Extended real number line

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The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). It is useful in describing various limiting behaviours in calculus and mathematical analysis, especially in the theory of measure and integration. The extended real number line is denoted R or [−∞, +∞].

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

Motivation

Limits

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

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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 1/x gets closer and closer to 0. This limiting behaviour is similar to the limit of a function at a real number, except that there is no "number" to 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

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n(x) = \begin{cases} 2n(1-nx), & \mbox{if } 0 \le x \le \frac{1}{n} \\ 0, & \mbox{if } \frac{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 extended real number line turns into a totally ordered set by defining −∞ ≤ a ≤ +∞ for all a. This order has the nice property that every subset has a supremum and an infimum: it is a complete lattice. The total order induces a topology on R. In this topology, a set U is a neighborhood of +∞ if and only if it contains a set {x : x > a} for some real number a, and analogously for the neighborhoods of −∞. R is a compact Hausdorff space homeomorphic to the unit interval [0, 1].

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 × −∞ = −∞ × a = −∞    if a > 0
  • a × −∞ = −∞ × a = +∞    if a < 0
  • a / ±∞ = 0    if −∞ < a < +∞
  • ±∞ / a = ±∞    if 0 < a < +∞
  • +∞ / a = −∞    if −∞ < a < 0
  • −∞ / a = +∞    if −∞ < a < 0

Here, "a + ∞" means "a + (+∞)" and "a − ∞" means both "a − (+∞)" as well as "a + (−∞)".

The expressions ∞ − ∞, 0 × ±∞ and ±∞ / ±∞ are usually left undefined. These rules are modeled on the laws for infinite limits.

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 neighbourhood 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 ab and if both a + c and b + c are defined, then a + cb + c.
  • if ab and c > 0 and both a × c and b × c are defined, then a × cb × c.

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

Miscellaneous

Several functions can be continuously extended to R by taking limits. For instance, one defines exp(−∞) = 0, exp(+∞) = +∞, ln(0) = −∞, ln(+∞) = +∞ etc.

Compare the real projective line, which does not distinguish between +∞ and −∞.

zh:扩展的实数轴