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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Write_Mikey_Write!

about 8 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.

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Adherennium Plotting something

about 10 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.

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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.

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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').

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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.

January 12, 2024 at 10:55am

#1062292

Inventive

It may surprise some of you to know that women can be clever. Examples, from Mental Floss:

20 Things You Might Not Know Were Invented by Women

Necessity isn’t the only mother of invention.

I'll say it again for y'all in the back: Necessity may be the mother of invention, but laziness is the milkman.

Though it wasn’t always easy to get patents or the credit they deserved, women are responsible for many items we use today.

In some cases, blame. Like when Lise Meitner invented the atomic bomb. Okay, men helped there.

As usual for these longer lists, I'm skipping a few; the link is there if you want to see more.

1. Paper Bags

A man named Charles Annan saw her design and tried to patent the idea first. [Margaret] Knight filed a lawsuit and won the patent fair and square in 1871.

The truly amazing thing here isn't that a woman came up with something, but that a 19th century court acknowledged it.

But the paper bag was only the first step. What I want to know is: Who was the absolute hero god(dess) who figured out how to give them handles relatively cheaply? That person deserves a statue, a commemorative coin, and a star named after them.

3. Foot-Pedal Trash Cans

For me, the best inventions are those that save time, giving me more opportunity to play video games. But this one's awesome for another reason: you can throw shit away while you're cooking, without washing your hands every time you lift a trash can lid. Or you can have a lidless trash can like I do, but if you live with someone, they inevitably complain about the odor.

In fairness, mine used to be one of those battery-operated sensor lid things, which is even lazier than a foot-pedal. But the problem with moving parts is that they tend to break, and then they need to be fixed or replaced. Or done without, in my case.

4. Monopoly

Pretty sure I've covered this one before in here.

Elizabeth Magie created The Landlord’s Game to spread the economic theory of Georgism—teaching players about the unfairness of land-grabbing, the disadvantages of renting, and the need for a single land value tax on owners.

From what I heard, she had two sets of rules: one for that particular flavor of socialism, and one for capitalism. For some reason, the company that eventually published the game only concentrated on the latter. The reason why escapes me.

6. Disposable Diapers

I'm sorry, but of all the things on this list, this one gets my vote for "most likely to have been invented by a woman."

7. Dishwashers

I know I covered this one before in here: "Pretty Petty"

One might argue that this would be a contender for "most likely to have been invented by a woman," but this was a case of "my servants are Doing It Wrong and I don't want to do it myself, so I'll invent a machine."

12. Circular Saws

And being a power tool, this would be my "least likely to have been invented by a woman" thing. But I guess that would make me sexist, so it isn't.

A weaver named Tabitha Babbitt is believed to be the first person to suggest that lumber workers use a circular saw instead of the two-man whipsaw that only allowed cutting when pulled forward.

That's "two-person whipsaw," you sexist.

15. Folding Cabinet Beds

Sarah E. Goode’s folding cabinet bed didn’t just maximize space in small homes. In 1885, it made her one of the first Black women with a U.S. patent, after Martha Jones, who received one in 1868 for her cornhusker design.

Yeah, but Jones cheated; she was a time traveler.

(I really hope that's not too obscure a reference.)

19. Computers

Women in computer science have a role model in Grace Hopper. She and Howard Aiken programmed and designed Harvard’s Mark I computer, a five-ton, room-sized machine in 1944.

Women have been involved in computer science since before there were computers. Not to minimize Hopper's contribution at all.

Like I said, there's more I didn't cover. It's not always necessary to know who invented the stuff we use (nor would it be practical, as most things were group efforts), but sometimes it's nice to acknowledge where they came from.

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