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Thursday, Apr. 25, 2024

10:12am EDT

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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

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Show Recent Comments

10 Most Recent Blog Comments:

Kåre Enga in Udon Thani

about 7 hours ago
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Udon Thani

about 7 hours ago
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

Kåre Enga in Udon Thani

about 7 hours ago
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

about 18 hours ago
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

a day ago
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

1 Like

Lazy Writer est 4/24/2008

a day ago
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

1 Like

Aennaytte: Free & Wild in GoT

a day ago
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

1 Like

Aennaytte: Free & Wild in GoT

a day ago
In response to "Cash Stash"

The discussion of money is an interesting one. Remember the recent collapse of a few banks? I think it was about a year or so ago?
Yes, deposits were FDIC insured up to $250,000 per depositor. But some people and companies had more money in that bank than the FDIC was insuring.
I never followed up on what happened to those people. I don't run in circles of people who own that much cash.

My accounting professor took half a period to explain to us what happened and how the bank went from Silicone Valley darling to dead so fast.

He also told us that the value we give money is not backed by anything except our hope and faith that our economy will keep going.

We didn't discuss whether the value of money needed a lower income class of people.

1 Like

Aennaytte: Free & Wild in GoT

a day ago
In response to "Revisited: "Hack This""

In my last psychology class, we discussed the evolution of the brain. It is now settled science that we evolved to have not only larger, but also vastly more powerful brains than most other animals because we learned how to cook (process) our food.
Although we are only about 1/4 to 1/3 of a normal gorilla, we eat half of the calories in a day. Not only that, we prepare the food in a way that we get to use more of the calories that we ingest.
The result is that our brain takes up more space in our noggin than a gorilla brain takes up in his noggin. Not only that, each of our brain cells is smaller, so we can fit exponentially more of those into our head.

Now, using all that goodness ...

1 Like

Aennaytte: Free & Wild in GoT

a day ago
In response to "Hope for Hops"

Wait! Why are beer and coffee affected by climate change? That wasn't supposed to happen. All we had to do was have a boat by the side of the house for when the sea levels rise and beg our property insurance to cover fire damage. How are we supposed to remember all of that if we don't get coffee to wake us up in the morning so that we can think and beer at night to dull our sense of doom?

#1978. Going Bananas
ID #1032870 entered on May 26, 2022 at 12:49am

#1977. Later
ID #1032826 entered on May 25, 2022 at 12:01am

#1976. Sam: I Am
ID #1032786 entered on May 24, 2022 at 12:03am

#1975. Annoying Things
ID #1032742 entered on May 23, 2022 at 12:02am

#1974. Speculations
ID #1032703 entered on May 22, 2022 at 12:10am

#1973. Sermon on the Mound
ID #1032660 entered on May 21, 2022 at 12:02am

#1972. Nerdy Words
ID #1032613 entered on May 20, 2022 at 12:01am

#1971. One Great Thing
ID #1032584 entered on May 19, 2022 at 12:01am

#1970. Little Water
ID #1032550 entered on May 18, 2022 at 12:02am

#1969. Industrious
ID #1032516 entered on May 17, 2022 at 12:05am

#1968. A Total Eclipse of the Moon
ID #1032463 entered on May 16, 2022 at 12:05am

#1967. Waters of Life
ID #1032420 entered on May 15, 2022 at 5:14am

#1966. Any Storm in a Port
ID #1032374 entered on May 14, 2022 at 12:01am

#1965. Hypothetical
ID #1032338 entered on May 13, 2022 at 12:02am

#1964. High-cal Local
ID #1032266 entered on May 12, 2022 at 12:07am

#1963. Doom and Gloom
ID #1032224 entered on May 11, 2022 at 12:01am

#1962. Hell is Empty, and All the Devils Are Here
ID #1032196 entered on May 10, 2022 at 4:53am

#1961. Shooting Tsar
ID #1032143 entered on May 9, 2022 at 12:01am

#1960. Sea What I Did There
ID #1032112 entered on May 8, 2022 at 12:05am

#1959. Dien Bien Phu
ID #1032067 entered on May 7, 2022 at 12:21am

#1958. Bore-dumb
ID #1032024 entered on May 6, 2022 at 12:01am

#1957. A Tour of the Town
ID #1031944 entered on May 5, 2022 at 12:03am

#1956. Oh, Canada...
ID #1031888 entered on May 4, 2022 at 12:01am

#1955. Just One Word: Plastics
ID #1031836 entered on May 3, 2022 at 12:01am

#1954. Charlottesville
ID #1031786 entered on May 2, 2022 at 12:01am

#1953. The Birds
ID #1031744 entered on May 1, 2022 at 12:01am

#1952. Conscious
ID #1031688 entered on April 30, 2022 at 12:01am

#1951. Garbage Patch Kids
ID #1031570 entered on April 29, 2022 at 12:01am

#1950. Influencers Under the Influence
ID #1031514 entered on April 28, 2022 at 12:01am

#1949. Donner's Blitzen
ID #1031448 entered on April 27, 2022 at 12:02am

#1948. Engage Tractor Beam
ID #1031393 entered on April 26, 2022 at 12:01am

#1947. Branding
ID #1031312 entered on April 25, 2022 at 12:02am

#1946. Mothman
ID #1031265 entered on April 24, 2022 at 12:04am

#1945. Swamp Life
ID #1031212 entered on April 23, 2022 at 12:04am

#1944. Unkosher
ID #1031161 entered on April 22, 2022 at 12:07am

#1943. The Forest for the Trees
ID #1031096 entered on April 21, 2022 at 12:01am

#1942. Advice Grip
ID #1031034 entered on April 20, 2022 at 12:02am

#1941. Save Game
ID #1030967 entered on April 19, 2022 at 12:01am

#1940. Legendaries
ID #1030917 entered on April 18, 2022 at 12:01am

#1939. Waisted
ID #1030863 entered on April 17, 2022 at 12:12am

#1938. Nothing to Celebrate
ID #1030814 entered on April 16, 2022 at 12:01am

#1937. Getting Cabbaged
ID #1030742 entered on April 15, 2022 at 12:02am

#1936. How To Write
ID #1030689 entered on April 14, 2022 at 12:01am

#1935. Zoned Out
ID #1030623 entered on April 13, 2022 at 12:02am

#1934. False
ID #1030583 entered on April 12, 2022 at 12:01am

#1933. Asking Persimmon
ID #1030515 entered on April 11, 2022 at 12:02am

#1932. Suffer the Little Children
ID #1030456 entered on April 10, 2022 at 12:14pm

#1931. Chat? Oh.
ID #1030357 entered on April 9, 2022 at 4:33am

#1930. Out of Phrase
ID #1030288 entered on April 8, 2022 at 12:02am

#1929. Your Mama
ID #1030231 entered on April 7, 2022 at 12:01am

#1928. Grow Slow
ID #1030155 entered on April 6, 2022 at 12:02am

#1927. Greenwashing
ID #1030097 entered on April 5, 2022 at 12:13am

#1926. Sleep Like a Baby
ID #1030030 entered on April 4, 2022 at 12:03am

#1925. Plowadise
ID #1029962 entered on April 3, 2022 at 12:02am

#1924. Suspect Motivations
ID #1029896 entered on April 2, 2022 at 12:02am

#1923. Steely Gazelle
ID #1029828 entered on April 1, 2022 at 12:02am

#1922. The Attempted Resurrection of Words
ID #1029770 entered on March 31, 2022 at 12:02am

#1921. Steely Gaze
ID #1029714 entered on March 30, 2022 at 12:02am

#1920. Beam Me Up
ID #1029660 entered on March 29, 2022 at 12:01am

#1919. Just Breathe
ID #1029594 entered on March 28, 2022 at 12:01am

#1918. Unmotivation
ID #1029527 entered on March 27, 2022 at 12:01am

#1917. Ill-Annoying
ID #1029484 entered on March 26, 2022 at 12:01am

#1916. Bite Me
ID #1029446 entered on March 25, 2022 at 12:02am

#1915. What Color is Your Cat?
ID #1029406 entered on March 24, 2022 at 12:01am

#1914. Unbound
ID #1029364 entered on March 23, 2022 at 12:01am

#1913. Unexplained is Not Unexplainable
ID #1029309 entered on March 22, 2022 at 12:02am

#1912. Isn't It Ironic?
ID #1029258 entered on March 21, 2022 at 12:01am

#1911. Luck Out
ID #1029210 entered on March 20, 2022 at 12:02am

#1910. Curious Waltz
ID #1029163 entered on March 19, 2022 at 12:03am

#1909. Poe Dunk
ID #1029114 entered on March 18, 2022 at 12:03am

#1908. We Have Met the Enemy
ID #1029050 entered on March 17, 2022 at 12:01am

#1907. It Was a Dark and Stormy Day
ID #1029004 entered on March 16, 2022 at 12:23am

#1906. Audience
ID #1028954 entered on March 15, 2022 at 12:11am

#1905. Grammar Police
ID #1028892 entered on March 14, 2022 at 12:02am

#1904. It's a Gas
ID #1028825 entered on March 13, 2022 at 12:01am

#1903. The Myths of Cheeses
ID #1028760 entered on March 12, 2022 at 12:02am

#1902. One of the Firsts
ID #1028715 entered on March 11, 2022 at 12:01am

#1901. Respair!
ID #1028667 entered on March 10, 2022 at 12:02am

#1900. Intolerable
ID #1028604 entered on March 9, 2022 at 12:02am

#1899. Real Men Don't Worry About What a Real Man Is
ID #1028548 entered on March 8, 2022 at 12:01am

#1898. Excessive
ID #1028470 entered on March 7, 2022 at 12:01am

#1897. Dedicated Indoorsman
ID #1028405 entered on March 6, 2022 at 12:01am

#1896. Unthinkable
ID #1028346 entered on March 5, 2022 at 12:04am

#1895. Back East
ID #1028252 entered on March 4, 2022 at 12:10pm

#1894. Goodnight L.A.
ID #1027926 entered on March 3, 2022 at 3:47am

#1893. Island Hopping
ID #1027749 entered on March 2, 2022 at 12:04am

#1892. No Fault of Mine
ID #1027699 entered on March 1, 2022 at 8:57am

#1891. Let Us Dance in Style, Let Us Dance for a While
ID #1027582 entered on February 28, 2022 at 10:06am

#1890. Escape to L.A.
ID #1027530 entered on February 27, 2022 at 9:36am

#1889. Degeneration
ID #1027449 entered on February 26, 2022 at 12:02am

#1888. Hot Tip
ID #1027390 entered on February 25, 2022 at 12:10am

#1887. Running for Precedent
ID #1027320 entered on February 24, 2022 at 12:01am

#1886. Inflate!
ID #1027271 entered on February 23, 2022 at 12:01am

#1885. It Snuck Up On Me
ID #1027192 entered on February 22, 2022 at 12:02am

#1884. Movie Knight
ID #1027122 entered on February 21, 2022 at 12:03am

#1883. I'm Game, Said the Deer
ID #1027069 entered on February 20, 2022 at 12:02am

#1882. Being Catty
ID #1027019 entered on February 19, 2022 at 7:10am

#1881. The Light
ID #1026940 entered on February 18, 2022 at 12:03am

#1880. Over the Rainbow
ID #1026872 entered on February 17, 2022 at 7:37am

#1879. Putting the Product in Productivity
ID #1026805 entered on February 16, 2022 at 9:17am

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