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 19 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 21 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.

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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').

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 20, 2020 at 12:30am

#975921

Easy Rider

This one has been on my list for a while, but it's appropriate that it came up by random selection now, as I spent a few hours alone on the road today.

https://www.afar.com/magazine/a-love-letter-to-driving-alone

A Love Letter to Driving Alone
Embarking on a road trip by yourself is solo travel taken to another level.

As I've mentioned before, I've taken numerous solo road trips: at least three halfway across the country, and at least four all the way (I live pretty close to the Atlantic Ocean). I might have lost track. And technically one of the halfway trips was with friends, but I did most of the driving.

And I like it.

When I was a kid, I had a picture in my mind of how my future adult self would look— a snapshot that never moved or changed. In it, I am driving a Jeep Wrangler with a removable top through a barren desert, Thelma and Louise–style, with a floppy-tongued dog riding shotgun as my partner in crime.

Except for the dog thing, Jamie Feldmar would be my spirit animal. Not that there's anything wrong with dogs; I just don't need the hassle of animal companions on a road trip.

But I have done fairly well for myself in the driving-alone-through-epic-scenery department. In fact, it’s one of my preferred ways to travel, ideally with the music cranked up, in a high-torque car, and several hours of daylight until my next destination. I am, in that moment, completely in control, cruising through remote and sometimes forbidding landscapes, marveling at the natural world and knowing that I—and I alone—get to have this moment.

I get it, Jamie.

Driving alone is not always smooth sailing. One night, in the foggy Pyrenees in southern France, my GPS died halfway to my farm stay, and I drove in a panicked circle across a cornfield as night fell

One thing I haven't done is drive in a foreign country. Well, Canada, but that hardly counts. But I never, never, never drive any appreciable distance without at least two GPS backups, one of them being a paper map.

I am aware that it is a great luxury to be able to do this. I am able, most of the time, to capably drive across these stunning surroundings without fear, though many of my female counterparts in other parts of the world would not.

It is not clear to me what gender Jamie is. With that name, it could be any. The "Thelma and Louise" reference makes me think one thing, but "female counterparts" seems to imply the other. Of course, it doesn't matter in terms of the feelings described in this (very short) article, but since they brought up the concept of privilege, there is a distinction.

To outward appearances, I'm a standard-issue American white guy. Right now my hair is fairly long (The Dude abides, after all), but I tend to keep it short on road trips. Why? There are still places in the US where Bob Seger songs and Dennis Hopper movies still ring true. Long-haired hippie-freak Waltz is in more danger than short-haired gender-conforming Waltz would be. Obviously, no road trip is risk-free; driving is one of the most risky things you can do, at least among the list of things that aren't overtly stupid like playing Russian Roulette. I think the number I heard quoted was something like a 1 in 800 chance of becoming an ex-human in any given year, which is pretty good odds compared to the one-in-six chance of kicking it in the Russian Roulette game, but lousy compared to, say, air travel or staying at home and quivering in fear (the latter also being more risky than air travel). And the risk of something bad but nonlethal happening, such as being robbed or getting a flat tire in the middle of nowhere on a dark and stormy night, is also nonzero.

But why marginally increase those odds by presenting myself as anything but a culture-conforming normal person?

Now, my perspective is, obviously, that of a standard-issue American white guy, and I can't presume to know what it might be like to be a chick, or an African-American, or a Hispanic person driving through southern Texas -- but the statistics imply that I have less to worry about. I can mosey into a Montana biker bar -- and I have -- without anyone batting an eye. There are few places that I don't feel, if not welcome, at least tolerated for the money I'm presumably about to spend.

So yeah, it is a great luxury to be able to do these things. I don't know if "without fear" is purely accurate, but I feel like I've at least made a risk assessment and determined that it's worth it. There's always risk; as I noted above, there's risk to just staying home. If I have an actual fear, though, it's that I'll crack my skull in the bathtub or something and my cat-chewn corpse won't be found for weeks. Because then I died doing something stupid instead of on an adventure, however tame that adventure is.

I always wonder, especially as I’m navigating some mildly treacherous path carved into the side of a mountain, how this road was built. Who drafted this route? Who blasted the raw earth and smoothed the asphalt? How long did it take? How did they know it would work?

I have to say, as a civil engineer, I appreciate these questions. Many people take roads for granted - and that's largely by design. We should be able to take them for granted; it means they're operating as designed. But most roads, especially the larger ones, had some engineering input, if not design from the ground up, so to speak. You only notice that, though, when something goes wrong: a flood indicates a failure of the drainage system, or rough asphalt may indicate a poorly thought out surfacing mixture. Or, you know, if a bridge that we don't have the money to maintain falls into the Mississippi.

On one of my epic road trips, I swung by the Hoover Dam. There's a plaque on it, right in the middle, on the state line (or at least close enough for government work):

A MODERN CIVIL ENGINEERING
WONDER OF THE UNITED STATES
––
ONE OF SEVEN SELECTED BY THE
AMERICAN SOCIETY
OF CIVIL ENGINEERS
• 1955 •
← NEVADA • ARIZONA →

(source)

And when you're standing on that amazing hunk of concrete, it's absolutely obvious that there were civil engineers who sweated over every last aspect of its creation, not to mention the hundreds of construction workers involved.

But I think I'd add the US road system to my list of Great Achievements of Civil Engineering.

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