Projectively extended real line

In real analysis, the projectively extended real line (also called the one-point compactification of the real line, or simply real projective line), is the extension of the number line by a point denoted ∞. It is thus the set (where is the set of the real numbers), sometimes denoted by The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line. More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded.

The projectively extended real line may be identified with the projective line over the reals in which three specific points (e.g. 0, 1 and ∞) have been chosen. The projectively extended real line must not be confused with the extended real number line, in which +∞ and −∞ are distinct.

Dividing by zero

Unlike most mathematical models of the intuitive concept of 'number', this structure allows division by zero:

for nonzero a. In particular 1/0 = ∞, and moreover 1/∞ = 0, making reciprocal, 1/x, a total function in this structure. The structure, however, is not a field, and division does not retain its original algebraic meaning in it, as witnessed for example by 0⋅∞ being undefined despite reciprocation being total. The geometric interpretation is this: a vertical line has infinite slope.

RP-1

RP-1 (alternately, Rocket Propellant-1 or Refined Petroleum-1) is a highly refined form of kerosene outwardly similar to jet fuel, used as rocket fuel. Although having a lower specific impulse than liquid hydrogen (LH₂), RP-1 is cheaper, stable at room temperature, far less of an explosion hazard and far denser. RP-1 is significantly more powerful than LH₂ by volume. RP-1 also has a fraction of the toxicity and carcinogenic hazards of hydrazine, another room-temperature liquid fuel. Thus, kerosene fuels are more practical for many uses.

Usage and history

RP-1 is most commonly burned with LOX (liquid oxygen) as the oxidizer, though other oxidizers have also been used. RP-1 is a fuel in the first-stage boosters of the Soyuz-FG, Zenit, Delta I-III, Atlas, Falcon 9, Antares and Tronador II rockets. It also powered the first stages of the Energia, Titan I, Saturn I and IB, and Saturn V. ISRO is also developing a RP-1 fueled engine for its future rockets.

During and immediately after World War II, alcohols (primarily ethanol, occasionally methanol) were the single most common fuel for large liquid-fueled rockets. Its high heat of vaporization kept regeneratively cooled engines from melting, especially considering that alcohols would typically contain several percent water. However, it was recognized that hydrocarbon fuels would increase engine efficiency, due to a slightly higher density, the lack of an oxygen atom in the fuel molecule, and negligible water content. Whatever hydrocarbon was chosen, though, would have to replicate alcohol's coolant ability.

RP 1

RP 1, RP-1, RP.1, RP1, or variant may refer to:

See also