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

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.

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

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.

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