Affinely Extended Real Number System Website Links For
Extended 		 

Information About

Affinely Extended Real Number System
  CATEGORIES ABOUT EXTENDED REAL NUMBER LINE
   
real numbers
   
calculus
   
mathematical analysis
   
infinity
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



When the meaning is clear from context, the symbol +∞ is often written simply as ∞.


MOTIVATION


Limits


We often wish to describe the behavior of a function ''f''(''x''), as either the argument ''x'' or the function value ''f''(''x'') gets "very big" in some sense. For example, consider the function

:f(x) = rac{1}{x^2}

The graph of this function has a horizontal Asymptote of ''y'' = 0. Geometrically, as we move farther and farther to the right down the ''x''-axis, the value of '' rac{1}{x^2}'' gets closer and closer to 0. This limiting behavior is similar to the limit of a function at a Real Number , except that there is no real number which ''x'' is approaching.

By adjoining the element +∞ to R, we allow ourselves to formulate a definition of such a "limit at infinity" which is Topologically identical to the usual definition at a real number.


Measure and integration


In Measure Theory , it is often useful to allow sets which have infinite measure and integrals whose value may be infinite.

Such measures arise naturally out of calculus. For example, if we are to assign a Measure to R that agrees with the usual length of intervals, this measure must be larger than any finite real number. Also, when considering infinite integrals, such as

:\int_1^{\infty} rac{dx}{x}

the value "infinity" arises. Finally, we often wish to consider the limit of a sequence of functions, such as

:f_n(x) = \begin{cases} 2n(1-nx), & \mbox{if } 0 \le x \le rac{1}{n} \ 0, & \mbox{if } rac{1}{n} < x \le 1\end{cases}

Without allowing functions to take on infinite values, such essential results as the Monotone Convergence Theorem and the Dominated Convergence Theorem would not make sense.


ORDER AND TOPOLOGICAL PROPERTIES


The affinely extended real number system turns into a Hausdorff Space Homeomorphic to the Unit Interval 1 . Thus the topology is Metrizable corresponding (for a given homeomorphism) to the ordinary metric on this interval. There is no metric which is an extension of the ordinary metric on R.

With this topology the specially defined Limits for ''x'' tending to +∞ and -∞, and the specially defined concept of a limit being +∞ and -∞, reduce to the general topological definition of limit.


ARITHMETIC OPERATIONS


The arithmetic operations of R can be partially extended to
R as follows:
  • ''a'' + ∞ = +∞ + ''a'' = +∞    if ''a'' ≠ −∞

  • ''a'' − ∞ = −∞ + ''a'' = −∞    if ''a'' ≠ +∞

  • ''a'' × ±∞ = ±∞ × ''a'' = ±∞    if ''a'' > 0

  • ''a'' × ±∞ = ±∞ × ''a'' = ∓∞    if ''a'' < 0

  • ''a'' / ±∞ = 0    if −∞ < ''a'' < +∞

  • ±∞ / ''a'' = ±∞    if 0 < ''a'' < +∞

  • ±∞ / ''a'' = ∓∞    if −∞ < ''a'' < 0


Here, "''a'' + ∞" means both "''a'' + (+∞)" and "''a'' - (−∞)", and "''a'' − ∞" means both "''a'' − (+∞)" and "''a'' + (−∞)".

The expressions ∞ − ∞, 0 × ±∞ and ±∞ / ±∞ are usually left undefined. These rules are modeled on the laws for . However, in the context of probability or measure theory, 0 × ±∞ is usually defined as 0.

Note that 1 / 0 is not defined as either +∞ or −∞, because although it is true that whenever ''f''(''x'') → 0 for a Continuous Function ''f''(''x''), we must have that 1/''f''(''x'') is eventually in every Neighborhood of the set {−∞, +∞}, it is ''not'' true that 1/''f''(''x'') must converge to one of these points. An example is ''f''(''x'') = 1/(sin(1/''x'')).


ALGEBRAIC PROPERTIES


Note that with these definitions, R is '''not''' a Field and not even a Ring . However, it still has several convenient properties:
  • ''a'' + (''b'' + ''c'') and (''a'' + ''b'') + ''c'' are either equal or both undefined.

  • ''a'' + ''b'' and ''b'' + ''a'' are either equal or both undefined.

  • ''a'' × (''b'' × ''c'') and (''a'' × ''b'') × ''c'' are either equal or both undefined.

  • ''a'' × ''b'' and ''b'' × ''a'' are either equal or both undefined

  • ''a'' × (''b'' + ''c'') and (''a'' × ''b'') + (''a'' × ''c'') are equal if both are defined.

  • if ''a'' ≤ ''b'' and if both ''a'' + ''c'' and ''b'' + ''c'' are defined, then ''a'' + ''c'' ≤ ''b'' + ''c''.

  • if ''a'' ≤ ''b'' and ''c'' > 0 and both ''a'' × ''c'' and ''b'' × ''c'' are defined, then ''a'' × ''c'' ≤ ''b'' × ''c''.


In general, all laws of arithmetic are valid in R as long as all occurring expressions are defined.


MISCELLANEOUS


Several (−∞) = 0, exp(+∞) = +∞, Ln (0) = −∞, ln(+∞) = +∞ etc.

Also, some discontinuities can be removed. For example, the function x^{-2} can be made continuous by setting the value to +∞ for ''x'' = 0, and 0 for ''x'' = +∞ and ''x'' = -∞. The function x^{-1} can ''not'' be made continuous (because the function approaches -∞ as x approaches 0 from below, and +∞ as x approaches 0 from above).

Compare the Real Projective Line , which does not distinguish between +∞ and −∞. As a result, on one hand a function may have limit ∞ on the real projective line, while in the affinely extended real number system only the absolute value of the function has a limit, e.g. in the case of the function x^{-1} at ''x'' = 0. On the other hand \lim_{x o -\infty}{f(x)} and \lim_{x o +\infty}{f(x)} correspond on the real projective line to only a limit from the right and one from the left, respectively, with the full limit only existing when the two are equal. Thus e''x'' and arctan (''x'') cannot be made continuous at ''x''= ∞ on the real projective line.


REFERENCES