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🎁

a day 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.

1 Like

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 9, 2024 at 11:19am

#1080964

Analog Log

This one's been hanging out in my queue for a long time, but it's not exactly time-sensitive. As they say, time and tide wait for no one.

Lord Kelvin and His Analog Computer
This tide-predicting machine was one of many advances he made to maritime tech

The source is a publication of the IEEE, the electrical engineering professional organization. But fear not; the article isn't very technical.

Civilizations recognized a relationship between the tides and the moon early on, but it wasn’t until 1687 that Isaac Newton explained how the gravitational forces of the sun and the moon caused them. Nine decades later, the French astronomer and mathematician Pierre-Simon Laplace suggested that the tides could be represented as harmonic oscillations. And a century after that, [William] Thomson used that concept to design the first machine for predicting them.

Thompson was Lord Kelvin and, yes, he's the one the temperature scale is named after. One wonders what it would have been called had Thompson not become a noble, because Thompson is a boring name for a unit of measure.

Thomson’s tide-predicting machine calculated the tide pattern for a given location based on 10 cyclic constituents associated with the periodic motions of the Earth, sun, and moon. (There are actually hundreds of periodic motions associated with these objects, but modern tidal analysis uses only the 37 of them that have the most significant effects.)

Translation: it's complicated.

The most notable one is the lunar semidiurnal, observable in areas that have two high tides and two low tides each day, due to the effects of the moon.

Which is what most of us think of when we think of tides, but it's not as simple as "it's high tide when the moon is directly overhead." There's a lag, and there are local conditions that affect the timing of tides (such as sea floor depth).

On Thomson’s tide-predicting machine, each of 10 components was associated with a specific tidal constituent and had its own gearing to set the amplitude.

Basically, it's a very complicated clock. Sure, the article calls it an analog computer, and I'm not going to argue with professionals (especially ones not in my field) but I think that's a categorization issue. At some point of increasing complexity, a clock stops being a clock and starts being an analog computer. But in my view, if it involves the timing of natural phenomena like the movement of solar system bodies, it's a clock.

The device marked each hour with a small horizontal mark, making a deeper notch each day at noon. Turning the wheel rapidly allowed the user to run a year’s worth of tide readings in about 4 hours.

But this bit, an output device, is probably what pushes it into the computer category. It also was designed for prediction, not for reading what the tide is right now.

As with many inventions, the tide predictor was simultaneously and independently developed elsewhere and continued to be improved by others, as did the science of tide prediction.

One thing I'm still unclear on when it comes to tidal prediction: the wind plays a role. And wind is way, way harder to predict than the future relative position of the sun and moon. I'd ask my dad, the sailor, but I seem to have misplaced my Ouija board.

In any case, mostly, I just liked the article and the history lesson, and I wanted to muse about the differences between clocks and analog computers.

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