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:

tj-Merry Mischief Maker

about 8 hours 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 ✍️

about 12 hours 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

about 15 hours ago
In response to "Gray Matters"

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

2 Like

tj-Merry Mischief Maker

a day ago
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

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

December 26, 2024 at 9:30am

#1081547

Gray Matters

Just about a month ago, I linked an article about going gray, in the entry "A Touch of Gray"

. The PopSci article proclaimed with great certainty that gray hair can never return to its original color. Well, here's SciAm going "nuh-uh" in 2021:

Gray Hair Can Return to Its Original Color—and Stress Is Involved, of Course
The universal marker of aging is not always a one-way process

And here I thought the universal marker of aging was yelling at kids to get off your lawn.

As we grow older, black, brown, blonde or red strands lose their youthful hue.

Which is why investments in hair dye companies are likely to be lucrative.

Although this may seem like a permanent change, new research reveals that the graying process can be undone—at least temporarily.

Obviously, everything is temporary, especially if you're old enough to have gray hair in the first place.

Hints that gray hairs could spontaneously regain color have existed as isolated case studies within the scientific literature for decades.

Sometimes, all it takes to disprove a hypothesis is a single counter-example. Like if you said "all planets have moons," and then someone showed that Venus is a planet that doesn't have moons, your hypothesis would be broken. In biology, though, things aren't always that clean-cut, and a single counterexample could just be a fluke or a hoax.

In a study published today in eLife, a group of researchers provide the most robust evidence of this phenomenon to date in hair from around a dozen people of various ages, ethnicities and sexes.

While a sample size of around 12 also doesn't do much to make the findings definitive, this phenomenon is apparently rare enough that to expect a bigger sample would be wishful thinking.

It also aligns patterns of graying and reversal to periods of stress, which implies that this aging-related process is closely associated with our psychological well-being.

On this point, the article is in alignment with the one I posted last month.

Around four years ago Martin Picard, a mitochondrial psychobiologist at Columbia University...

Now, that's someone who either hates Star Trek with a burning passion, or takes every opportunity to tell his underlings to "make it so."

...was pondering the way our cells grow old in a multistep manner in which some of them begin to show signs of aging at much earlier time points than others.

I don't really have anything to say about that process or the science behind it; I just wanted to make a Picard joke in an entry about hair.

These patterns revealed something surprising: In 10 of these participants, who were between age nine and 39, some graying hairs regained color.

So, this is highly unlikely to apply to older folks who go gray. Still, research like this might lead to a way to artificially restore hair melanin without dyes. Not that I care. I think I'd look awesome with a gray mane.

Most people start noticing their first gray hairs in their 30s—although some may find them in their late 20s.

That early? This surprises me, though I've known quite young gray-haired people. But one never knows: those who turn gray then could be dyeing their hair, or shaving it off entirely.

The team also investigated the association between hair graying and psychological stress because prior research hinted that such factors may accelerate the hair’s aging process. Anecdotes of such a connection are also visible throughout history: according to legend, the hair of Marie Antoinette, the 18th-century queen of France, turned white overnight just before her execution at the guillotine.

If true, for a queen, that had to add insult to injury.

Eventually, Picard says, one could envision hair as a powerful tool to assess the effects of earlier life events on aging—because, much like the rings of a tree, hair provides a kind of physical record of elapsed events.

Or, you know, a sorcerer could use it to control you. You never know.

5 Like • Comment • View 3 comments

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