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Rational number

In mathematics, a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number; the set of all rational numbers referred to as "the rationals", the field of rationals or the field of rational numbers is denoted by a boldface Q. The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number; these statements hold true not just for base 10, but for any other integer base. A real number, not rational is called irrational. Irrational numbers include √2, π, e, φ; the decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, the set of real numbers is uncountable all real numbers are irrational. Rational numbers can be formally defined as equivalence classes of pairs of integers such that q ≠ 0, for the equivalence relation defined by ~ if, only if p1q2 = p2q1.

With this formal definition, the fraction p/q becomes the standard notation for the equivalence class of. Rational numbers together with addition and multiplication form a field which contains the integers and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of Q are called algebraic number fields, the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the rational numbers form a dense subset of the real numbers; the real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals. The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, "rational" is used as a noun abbreviating "rational number"; the adjective rational sometimes means. For example, a rational point is a point with rational coordinates.

However, a rational curve is not a curve defined over the rationals, but a curve which can be parameterized by rational functions. Every rational number may be expressed in a unique way as an irreducible fraction a/b, where a and b are coprime integers, b > 0. This is called the canonical form. Starting from a rational number a/b, its canonical form may be obtained by dividing a and b by their greatest common divisor, and, if b < 0, changing the sign of the resulting numerator and denominator. Any integer n can be expressed as the rational number n/1, its canonical form as a rational number. A b = c d if and only if a d = b c. If both fractions are in canonical form a b = c d if and only if a = c and b = d If both denominators are positive, and, in particular, if both fractions are in canonical form, a b < c d if and only if a d < b c. If either denominator is negative, each fraction with a negative denominator must first be converted into an equivalent form with a positive denominator by changing the signs of both its numerator and denominator.

Two fractions are added as follows: a b + c d = a d + b c b d. If both fractions are in canonical form, the result is in canonical form if and only if b and d are coprime integers. A b − c d b c b d. If both fractions are in canonical form, the result is in canonical form if and only if b and d are coprime integers; the rule for multiplication is: a b ⋅ c d = a c b d. If both fractions are in canonical form, the result may be a reducible fraction; every rational number a/b has an additive inverse called its opposite, − ( a

O2

O2, O-2, o2, O2, or O2 may refer to: O2, the common allotrope of the chemical element oxygen O2, an EEG electrode site according to the 10–20 system SGI O2, a Unix workstation computer UOC O2, institutional repository of the Open University of Catalonia The O2, an entertainment district in London, England The O2 Arena, the arena within The O2 O2 Arena or Sazka Arena, an arena in Prague, Czech Republic O2 Centre, an indoor shopping and entertainment centre on Finchley Road, England O2 Residence, a part of the Jumeirah Lake Towers in Dubai, United Arab Emirates O2 World, an indoor arena in Berlin, Germany O2 World, an indoor arena in Hamburg, Germany Otoyol 2, a motorway in Turkey called "O2" The O2, the former name of 3Arena O2 album of the rock band FireHouse O2 album by American boy band O-Town O2 album by Gospel singer Tonéx O2: Avalon Remixed, a 2002 album by Avalon O2, 2006 album by Son of Dave "O2", a song by Orange Range from Panic Fancy "O2", a 2002 song by Sleater-Kinney from One Beat Ö2, a radio service for Austria and South Tyrol O2, a character in Kirby 64: The Crystal Shards O2, trading name for Telefónica Europe, a European telecommunications provider O2 Store, a chain of retail stores owned and operated by Telefónica Europe, with regional subsidiaries: O2, Telefónica mobile network global brand name O2, merged into Three Ireland O2 O2 Czech Republic Telefónica Germany Telefónica Slovakia O2 Academy, a chain of music venues O2 Global Network, an international network for sustainable design O2 Wireless USA or H2O Wireless, a prepaid wireless service by Locus Telecommunications O2TV, a Russian independent political TV channel Cessna O-2 Skymaster, a military twin-engine light aircraft Douglas O-2, a military single-engine observation biplane O 2-class submarine, a class of submarines of the Royal Netherlands Navy Oldershaw O-2, a glider SP&S Class O-2, a 1910 steam locomotives class USS O-2, a 1918 United States O class submarine O-2, a pay grade in the US uniformed services First Lieutenant in the Army, Air Force, Marine Corps Lieutenant in the Navy, Coast Guard, Public Health Service Commissioned Corps, NOAA Commissioned Officer Corps 02 Oz Oxide ion, chemical formula O2− Superoxide, chemical formula O−2 Dioxygenyl, O+2 2O 2Q Q2

Surfin' on a Backbeat

Surfin' on a Backbeat is the third studio album by German pop singer Sasha, released by Warner Music on October 29, 2001 in German-speaking Europe. Taking Sasha's work further into pop rock, Surfin' on a Backbeat reached the top ten of the albums charts in Germany, but charted lower than its predecessors, it was however, certified gold by the German leg of the IFPI for more than 200,000 sold copies. The album and its re-release produced four singles, including leading single "Here She Comes Again" and "This Is My Time," the 2002 FIFA World Cup television hymn. A reissue of the album was released on July 22, 2002, containing two unreleased tracks. "Rooftop" – 4:09 "Blown Away" – 3:21 "Turn It Into Something Special" – 4:41 "One Look in Your Eyes" – 4:13 "Here She Comes Again" – 3:55 "Everybody Loves You" – 4:31 "On And On" – 4:02 "Let's Get Closer" – 3:17 "Just a Second Away" – 4:14 "Drive My Car" – 3:45 "Days Like These" – 3:54 "Why Does Everybody Hurt" – 3:35 "This Is My Time" – 3:37 "Rooftop" – 4:09 "Blown Away" – 3:21 "Turn It Into Something Special" – 4:41 "One Look in Your Eyes" – 4:13 "Here She Comes Again" – 3:55 "Everybody Loves You" – 4:31 "On And On" – 4:02 "Let's Get Closer" – 3:17 "Just a Second Away" – 4:14 "Drive My Car" – 3:45 "Days Like These" – 3:54 "Why Does Everybody Hurt" – 3:35 "Through the Barricades" – 6:16 Official website