Complex Numbers (Book)

Blog Calendar

◄ April ►
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
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:

tj-Merry Mischief Maker

about 7 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

about 12 hours ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

about 14 hours ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

a day ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

a day ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

April 12, 2024 at 11:00am

#1068587

Coincidences

Gonna have to contradict folk wisdom again: there really is such a thing as coincidence. Cracked has a few examples here:

6 Coincidences Where Different Stuff Developed the Same Exact Way

...except "same exact" is, for most of these, a stretch.

In March 1951, a new comic called Dennis the Menace debuted in the U.S. That very week, a different comic about a different mischievous boy debuted in Britain, and it was also called Dennis the Menace. Both became hugely popular, and neither adapted the other, and neither ripped off the other.

It would be weirder if coincidences never happened. Other things invented in disparate places simultaneously include calculus and the theory of evolution. Those are less coincidence, though, and more about the background having been laid out, setting the stage for ideas that were ready to be invoked.

6. The Tale of Hershey’s and Hershey’s

These two brands didn’t, say, form on different continents with the same name, like how there was one restaurant in Australia called Burger King and was unrelated to the famous burger chain. The two Hershey companies both formed in Pennsylvania, in the same county, within a decade of each other, by unconnected men named Hershey.

Trademark law is complicated and way outside my expertise, but from what I understand, it gets even more complicated when actual peoples' names are involved.

Milton Hershey created the Hershey Chocolate Company in Lancaster, Pennsylvania, in 1884. Jacob Hershey and his four brothers created the Hershey Creamery Company in Lancaster, Pennsylvania, in 1894. The Hershey brothers were not related to Milton Hershey, and they didn’t form a creamery to piggyback off Hershey chocolate. They formed a creamery because they were a family of farmers.

Still, ten years apart, same county? You'd think even back then some lawyer would have advised one of them to change their name.

5. The True Name of Dogs

We’re playing around with exactly how the conversation went down, but the basics are true: The host kept saying “dog,” and Dixon assumed he’d misunderstood the question. Against all likelihood, “dog” really was the Mbabaram word for dogs, though the language shares no other roots with English or with any of the many languages related to English.

There exist other linguistic coincidences, though they escape me at the moment and I can't be arsed to look for them. But given the number of extinct and extant languages in the world, combined with the relatively limited number of sounds a human can make, it would surprise me more if there were no such coincidences.

4. We’re Stuck With Two Different Calories

We measure food using what’s called the large calorie, the dietary calorie or (most confusing of all) Calorie with a capital C. The other type of calorie is the small calorie, or calorie with a lowercase c.

And yet, people act like calories are an ingredient in food, not a measure of energy.

Obviously, we should not use the same word to describe these two different measures, and for a while, people tried calling Calories “kilocalories.” That made far too much sense, so it never caught on.

Pretty sure I've seen kCal in European food labeling, but I could be wrong. In any case, again, this is less coincidence and more sharing a common origin, unlike the dog example above.

Also, it's not that hard to determine which one is meant from context. If it's about stuff you're going to put into your gaping maw, it's Calories. If it's not, it's calories.

3. Nacho, and His Nacho Formula

Nachos were invented by Ignacio Anaya. “Nacho” was his nickname and has traditionally been the nickname of various people named Ignacio. You know that already if you have any friends named Ignacio, if you saw that Jack Black luchador movie or if you watched Better Call Saul.

Okay, gotta admit I didn't know that. I like Jack Black, but haven't seen that movie. Nor have I seen the referenced TV show.

In this example, the coincidence isn't, as one might expect, two dudes named Ignacio coincidentally inventing nachos at the same time. No, it's way cooler than that, at least if you know the slightest bit about chemistry:

The emulsification agent used in nacho cheese today is sodium citrate. Its chemical formula is Na₃C₆H₅O₇. NaCHO. It was destiny.

No, it's coincidence. Stay focused, okay?

2. We Keep Lolling Over Lol

Lol is Dutch for “laugh,” or “fun” or “joke.” This probably derives from an earlier word lallen, which referred to drunken slurring.

While it shouldn't be surprising that there are similar words in English and Dutch, which are, unlike with the dog thing above, related languages, the similarity here does seem to be coincidence in that LOL is an English acronym that happens to mean something similar to what this article claims is a Dutch word for laugh.

But, again, it's not that farfetched.

1. Everything Ends Up Crabs

No, everything really doesn't. Crabs are no more an inevitable end product of evolution than humans or bees. Some scientists a while back pointed out that different evolutionary lineages produced crablike forms, and the popular media ran with that and exaggerated it to "everything ends up crabs."

I can deal with some hyperbole on a comedy site, though.

You see, we have a lot of different types of crab, and they didn’t all form from one crab diverging into different species as evolution progressed. They formed because a bunch of different species all independently evolved into the same basic animal: the crab. Biologists call this process crabification.

Coincidentally (heh), "crabification" is what I call what people do at certain all-you-can-eat seafood buffets.

Often, when different species evolve in some convergent way, we can point to what’s desirable about that trait, which makes everyone naturally acquire it. When a bat and a dolphin each evolve methods to navigate through chirping, that’s weird, but we can easily say why multiple species would evolve echolocation: because it helps them get around. With crabs, we can only speculate on why creatures keep evolving these flat bodies and tiny tails instead of, say, all turning into lobsters.

Fair enough, but it doesn't rise to the level of coincidence if there's something in the marine environment that makes it useful to have a hard shell and pincers with a flattish body plan, even if we can't quite point to what that something is.

This would be like saying "what a coincidence that fish and marine mammals both have fins." Except we know that fins are useful appendages for underwater locomotion, and that underwater locomotion is helpful for underwater survival.

So, honestly, I mostly kept this article in the queue because of the sodium citrate nacho thing. But it's all interesting, as far as I'm concerned.

Because sometimes, there really is such a thing as coincidence.

3 Like • Comment • View 1 comment

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