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:

tj-Merry Mischief Maker

about 20 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

about 24 hours ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

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tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

December 3, 2024 at 8:51am

#1080769

Coasting Along

I'm not entirely sure why I saved this particular Live Science article, and it's only been like three weeks.

What's the 'coastline paradox'?
Coastlines are challenging to measure, but why?

Yeah... it's not really a paradox.

Flanked with fjords and inlets, Alaska is the state with the most coastline in the United States.

Easy to accomplish when you're a giant peninsula with many craggy islands offshore.

But what is the length of its oceanic coast?

It depends on whom you ask. According to the Congressional Research Service, the number is 6,640 miles (10,690 kilometers). But if you consult the National Oceanic and Atmospheric Administration (NOAA), the coastal edges of the state total 33,904 miles (54,563 km).

Yep, that's a big difference, all right. Not to brag, but I knew the answer. Still, I want to say that, obviously, the former number is incorrect, because Congress always lies.

The coastline paradox occurs because coasts are not straight lines, and this makes them difficult, or impossible, to measure definitively. From an aircraft, you can see that the coast has many features, including bays, inlets, rocks and islands. And the closer you look, the more nooks and crannies you'll find.

Oh, now I remember why I saved it. It's related to fractals like the Julia set or Mandelbrot set, which involve complex numbers, and, well... you know.

As a result, the length of a coastline depends on the size of the ruler you use.

This isn't just a coastline issue. Lots of survey boundaries follow the thread of a river (or, in the case of VA/MD, the low-tide shoreline of the Potomac), which has similar characteristics.

But if you used a smaller ruler, you'll capture more complexity, resulting in a longer measurement. Hence, a paradox.

Okay, I suppose, for some definitions of paradox. Regardless, that's what it's called, so I'll run with it.

According to work published in 1961, English mathematician Lewis Fry Richardson noted how different countries had different lengths for the same shared border because of differences in scales of measurement. In 1967, mathematician Benoit Mandelbrot expanded on Richardson's work, writing a classic Science paper on the length of Britain's coastline. This later led him to discover and conceptualize the shape of fractals, a curve that has increased complexity the more you zoom in.

This is also related to why no one can agree whether China or the US is the third-largest country by area: it depends how you measure some of the boundaries, including the coasts. Also, vertical differences get thrown in; this is the same fractal problem, only in two dimensions (surface), not one (boundary line).

The concept of "dimensions" also gets modified when you're dealing with fractals; you can get fractional dimensions. Which, it should come as no surprise, gave fractals their name.

I object, however, to the idea of "increased complexity the more you zoom in." I'd argue that you get the same complexity, just at different scales. I guess the sentence can be read like zooming in reveals greater complexity.

The article also features some nice Mandelbrot set zoom animations, which I always find fascinating.

This can hold true for coastlines. You could technically measure a coastline down to the grain of sand or atomic level or smaller, which would mean a coastline's length could be close to infinity, Sammler said.

Another nitpick: no such thing as "close to infinity." In the real world, as opposed to a purely mathematical construct, there's a minimum length (it's very, very small). That minimum length implies an upper bound to how long a fractal boundary can be. It can be very, very long... but that's still not infinity.

Coastlines are also shifting entities. Tides, coastal erosion and sea level rise all contribute to the fluctuating state of coastlines. So maps from the 1900s, or even satellite imagery from a few years ago, may not resemble what coastlines really are today.

And if you want to get really technical, it changes from moment to moment, as portions are eroded and others built up. Not to mention general changes in sea level. As I've noted before, you can't step into the same river once.

So how much coastline does Alaska, the United States, or our entire planet, have? We may never know the accurate number. It's a paradox, and like many things in nature, escapes our ability to define it.

Which shouldn't mean we throw up our hands and give up. I like to think of it as a metaphor for life itself: always approaching an answer, never quite getting there. But learning more and more along the way.

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