Complex Numbers (Book)

Blog Calendar

◄ December ► 2020
S	M	T	W	T	F	S
		1	2	3	4	5
6	7	8	9	10	11	12
13	14	15	16	17	18	19
20	21	22	23	24	25	26
27	28	29	30	31
Archive				RSS

About This Author

Contact Me

Come closer.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

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.

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.

1 Like

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.

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

1 Like

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.

2 Like

December 22, 2020 at 12:11am

#1000643

Tunnels Aren't Boring, or Are They?

History is cool and never really boring, unless you approach it like my teachers did in school, with endless memorization of dates that shit happened, without any context. I think it takes some experience and lifelong learning to be able to have a syncretic approach to historical events.

"30-Day Blogging Challenge ON HIATUS" [13+]:
22. Today in History
Look up some events that happened on December 22nd,
select one, and tell us about it.

That is, with the possible exception of December 22nd, which, while Wikipedia lists a whole slew of events that took place

on this day in history, it turns out that nearly every single one of them is yawn-inducing.

For example:

1920 – The GOELRO economic development plan is adopted by the 8th Congress of Soviets of the Russian SFSR.

Who did what to which now?

1965 – In the United Kingdom, a 70 mph speed limit is applied to all rural roads including motorways for the first time.

And so? A dozen years later, the US adopted a 55 mph speed limit, and our roads are straighter, wider, and feature driving on the correct side.

1891 – Asteroid 323 Brucia becomes the first asteroid discovered using photography.

I mean, two of the few subjects that interest me more than history are photography and astronomy, and this is still meh.

1807 – The Embargo Act, forbidding trade with all foreign countries, is passed by the U.S. Congress, at the urging of President Thomas Jefferson.

If there is anything more boring than international trade negotiations, I haven't discovered it.

It is true that, per the link above, some cool shit happened too, like the reopening of the Brandenburg Gate in Berlin, but there really isn't much there for me to want to write about. Even the births and deaths lists are largely boring. Also, can the dates really be trusted before the adoption of the Gregorian calendar?

But if I had to pick one -- and I don't, really, but it's only polite to address the prompt -- I'm going to go with one related to civil engineering.

1937 – The Lincoln Tunnel opens to traffic in New York City.

...which led to a traffic jam that persists to this day.

The Lincoln Tunnel is an approximately 1.5-mile-long (2.4 km) tunnel under the Hudson River, connecting Weehawken, New Jersey, to the west with Midtown Manhattan in New York City to the east.

Can I just point out how much sheer fun it is to say "Weehawken?" I do that every time I go through the Lincoln Tunnel, which may help to explain why I'm single.

Turns out that the 1937 date is just for one of the current three tubes of the tunnel:

Construction of the central tube, which originally lacked sufficient funding due to the Great Depression, started in 1934 and it opened in 1937. The northern tube started construction in 1936, was delayed due to World War II-related material shortages, and opened in 1945. Although the original plans for the Lincoln Tunnel called for two tubes, a third tube to the south of the existing tunnels was planned in 1950 due to high traffic demand on the other two tubes. The third tube started construction in 1954, with the delay attributed to disputes over tunnel approaches, and opened in 1957.

I would have hated to be the guy supervising the construction of the other tubes while the first one was in operation. "Now, don't screw this up or you'll flood the tunnel."

The tolls on each crossing are only collected in the New York-bound direction.

That's because if you charged people to go into New Jersey, no one would go into New Jersey.

As of 2016, both directions of the tunnel carry a combined average of 112,995 vehicular crossings every day.

Mostly at the exact same time I'm trying to cross.

The article goes on to describe the tunnel's specifications and the negotiations leading up to its construction, which I find fascinating but everyone else would probably feel about it the way I feel about trade negotiations, so I'll skip most of the commentary on it.

Officials from the federal, state, and city levels were in attendance at the ceremony on the New York side, where New York City Mayor Fiorello H. La Guardia and New Jersey Governor A. Harry Moore wielded picks to dig up the ceremonial first mound of dirt.

Personally I think it would have been better if the governors had done most of the work themselves. You know, give them something to do besides trade negotiations.

The first tube was formally dedicated on December 21, 1937. The opening ceremony was accompanied by a military parade on the New Jersey side, as well as the detonation of a series of aerial bombs launched from military ships.

'Muricans gonna 'Murica.

The first vehicles began passing through the tube at 4:00 AM the next day.

Look, I know the construction crews most likely drove vehicles through the tunnel before that, but still, imagine being the first family through a new tunnel bore. "Hey, look, Mildred. Is that... a crack? Leaking water?"

Due to the limited capacity of the new tube, heavy trucks were temporarily banned, and a minimum speed limit of 20 miles per hour (32 km/h) was imposed until a second tube could be completed.

The current minimum speed is 0 mph and the max is about 3. Hell, Google usually routes me to the GWB instead. At least that way I get a view of something besides 80-year-old tunnel walls.

5 Like • Comment • View 4 comments

© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.