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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 9 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 11 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.

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.

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