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:

Write_Mikey_Write!

about 10 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 12 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.

March 24, 2022 at 12:01am

#1029406

What Color is Your Cat?

As you know, sometimes I like to address the Big Questions in here.

Why do cats—and so many other animals—look like they’re wearing socks?
The possibilities of pigmentation are endless.

This is one of those times.

Grumpy Cat. Lil Bub. Maru. What do all of these internet-famous cats have in common? From ankle down, their paws are as white as the trendy marble countertops vying for attention in the very same Instagram feed.

I don't know about Instagram feeds, but I do have some experience with trends. Those marble countertops? One day, y'all will look back on those and they will seem just as outdated as Formica, or wooden wall paneling. All of those things that you laugh at now from the 70s and 80s? Yep, that's the fate of marble countertops, eventually.

The phenomenon of pigment mixed with white splotches can occur in pigs, deer, horses, dogs, guinea pigs, birds, and, in rare cases, humans. But it’s particularly prominent in cats, as evidenced by the fact that Socks consistently ranks in the top names for felines.

Me, I always wanted to name a pet Spot. Not just in honor of Data's cat from ST:TNG, but because when they start to annoy me I can open the door and go, "Out! Out, damned Spot! Out, I say!"

If the melanocytes are evenly distributed, the cat could have a unicolor coat, like Sabrina the Teenage Witch‘s all-black cat, Salem, or the all-white Hello Kitty.

Right, because the Evil Feline is totally a biological entity and not the product of some Japanese person's nightmare.

Piebaldism isn’t the only genetic quirk that can alter an animal’s fleece, according to the UC Davis Veterinary Genetics Lab. The tabby cat’s signature look is served up by the agouti gene, which determines the distribution of black pigment.

Another common name for a cat is "Tiger," because some cats have stripes, like tabbys. I remember reading somewhere once that back when there were a lot of tigers, before they were hunted to near-extinction by homo sapiens, they had the capacity to possess a calico coat, like the ones you find in domestic cats sometimes. I have no idea how true this is, but I find it hard to believe that it would have been conducive to survival.

Maybe the next meme-worthy cat should be named for a geneticist. Gregor Meowndel, anyone?

No.

One of my first forays into science was a high school project wherein I looked into cat genetics. Cat coat color isn't Meowndelian. Er, I mean, Mendelian. That is, sometimes you get strange results from breeding cats, as with calicos. But as it turns out, apparently, that's because the cat's pattern isn't strictly genetic, but also a matter of gene expression and conditions in the kitty womb. Kind of like with our fingerprints.

All very interesting, but you have to wonder what the point is, when it's well-known that whatever their fur patterns, they'll still knock things off of tables, beg for food, and try to fit into boxes.

Perhaps that last bit is because their genetic inheritance doesn't fit neatly into boxes.

4 Like • Comment • View 4 comments

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