman (opens a new tab), a paleontologist who studies early life at the University of California, Santa Barbara.

It took me a minute to grok that "bananas" in this case was used in its metaphorical sense of "wild and crazy." Though bananas are eukaryotic life, too.

The eukaryotes invented organization, if we use the literal definition of “organize”: to be furnished with organs.

It's not like that was directed by consciousness, but okay.

For many decades, biologists considered eukaryotes to be one of three main domains of life on Earth. Life is composed of three distinct cell types: bacteria and archaea, which are both prokaryotic cells with some key differences — for example in their cell membranes and reproductive strategies — and then there are eukaryotes, which are a much different kind of cell. Experts believed that bacteria, archaea and eukaryotes each evolved independently from a more ancient ancestor.

I know some people might be upset when science changes its mind. They want everything to be known, settled, certain. I kind of get that. But life doesn't work that way. Science correcting its own misconceptions is part of why it's awesome.

Then there's a bit in there about some researchers finding some archaea that might be like the ones that first became eukaryotes, and the whole family is named from Norse sagas, which I find amusing.

There's speculation out there that the origin of eukaryotic life is an extremely unlikely, once-in-a-planetary-lifetime process, and one that would have to take place for complex life to evolve on other planets. As we have not found even bacteria-equivalents on other planets yet, that's largely speculation. But we owe our existence to the microbial equivalent of a Reese's Cup.

4 Like • Comment • View 5 comments

© Copyright 2025 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.