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 16 hours ago
In response to "Extro Special"

I'll also use the spelling extrovert because if I use extravert my spell checker flags it. In college psychology our professor termed his own words, innervert and outervert because he thought they fit better. I, however, thought they sounded like definitions of belly buttons.

1 Like

Kåre เลียม Enga

about 22 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

a day 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

a day 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

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

Dec 26, 2024 at 4:20pm
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 ✍️

Dec 26, 2024 at 1:07pm
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

Dec 26, 2024 at 10:00am
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

February 26, 2024 at 10:11am

#1064910

Round and Round

A rare case of me sharing something relevant to my actual education and career. From The Conversation:

What are roundabouts? A transportation engineer explains the safety benefits of these circular intersections

Not that I can claim to be an expert at this. Mostly, I designed small subdivisions and commercial site plans, along with their unseen infrastructure. But not being an expert has never stopped me before in here; why start now? I know I've talked about them before, but this is, as far as I can tell, a fresh take.

If you live on the East Coast, you may have driven through roundabouts in your neighborhood countless times. Or maybe, if you’re in some parts farther west, you’ve never encountered one of these intersections. But roundabouts, while a relatively new traffic control measure, are catching on across the United States.

I've seen a few in what you lot call flyover states, too.

Roundabouts, also known as traffic circles or rotaries, are circular intersections designed to improve traffic flow and safety. They offer several advantages over conventional intersections controlled by traffic signals or stop signs, but by far the most important one is safety.

While it's possible to go overboard with safety (that is, you reach a point of diminishing returns; for example, why buses don't have seat belts), I don't think that's the case here.

As early as the 1700s, some city planners proposed and even constructed circular places, sites where roads converged, like the Circus in Bath, England, and the Place Charles de Gaulle in France. In the U.S., architect Pierre L'Enfant built several into his design for Washington, D.C.. These circles were the predecessors to roundabouts.

I think the DC traffic circles are one of the reasons some people freak out about roundabouts. They are, in a word, messy. It's important to remember that they were originally designed for horses, not vehicular traffic.

Anyway, the article delves more into the history, and discusses a lot of their benefits, concluding with:

The Federal Highway Administration estimates that when a roundabout replaces a stop sign-controlled intersection, it reduces serious and fatal injury crashes by 90%, and when it replaces an intersection with a traffic light, it reduces serious and fatal injury crashes by nearly 80%.

I didn't follow the links to those numbers, but it tracks with what I'd already heard.

One advantage that I think should be noted, but I didn't see in the article: Traffic light installation is expensive, and it incurs ongoing maintenance and operating costs. While a roundabout often takes up more space, usually requiring the purchase of additional right-of-way, I tend to think the life cycle costs are lower, considering that you're going to be doing things like mowing and repaving anyway. This may vary depending on location; rural right-of-way is generally cheaper and easier to obtain than urban.

Another thing kind of glossed over is the psychological aspect. People who are used to stoplights don't necessarily want to, or know how to, deal with this weird new thing. Well, part of that can only be overcome through time and familiarity. I'm sure it took a while to get used to traffic lights and highway cloverleafs, too.

An objection that I've heard is something along the lines of "I used to just go through that intersection, but now I have to slow down." I think some of that is selective memory. You might remember when you approached on a green light, breezing right across the intersection, but not so much the multiple times you've been stuck at a red light, fuming, willing the light to turn green through the power of mind alone.

Slowing down every time is still, in my view, superior to sometimes having to stop and wait.

As with all new things, there's a period of adjustment. If we still have cars in 50 years, I'm sure the future people will view stoplights as an unnecessary and hazardous anachronism.

2 Like • Comment • View 5 comments

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