Extended Real Number Line - Arithmetic Operations

Arithmetic Operations

The arithmetic operations of R can be partially extended to R as follows:

\begin{align} a + \infty = +\infty + a & = +\infty, & a & \neq -\infty \\ a - \infty = -\infty + a & = -\infty, & a & \neq +\infty \\ a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\ a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\ \frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\ \frac{\pm\infty}{a} & = \pm\infty, & a & \in \mathbb{R}^+ \\ \frac{\pm\infty}{a} & = \mp\infty, & a & \in \mathbb{R}^- \end{align}

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

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

The expression 1/0 is not defined either as +∞ or −∞, because although it is true that whenever f(x) → 0 for a continuous function f(x) it must be the case that 1/f(x) is eventually contained in every neighborhood of the set {−∞, +∞}, it is not true that 1/f(x) must tend to one of these points. An example is f(x) = 1/(sin(1/x)). (Its modulus 1/| f(x) |, nevertheless, does approach +∞.)

Read more about this topic:  Extended Real Number Line

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