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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Brandiwyn🎶

a day ago
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.

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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March 28, 2023 at 9:39am

#1047081

Measure by Measure

Ever wonder why we here in the US use an archaic system of measurements? This article Britsplains it.

It's been 230 years since British pirates robbed the US of the metric system
How did the world's largest economy get stuck with retro measurement?

Surprisingly, the answer isn't "because apart from the Revolution, we don't like change."

I don't know how historically accurate this is, but it's a good story.

In 1793, French scientist Joseph Dombey sailed for the newly formed United States at the request of Thomas Jefferson carrying two objects that could have changed America. He never made it, and now the US is stuck with a retro version of measurement that is unique in the modern world.

That's rich coming from Brits who a) definitely do use miles to talk about distance, while measuring petrol in liters and b) didn't decimalize their currency until like 1970.

The first, a metal cylinder, was exactly one kilogram in mass. The second was a copper rod the length of a newly proposed distance measurement, the meter.

Since that time, the way SI units are defined has changed. The copper rod, especially, would have been subject to thermal expansion and contraction. Sure, nowadays you can keep it in a climate-controlled room, but in the 18th century?

Jefferson was keen on the rationality of the metric system in the US and an avid Francophile. But Dombey's ship was blown off course, captured by English privateers (pirates with government sanction), and the scientist died on the island of Montserrat while waiting to be ransomed.

I've been to Montserrat. There are worse places to spend your last days. Well... there were, before a volcano devastated the island.

The reason for this history lesson? Over the last holiday period this hack has been cooking and is sick of this pounds/ounces/odd pints business – and don't even get me started on using cups as a unit of measurement.

You know, actually, I get it. I've mentioned before the difficulty in measuring, say, one cup of broccoli. And you're supposed to use a different "cup" for liquids and solids. So much better to use grams (or even ounces) for more consistency. Some recipes do that, and I'm grateful for it.

It's time for America to get out of the Stone Age and get on board with the International System of Units (SI), as the metric system used to be known.

That's... not precisely true, as I understand it. Which is to say, any measurement system that's self-coherent and is decimal (using powers of ten) is "a" metric system. SI is a particular metric system based on, primarily, the meter, the kilogram (not the gram, for some arcane reason) and the second.

There's a certain amount of hypocrisy here – I'm British and we still cling to our pints, miles per hour, and I'm told drug dealers still deal in eighths and 'teenths in the land of my birth.

Thank you for acknowledging that. Also, a British pint isn't the same as an American pint.

The cylinder and rod Dombey was carrying, the former now owned by the US National Institute of Standards and Technology, was requested by Jefferson because the British system in place was utterly irrational.

Perhaps, but once you learned it, you didn't want to change.

To make things even more confusing, individual settlements adopted their own local weights and measures. From 1700, Pennsylvania took control of its own measurements and other areas soon followed. But this mishmash of coins, distances and weights held the country back and Jefferson scored his first success in the foundation of a decimal system for the dollar.

So if you've ever wondered why we managed to use decimal currency while the UK held off for nearly 200 years, well, that's on Mr. Jefferson.

"I question if a common measure of more convenient size than the Dollar could be proposed. The value of 100, 1,000, 10,000 dollars is well estimated by the mind; so is that of a tenth or hundredth of a dollar. Few transactions are above or below these limits," he said

Which is kind of like Bill Gates' possibly apocryphal saying that no one would ever need more than 32 megabytes in computing power. Or whatever; can't be arsed to look it up.

Incidentally, though they share the same prefixes, computer memory isn't decimal; it's based on powers of two. Like, a kilobyte isn't 1000 bytes; it's 1024 (which is a power of 2). You know, just in case we weren't confused enough.

America was a new country, and owed a large part of the success of the Revolutionary War to France, in particular the French navy. The two countries were close, and the metric system appealed to Jefferson's mindset, and to many in the new nation.

I'm sure "many" others were vehemently opposed.

And this desire for change wasn't just limited to weights and measures. Also in 1793, Alexander Hamilton hired Noah Webster, who as a lexicographer and ardent revolutionary wanted America to cast off the remnants of the old colonial power. Webster wrote a dictionary, current versions of which can be found in almost every classroom in the US.

Yes, and from what I understand, that's why we spell "color" without a "u." Wow. Much change. Such revolution.

What has kept the metric system going is its inherent rationality. Rather than use a hodgepodge of local systems, why not build one based on measurements everyone could agree on configured around the number 10, which neatly matches the number of digits on most people's hands?

I would argue that base-12 makes much more sense. What's a quarter of 10? 2.5. What's a quarter of 12? A much more easily comprehended whole number, 3.

Above all it's universal, a gram means a gram in any culture. Meanwhile, buy a pint in the UK and you'll get 20oz of beer, do the same in America and, depending where you are, you'll likely get 16oz – a fact that still shocks British drinkers. The differences are also there with tons, and the odd concept of stones as a weight measurement.

Yeah, except a standard gram is a unit of mass, so its weight is different at the South Pole than it is at the equator, or in an airplane. As for the beer thing, most US breweries measure in ounces. Ask for a pint, and they'll verify if you mean "16 ounces."

As for stones, it took me an embarrassingly long time to figure that one out.

This is down to convoluted systems like 12 inches in a foot, three feet in a yard, 1,760 yards in a mile, compared to 100 centimeters in a meter and 1,000 meters to a kilometer. A US pound is 0.453592 kilograms, to six figures at least, these are the kind of numbers that cause mistakes to be made.

What's less well known is that the US actually uses SI standards. All of our common measurements are defined by SI units. This doesn't help with conversion issues, of course; as with the Mars probe that missed Mars (the author goes into this further).

And don't even get me started on Celsius and Fahrenheit. With Celsius water freezes at 0 degrees and boils at 100 at ground level, compared to 32 and 212 for Fahrenheit. It's a nonsensical system and the US is now the only nation in the world to use Fahrenheit to measure regular temperatures.

It's not "nonsensical;" it just uses different benchmarks. The advantage of Fahrenheit is its ability to easily record subtle differences in temperature. To get a similar granularity with Celsius, you have to use half-degrees. Also, it's a lot more impressive to say "it's 100 degrees out!" than "it's 38 degrees out!"

But yeah, okay, Celsius makes more sense.

The article delves into more history, and I won't repeat more of it, but it's there if you're interested.

I'll just share what, to me, is the stupidest reason for not converting (there are less stupid reasons, like retrofitting old piping systems): "We'd have to change football fields!"

For fuck's sake. It's not as if sports can't continue to use yards, feet, miles, whatever. Nothing says that abomination called American football can't continue to talk about the 30-yard line or whatever. Hell, there's even precedent: horse races are still measured in furlongs.

And have been fur a long time.

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