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:

Turkey DrumStik

about 12 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

September 25, 2018 at 1:13am

#941992

I'm okay with this

When the British occupied India, many things caught them unprepared. To name but a few: the heat; Indians; lack of beer; and malaria.

But the Brits are nothing if not inventive, so they figured out ways to deal with most of these issues. For example, malaria, which is a disease spread by mosquitoes: they realized that quinine makes an effective anti-malarial treatment. Quinine is the active ingredient in tonic water (that is, the "tonic" part), but the problem was that tonic water tastes remarkably like ass.

So in order to make it more palatable, they invented the gin and tonic. Leave it to the Brits to add gin to something to make it taste better.

Remarkably, the invention of the gin and tonic represents the one and only positive thing to result from mosquitoes. Since that has already been developed, though, it's not going away.

Neither are mosquitoes.

Or... are they?

https://www.bbc.com/news/science-environment-45628905

Gene editing wipes out mosquitoes in the lab

Researchers have used gene editing to completely eliminate populations of mosquitoes in the lab.

The team tested their technique on the mosquito Anopheles gambiae, which transmits malaria.

They altered part of a gene called doublesex, which determines whether an individual mosquito develops as a male or as a female.

This allowed the Imperial College London scientists to block reproduction in the female mosquitoes.

Now, normally, I'm against genocide. It's rude. But for mosquitoes, I'll make an exception.

And yeah, I know, the risks, unintended consequences, etc. I've been reading science fiction most of my life, so I know that there might be some downsides. Take out a species, and we can't always predict what follows. Still, I'm pretty sure the ecosystem would adjust, eventually, to having no bloodsucking pests around. Well, other than politicians, anyway - still no way to eliminate them entirely.

Or maybe something more catastrophic could occur. According to the gospel of science fiction, maybe the gene editing will take on a life of its own, eventually spreading to humanity. But you know what? We're all going to die eventually, anyway, and I promise you that the last words of the last human on earth will be "At... least... there... are... no... more... mosquitoes... croak."

Worth it.

In fact, about the only downside I can see would be if the cure gets out of control chaotically, attracting more dark matter to Earth, increasing its gravity such that it spirals into the sun. Millions of years later, flaming fusion cockroaches swarm from the solar orb and take over the galaxy.

But hey, at least there won't be mosquitoes.

3 Like • Comment • View 1 comment

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