This article refers to the R.E.M. live recording. For the mathematical term, see Perfect square.

Perfect Square
Perfect Square cover
Live performance by R.E.M.
Released	March 9 2004
Recorded	Bowling Green Wiesbaden, Germany
Genre	Rock music
Label	Warner Bros.
R.E.M. video chronology
(2003)	Perfect Square (2004)	(2006)

Perfect Square is a video recording of a concert by rock band R.E.M., filmed on 19 July 2003 at the Bowling Green in Wiesbaden, Germany. It was released in DVD format on the Warner Brothers label on March 9, 2004.

This concert is perhaps most notable for a performance of the song "Country Feedback", which Michael Stipe opens by declaring it his "favourite song". The rendition features appended lyrics from another song, Reveal's "Chorus and the Ring", and a guitar solo by Peter Buck, neither of which are in the original. The performance (in audio form) also found its way onto the limited edition issue of the In Time compilation, which was released a few months after the Wiesbaden concert.

Other concert highlights include the appearance of two then-new songs, "Bad Day" and "Animal", and the re-emergence of the long-lost song "Permanent Vacation". "Permanent Vacation" dates back to the band's Murmur days. This DVD remains the song's sole official release.

It is believed that the title Perfect Square is a play on one of R.E.M.'s earlier songs, "Perfect Circle", although that song does not appear on the DVD.

Included on the DVD release is a bonus documentary, A Stirling Performance, chronicling the band's three-night stand at Stirling Castle in Scotland in July 1999.

Audio

• Dolby Digital Stereo
• Dolby Digital 5.1

Track listing

1. 'Begin the Begin' (4:04)
2. 'What's the Frequency, Kenneth?' (4:12)
3. 'Maps and Legends' (3:28)
4. 'Drive' (4:44)
5. 'Animal' (4:22)
6. 'Daysleeper' (3:46)
7. 'The Great Beyond' (4:17)
8. 'Bad Day' (4:24)
9. 'The One I Love' (3:22)
10. 'All the Way to Reno (You're Gonna Be a Star)' (5:00)
11. 'Orange Crush' (3:59)
12. 'Losing My Religion' (4:45)
13. 'At My Most Beautiful' (3:37)
14. 'Electrolite' (4:28)
15. 'She Just Wants to Be' (5:58)
16. 'Walk Unafraid' (5:16)
17. 'Man on the Moon' (5:57)
18. 'Everybody Hurts' (6:26)
19. 'So Fast, So Numb' (4:24)
20. 'Country Feedback' (6:11)
21. 'Permanent Vacation' (2:52)
22. 'Imitation of Life' (3:55)
23. 'It's the End of the World as We Know It (and I Feel Fine)' (5:55)

Personnel

R.E.M.

• Peter Buck - guitar, bass
• Mike Mills - bass, vocals, piano
• Michael Stipe - vocals

Auxiliary musicians

• Scott McCaughey - guitar, keyboard, vocals
• Bill Rieflin - drums, percussion
• Ken Stringfellow - keyboard, banjo, vocals

External links

• Perfect Square at the IMDb

•  • [ e] R.E.M.

Peter Buck Mike Mills Michael Stipe Bill Berry Accompaniment: Scott McCaughey Bill Rieflin Ken Stringfellow Amanda Brown Nathan December Buren Fowler Peter Holsapple Barrett Martin Joey Waronker Management and producers: Bertis Downs, IV Jacknife Lee Joe Boyd Don Dixon Mitch Easter Don Gehman Jefferson Holt Scott Litt Pat McCarthy Discography Studio albums and EPs: Chronic Town Murmur Reckoning Fables of the Reconstruction Lifes Rich Pageant Document Green Out of Time Automatic for the People Monster New Adventures in Hi-Fi Up Reveal Around the Sun Upcoming studio album #14 Singles Compilations: Dead Letter Office Eponymous The Best of R.E.M. ''And I Feel Fine... The Best of the I.R.S. Years 1982-1987 Other albums: Athens Andover Man on the Moon Not Bad for No Tour r.e.m.IX R.E.M. Live Videos: Succumbs Tourfilm Pop Screen This Film Is On Parallel Road Movie Perfect Square Related articles Alternative rock Hindu Love Gods I.R.S. Records Warner Bros. Records Homer the Moe

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The term perfect square is used in mathematics in two meanings:

• an integer which is the square of some other integer, i.e. can be written in the form n² for some integer n (and because of this a square is always nonnegative). Thus a perfect square always has a square root that has no decimal expansion.
• Examples: 0, 1, 4, 9, 16, 25, 36, 49, of which the square roots are 0, 1, 2, 3, 4, 5, 6 and 7 respectively... See square number.
• an algebraic expression that can be factored as the square of some other expression, e.g. a² ± 2ab + b² = (a ± b)² (see square (algebra)).

This is not the same as a magic square.

Using differences of squares as multiplication

Integer multiplication can be done entirely by a difference of two squares.

Examples:

• 
• 
• 
• 

In general, the product of two numbers is equal to the square of their average minus their difference from the average squared.

• 

A geometric constructive "proof" of this relation is shown in the following animation:

The starting rectangle is A by B. The resulting large square is length (A+B)/2, and the smaller gray square (remainder being subtracted) is length |A-B|/2.

Using this relation, you can multiply relatively large numbers more quickly if you memorize a relatively small list of squares.

If you're multiplying an even by an odd, you can avoid "halves" by adjusting one number, requiring one more addition at the end

• 

Example:

• 

Last Digits of Perfect Squares in Base 10

In base 10, the last digit of every perfect square is either 0, 1, 4, 5, 6, or 9. If you know a number is a perfect square then you can determine the last digit of its square root as follows:

• If the last digit of the perfect square is 0, then the last digit of the square root is 0.
• If the last digit of the perfect square is 1, then the last digit of the square root is either 1 or 9.
• If the last digit of the perfect square is 4, then the last digit of the square root is either 2 or 8.
• If the last digit of the perfect square is 5, then the last digit of the square root is 5.
• If the last digit of the perfect square is 6, then the last digit of the square root is either 4 or 6.
• If the last digit of the perfect square is 9, then the last digit of the square root is either 3 or 7.

Determining the Square Root of a Perfect Square More Than or Equal to 100 but Less Than 10000

Suppose you are given a perfect square that is greater than or equal 100 but less than 10,000. All of these perfect squares have two digit square roots. A simple party trick algorithm exists for determining its square root with one or two guesses.

First, separate the digits into pairs of digits starting from the right and moving to the left. For examples, consider the perfect squares 4624, 729, and 1600. Break 4624 into 46|24, or break 729 into 7|29, or 1600 into 16|00.

Second, we find the first digit of the square root. If the leftmost pair is a perfect square, find its square root. If the leftmost pair is not a perfect square, find the square root of the largest perfect square smaller than the leftmost pair. That square root will be the leftmost digit of the square root. For examples, for 46|24, the leftmost pair is 46, which is not a perfect square; but the largest perfect square less than 46 is 36 which has a square root of 6. For 7|29, 7 is not a perfect square, but 4 is the largest square less than 7, and its square root is 2. For 16|00, the square root of 16 is 4.

Third, as mentioned in an earlier section, if we know the last digit of a perfect square in base ten, then we can also know the last digit of its square root in two guesses. Thus, we know the last digit of the square root of 4624 is either 2 or 8; for 729 it is either 3 or 7; for 1600 it is 0.

Fourth, we can guess the square root by conjuncting the second and third steps. The square root of 4624 is either 62 or 68; the square root of 729 is either 23 or 27; the square root of 1600 is 40.

If you use a calculator, you will find that the square roots of 4624, 729, and 1600 are actually 68, 27, and 40.

See also

• Perfect power

External links

• Lists of the first 100, the first 1,000, the first 10,000, the first 100,000, and the first 1,000,000 perfect squares
• JavaScript code for finding ten-digit numbers, consisting of distinct digits, that are perfect squares