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 2 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 4 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').

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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.

August 12, 2021 at 12:06am

#1015527

Furlonger Than You Thought

Low energy tonight. Worn out from playing D&D earlier. Am I getting too old for this shit? ....Nah.

So it's good that the random number generator gave me an easy, uncontroversial, and informational article today.

Why Are There 5,280 Feet in a Mile?

Because measurement units are completely arbitrary? Yes, even SI ones. I'd be more interested in why the meter is defined as the distance light travels in 1/299,792,458 seconds instead of a nice round 1/300,000,000. Okay, yes, I know the history of the meter; I'm just saying it makes just as much sense as defining a foot as 1/5280 of a mile.

Why are there 5,280 feet in a mile, and why are nautical miles different from the statute miles we use on land? Why do we buy milk and gasoline by the gallon? Where does the abbreviation "lb" come from? Let's take a look at the origins of a few units of measure we use every day.

Smugly, I knew most of these, but it never hurts to have a refresher.

If the mile originated with 5,000 Roman feet, how did we end up with a mile that is 5,280 feet? Blame the furlong.

I just like the word "furlong." Say it. Say it out loud. Right now. Absorb the laughter from those around you. Embrace it.

So if the statute mile is the result of Roman influences and plowing oxen, where did the nautical mile get its start? Strap on your high school geometry helmet for this one. Each nautical mile originally referred to one minute of arc along a meridian around the Earth. Think of a meridian around the Earth as being made up of 360 degrees, and each of those degrees consists of 60 minutes of arc. Each of these minutes of arc is then 1/21,600th of the distance around the earth. Thus, a nautical mile is 6,076 feet.

And I just lost half my audience.

Seriously, though, that sounds a little bit like how the meter was originally defined.

Like the mile, the acre owes its existence to the concept of the furlong... That's how we ended up with an acre that's equivalent to 43,560 square feet.

I spent my college years learning how to do engineering in both SI and Imperial units. I figured by the time I got out of college, we'd be using SI exclusively. Silly me. By the time I was a month (another arbitrary unit of measurement as we define it today) into my first job at a civil engineering firm, I had the area of an acre in square feet etched into my brain.

As the name implies, scholars think that the foot was actually based on the length of the human foot.

More like the sasquatch foot.

The wine gallon corresponded to a vessel that was designed to hold exactly eight troy pounds of wine.

Oh, that's helpful.

When I was in England, I noticed that they quote distances in miles, petrol in liters, and the price of petrol in pounds (the money, not the weight) per liter. Trying to convert all of that to all-Imperial or all-SI and dollars broke my brain (the best I could figure out is that they pay A LOT more for petrol there than we do for gas in the US). Fortunately, they still sell ale in pints so I could easily forget trying to figure it out.

Like several other units, the pound has Roman roots. It's descended from a roman unit called the libra. That explains the "lb" abbreviation for the pound, and the word "pound" itself comes from the Latin pondo, for "weight." The avoirdupois pounds we use today have been around since the early 14th century, when English merchants invented the measurement in order to sell goods by weight rather than volume.

As an aside, I wish more recipes would specify quantities by weight rather than by volume. The amount of flour in a cup, for example, depends on the amount of air in the sifted flour (even if you level it off like you're supposed to but I'm the only one who ever actually does that). But, for all practical purposes, on the surface of the Earth, weight is weight. You might have to adjust some shit if you're on top of a mountain, but that's more because of water's lower boiling point there and not gravity differential. Yes, gravity is slightly different at the Equator than at the South Pole, but if you're cooking at the South Pole, you have bigger problems than determining food quantity, like how to avoid freezing solid.

Also, how much broccoli is in a cup? Do you fill the cup to the rim, then stop? That doesn't seem like a lot of broccoli. Stuff all the broccoli you can into the cup, heaping it up? You get more like two cups that way. And it's heavily dependent on how finely chopped the nasty buggers are. If two people can't come up with the same amount of broccoli when cooking, then the cooking method is suspect. Specify it by weight, and there's no question.

Don't tell me "but it doesn't matter; just eat the damn broccoli already." I'm a goddamn engineer; quantities matter.

Early 18th-century steam engine entrepreneurs needed a way to express how powerful their machines were, and the industrious James Watt hit on a funny idea for comparing engines to horses.

Ironically, Watt's name ended up becoming a unit of measure (for power, which is energy per unit time; it's an SI standard).

Anyway, like I said, informational article. There's even stuff in there I didn't know, but now I do. Win.

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