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(thing) by quantumet (3.2 y) (print)   ?   (I like it!) 1 C! Sat Mar 11 2000 at 16:10:33

0! = 1
1! = 1
1 * 1 = 1
1 + 0 = 1
1^n = 1 for any n, even complex
n/n = 1 for any n, even complex, except for 0.
n^0 = 1 for any n, even complex, except for 0.
-e^(pi*i) = 1
sqrt(sqrt(sqrt.... ( n ) ... ) ) = 1 for any n, even complex ones, except for 0

(thing) by pealco (8.5 mon) (print)   ?   (I like it!) Mon Aug 14 2000 at 20:00:56

The number one, also called "unity" is the first positive integer. It is an odd number. Although the number 1 used to be considered a prime number, it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. The number 1 is sometimes also called "unity," so the nth roots of 1 are often called the nth roots of unity. Fractions having 1 as a numerator are called unit fractions. If only one root, solution, etc., exists to a given problem, the solution is called unique.

(thing) by Adam Walker (1.5 mon) (print)   ?   (I like it!) 1 C! Wed Nov 15 2000 at 23:43:42

After the release of The Beatles Anthologies (1, 2 and 3), as well as the Anthology miniseries in 1995/96, it seemed like the Beatles had completely exhausted their supply of material for release. Not so. With the release of 1 in 2000, The Beatles topped the charts again, with an album made up of all their previous number one singles. Spanning eight years, between 1962 and 1970, this album shows the Beatles at their popular best. From the tight pop sound of Love Me Do and Can't Buy Me Love, to the experimental sound of Eleanor Rigby and Penny Lane, to the straight forward rock and roll of Get Back, this is a chronicle of the most famous band in the world in each stage of its life.

Because of the global fan base, it was released simultaneously in North America and Europe in November 2000 and became the #1 Album in 35 different countries (as recorded in the Guiness Book of World Records).

I went and bought this album a few days after it came out. Already a fan of the Beatles, the endurance of these songs impressed me the most; they're all still really, really good. The production of George Martin and talent of Lennon, McCartney and Harrison has never been more evident than with all these songs stacked back to back. Even the "Spectorized" Long and Winding Road fits here as the finale, completing the album with a great cascade of sound.

Below is the track list, with the duration of each single's stay on the chart. (This is from the booklet that came with the album). Track list:
Love Me Do
US: 1 week
From Me to You
UK: 7 weeks
She Loves You
UK: 7 weeks (total) US: 2 weeks
I Want To Hold Your Hand
UK: 7 weeks US: 5 weeks
Can't Buy Me Love
UK: 3 weeks US: 5 weeks
A Hard Day's Night
UK: 3 Weeks US: 2 weeks
I Feel Fine
UK: 5 weeks US: 1 Week
Eight Days A Week
US: 2 weeks
Ticket To Ride
UK: 3 weeks US: 1 week
Help!
UK: 3 weeks US: 3 weeks
Yesterday
US: 4 weeks
Day Tripper
UK: 5 weeks
We Can Work It Out
UK: 5 weeks US: 3 weeks
Paperback Writer
UK: 2 weeks US: 2 weeks
Yellow Submarine
UK: 4 weeks
Eleanor Rigby
UK: 4 weeks
Penny Lane
US: 1 week
All You Need Is Love
UK: 3 weeks US: 1 week
Hello, Goodbye
UK: 7 weeks US: 3 weeks
Lady Madonna
UK: 2 weeks
Hey Jude
UK: 2 weeks US: 9 weeks
Get Back
UK: 6 weeks US: 5 weeks
The Ballad of John and Yoko
UK: 3 weeks
Something
UK: 1 week
Come Together
US: 1 week
Let It Be
US: 2 weeks
The Long and Winding Road
US: 2 weeks

Originally posted Nov. 15 2000, buffed up Jan. 30 2007


(idea) by irexe (1.3 y) (print)   ?   (I like it!) Sun Dec 01 2002 at 15:23:10

Also known as 0.999999........

It is a little known, yet very important fact that the Arab numeral system, which is universally used nowadays, is not unambiguous. Any finite decimal fraction can be represented by an infinite decimal fraction by decrementing the last digit and adding a trail of 9's. Thus, the number 0.9999... is the same number as the number 1. This also holds for numbers like 0.4, which can be represented by 0.39999999...

Note that this is not some kind of limiting property! It is not so that the number 0.99999... 'approaches' 1. It is, most definitely the same number. In some mathematical proofs, like the proof that the set of real numbers is uncountable, special measures have to be taken to eliminate this ambiguity. That is, the existence of both 0.99999.. and 1 seems to break the proof, but since both are identical, it does not.

Casual readers may stop here. Pedants read on..

It has been argued by some that this is in fact a limiting property. To see why this is utterly nonsensical, let us review what the definition of a limit is:

Let f be a function from R to R. If, for some a there exists an L such that for every ε > 0 there exists a δ such that:

|x - a| < δ ⇒ |f(x - a) - L| < ε
..then L is defined as the limit of f in a.

From this definition it should be unambiguously clear that the concept of a limit applies to functions and not to constants. You could argue that a number is a constant function, but then you would have your definitions seriously mixed up. To illustrate that last point: even if we do not define the concept of 'function' at all, 0.99999... is still 1. And, since the definition of a limit requires the definition of the function concept, this implies that the limiting construct need not be defined either for 0.99999.. to be 1 QED.

I'll Fitch it for you if you want! ;-)

Note: This fact is not a theorem! It is just a commonly accepted sloppy notation of "..." meaning ad infinitum and, worse, that a series like this is actually equivalent to a number. Because of this there is no 'proof' and so this and the 'proofs' in that are exercises in futility.

..if you refuse to understand this, I cannot help you. Stop msging me. You are actually wrong and not because I am being pedantic. Please read a book before you downvote me.

(fiction) by stainedglass (14.1 hr) (print)   ?   (I like it!) 1 C! Wed Jul 23 2008 at 18:47:49

The tracework of thin black lead sailed in and out of a sea of colour. It was a pure colour, virgin and untainted, that was the only way to describe it. There, in its depths, washed unutterably beautiful crystal-shard sparklings, born of a single split ray of sunlight. It took Peg's breath away, quite unexpectedly.

All her life, she'd imagined that to be an exaggeration. 'It took my breath away', people would say to her, and she'd nod in response. Up until now it had meant little more to Peg than when somone couldn't find the right words and blurted 'really, really' into the middle of a sentence. She'd done it herself, wishing words like 'exquisite' and 'draped' existed comfortably in her head outside of Jackie Collins novels. Up until now, that is. For now it meant this, and it was perfect.

'It is, it's perfect,' she breathed to herself, a hand creeping to her mouth. Perfect meant glimmering pools of colour, rippling like stones had skimmed a still body of petrol-slicked water. Layers of detail rushed at her in overwhelming ranks then retreated to form a backdrop for the next phalanx of components, bursting on the surface of the stained glass lake and dragging her in with siren calls of translucence.

'Beautiful,' she said. 'Truly beautiful.'

But this beautiful translucency is misleading. Its beauty hides a single, near ironic secret, for it's hard to believe that the grit that fills shoes and beats endlessly against the cliffs, such commonplace oxidised silicon - the endless tons that cover the sea-bed - can be changed so utterly merely by heating to an appropriate temperature, perhaps adding sodium to temper the ferocity of this newly-made, brittle crystal. But that's all it is. The grit in the oyster, the scratch in the corner of the eye. Peg knew all this - she'd spent many an afternoon on Brighton beach when the kids were still kids, chewing grimly on sand sandwiches in the shelter of a hired windbreak, washing it down with luke-warm tea from a twenty-five pence deposit mug.

It's all just sand, she knew that, and yet despite this she allowed the beauty of the translucency to lure her, to draw her in. She reached out, and in an instant's slow-ticked space she found her fingers plunged into the liquid depths of a single plane. Not merely stroking the surface, Peg saw, to her amazement, that her fingers intruded into some new, crystalline space. The sensation, or rather the detached knowedge that her hand was somehow inthe glass, was surreal. Her fingers were immersed in ice and she gazed at them and gave them an experimental flex, entranced by the ripples that moved away from her fingers and circled into the tight, lead-enclosed space. They even reflected back from the lead tracework, she noticed, as though the deep cobalt were nothing more than a perfect quadrilateral of thick, mulish water. It was only when her whole hand began to feel like ice, as though it were being eaten alive by the cold and hungry glass, that she withdrew her hand and put the fingers in her mouth.

She felt the experience settling down inside her. It left her feeling giddy, and she whirled round like an excited child, ostensibly to survey the rest of the chamber. She felt strange. Full, and yet empty. Hanging onto the edge of something, like this point in her life were a cliff and she were halfway over the edge. She could clamber back up onto the clifftop, or allow herself to drop down into the depths. Which way was right, she wondered, which way was she trying to go?

Most of the time Peg felt such a small, small part of the world. Dipping a finger into a scrap of glass probably wasn't big-time news on the universal scale of things, she was sure, but then her finger did feel like it had delved into the secret place of something precious. Even if all she had to show for it was a slight feeling of cold, perhaps even a little residual tingling, on her own, insignificant scale of things it felt like pretty big news. She wondered if she could ever tell Alan. Probably not. She could see it in her head, how he'd be when she got home that night, slopped before the screen like spilt oil, clad only in his pants and a thin layer of stale sweat from the heat, his belly spilling out over the top of his waistband and his hand jammed down the front, cupping his balls, perhaps for comfort or maybe just that he might scratch himself in accompaniment to his favourite programmes with minimal effort.

No, she could never tell him - miracles became an everyday mundanity in the face of such pale and naked flesh; if she ever tried to tell him there'd be a moment of scorn that crossed his face and a comment. His comments confused her; they were often so cutting that she could never quite be sure that he'd meant it that way, because surely no-one could be so unforgivably mordant one minute and so compassionate the next. He was capable of sensitivity, she knew, expressing understanding way beyond that which his appearance might suggest she should expect. It didn't happen often, but it happened often enough to keep her with him. On occasion, though, she still remembered the day they married and somehow regretted it all a little.

Shaking her head, ever so slightly, she caught fields of cerulean blue and her head, drawn upwards, began to nod in accordance. Peg smiled, despite herself. She'd find someone to tell, she simply had to. And maybe, you never know, maybe Alan would catch something in her smile, something special... something magical. Perhaps it would spill over into him and it'd all be like the wedding, back before his gut got the better of his romance. You never knew...

Part 2 to follow


printable version
chaos

2 1, 11, 21, 1211 Statements that would be shocking to people living in 1975 One equals point nine repeating
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1 is not prime I Random number node surfing Collatz conjecture
Snuffle spiffy It's better to be solitary than with incompatible people Hell
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