|
About This Author
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.
October 29, 2007 at 9:49pm October 29, 2007 at 9:49pm
| |
In Virginia, the "Fall Line" is a fairly well-defined boundary between tidal and upland rivers and streams. I grew up spent my childhood very close to the Fall Line, on the tidal side. The town of Falmouth, in the county where I (and George Washington) lived, is derived from this geographical feature: literally, the "fall mouth" of the Rappahannock. The line roughly follows U.S. Route 1 through most of the state.
The historical date of first frost in most of the non-mountainous regions of Virginia is October 15 - that's what they tell you in almanacs and such. At one point, this had real significance, as much of Virgina was given over to agriculture. Now, the only real effect it has on most people is in helping to choose clothes to wear, and preparing those of us too poor, cheap or stubborn to have a garage to scrape frost off our windshields in the morning.
Today, October 29, was the first day I had to scrape frost off my pickup's windshield. There wasn't much of it, and it came away easily, but it reminded me that we're actually in Fall. You wouldn't know it from the temperatures around here lately, which have alternated between sweltering and cool/rainy for most of the month.
In short, it hasn't felt much like October.
I was never a big fan of Fall - or Winter, for that matter; Fall being little more to me than a precursor of Winter. When I was a kid, living on a farm where the date of first frost still had some real meaning, Fall meant two things: going back to a school I didn't much like, and being cold all the damn time.
My dad, who grew up (and in his case, I use the term precisely) during the Great Depression, was a cheap son of a bitch, especially when it came to climate control. Heating was kept to a bare minimum, and consisted primarily of wood stoves in the basement supplemented by oil heat set in the low 60s F. This in and of itself wouldn't be so bad, but Dad cannily kept the thermostat in the hottest part of the house, tricking it into thinking it was a lot warmer in the house than it actually was.
And then at night he'd drop the fucker to the mid-fifties.
My chief memory of childhood winter mornings consists of him trying to wake me up ("Rise and shine!" "Pick one!") and, when that finally succeeded, of me sitting in front of the heater vent, curled up into a little ball as the first blast of mildly warm air resulting from his setting the thermostat back to a balmy 62 shook off my overnight chills.
Dad wasn't poor, mind you. Just cheap. Paid for my college education in full, however, but as a 9 year old kid huddling near a stingy register, I couldn't have appreciated that.
I hated fall. I especially hated winter, except on those occasions when the snows would cancel school and I got to stay home, alternately playing in the snow and draping my exhausted, frozen self over the wood stove in the basement.
When I was much younger, he'd wrap me in a blanket and take me out to look at the stars. He had a sailor's knowledge of the constellations and the twinkling lighthouses in the sky - mainly because he was, after all, a sailor, but partly because he loved knowledge for the sake of knowledge. Never could pronounce Betelgeuse correctly, but could find it in a heartbeat, the orange beacon in the shoulder of Orion.
Orion became, to me, the harbinger of winter. I'd see its familiar asterism shining palely over the river in the early evening, and brace myself for quality time with the heat register in the coming mornings. The Hunter's belt became, to me, the Fall Line.
It will still be a while before I'm as old as my dad was when he took me out on those cold, winter nights to view the stars that guided him and serve me as a rich source of metaphor. But somewhen between then and now, I ceased to love summer and began to appreciate the fall and winter. Partly, I know, this is because I'm not as cheap as my dad was, and we keep our house at a civilized temperature year-round. In part, though, it's because the memory of my first glimpse of the stars is covered with a skin of frost, wrapped in a blanket of warmth and mellowed by a sailor's patient lessons. |
October 29, 2007 at 12:52pm October 29, 2007 at 12:52pm © Copyright 2025 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.
|
I'm more than happy with Carrion Luggage, however.
Comment • View 5 comments 
![Merit Badge in Quill Award
[Click For More Info]
Congratulations on winning Best Blog in the 2021 edition of [Link To Item #quills] !](https://images.Writing.Com/imgs/writing.com/writers/badges/group-Quill_Award-1375927_6.gif)
![Merit Badge in Quill Award 2
[Click For More Info]
2022 Quill Award - Best Blog - [Link To Item #1196512] . Congratulations!!!](https://images.Writing.Com/imgs/writing.com/writers/badges/group-Quill_Award_2-1375927_6.gif)
![Merit Badge in 30DBC Winner
[Click For More Info]
Congratulations on winning the Jan. 2019 [Link To Item #30dbc] !!](https://images.Writing.Com/imgs/writing.com/writers/badges/group-30DBC_Winner-2064535_14.gif)
![Merit Badge in Blogging
[Click For More Info]
Congratulations on winning an honorable mention for Best Blog at the 2018 Quill Awards for [Link To Item #1196512] . *^*Smile*^* This award was sponsored by the blogging consortium including [Link To Item #30dbc] , [Link To Item #blogcity] , [Link To Item #bcof] and [Link To Item #1953629] . For more details, see [Link To Item #quills] .](https://images.Writing.Com/imgs/writing.com/writers/badges/type-Blogging-2180.gif)
![Merit Badge in Highly Recommended
[Click For More Info]
I highly recommend your blog.](https://images.Writing.Com/imgs/writing.com/writers/badges/Kudos-Highly_Recommended.gif)
![Merit Badge in Opinion
[Click For More Info]
For diving into the prompts for Journalistic Intentions- thanks for joining the fun!](https://images.Writing.Com/imgs/writing.com/writers/badges/genre-Opinion-1150.gif)
![Merit Badge in High Five
[Click For More Info]
For your inventive entries in [Link To Item #2213121] ! Thanks for the great read!](https://images.Writing.Com/imgs/writing.com/writers/badges/Kudos-High_Five.gif)
![Merit Badge in Enlightening
[Click For More Info]
For winning 3rd Place in [Link To Item #2213121] . Congratulations!](https://images.Writing.Com/imgs/writing.com/writers/badges/general-Enlightening.gif)
![Merit Badge in Quarks Bar
[Click For More Info]
For your awesome Klingon Bloodwine recipe from [Link to Book Entry #1016079] that deserves to be on the topmost shelf at Quark's.](https://images.Writing.Com/imgs/writing.com/writers/badges/group-Quarks_Bar-2046190_12.gif)
