Solution for indeterminate forms using interval numbers

by Norbert Nopper

Credits

Thanks to Eric Lengyel for asking the initial right questions.

Thanks a lot, to my family and their patience having me as a 🤓.

Note

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For properly displaying the formulas, please use another editor like Visual Studio Code.

Motivation

The expressions $0 \cdot ∞$ and implicitly $0 \cdot -∞$ are indeterminate forms [1]. It is not possible to do any calculus on these expressions like $-1 \cdot (0 \cdot -∞)$.
However, compared to $x = -1 \cdot -x$, intuitively the following equation should be true:

$$0 \cdot ∞ = -1 \cdot (0 \cdot -∞)$$

Problem

A naive approach like $-1 \cdot (0 \cdot -∞) = (-1 \cdot 0) \cdot -∞$ with math limits results again in $0 \cdot -∞$ even when the associative law is allowed.

Investigation

Existing required math limits for above assumption:

$\lim_{n\to\infty}\sqrt[n]{n}=1$
$\lim_{n\to\infty}\frac{1}{n}=0$

Assume the usage of the associative law is allowed:

$\lim_{n\to\infty}-\sqrt[n]{n} \cdot \lim_{n\to\infty}\frac{1}{n} = \lim_{n\to\infty}-\sqrt[n]{n} \cdot \frac{1}{n} = \lim_{n\to\infty}\frac{-1}{n}$

The result is again 0 and does not provide the expected result.

New interval number

Assumption

In the case of the given two indeterminate forms, the result could be any number. However, the resulting number can be enclosed in an interval.

Using extended real number

Using the extended real number system $\overline {\mathbb R}$ [2], the intervals $[0, ∞]$ and $[-∞, 0]$ are allowed.

Enclosing the result in an interval

Following is given:

$a, b \in \overline {\mathbb R} \land a \le b$

Interval, where the result for $a \cdot b$ is enclosed in the interval:

$[x_0, x_1] = { x_0 = a \cdot \beta | \beta \in [a, b[ \land x_1 = \alpha \cdot b | \alpha \in ]a, b] }$

Definition

As the result is probably not all numbers in the interval, any or at least one number in the interval must be expressed as a new interval number in:

$[x_0, x_1]in := { x \in \overline {\mathbb R} | \exists x \in [x_0, x_1] }$

The indeterminate form of $0 \cdot ∞$ can be expressed as the first rule:

Rule I
$0 \cdot ∞ = [0, ∞]in$

Similar, the expression for the indeterminate form of $0 \cdot -∞$ is the second rule:

Rule II
$0 \cdot -∞ = [-∞, 0]in$

Interval number operation

These is the given mathematical operation.

Multiplication

$[x_0, x_1]in \cdot [y_0, y_1]in := [\min(x_0 \cdot y_0, x_1 \cdot y_1), \max(x_0 \cdot y_0, x_1 \cdot y_1)]in$

Regarding the algebraic structure [3], only the required multiplication for the given rules is investigated.

This algebraic structure of the interval numbers is at least a Magma [4], as all multiplications in $\overline {\mathbb R}$ including Rule I and Rule II are defined.

Deduction

	Operation or rule
$-1 \cdot (0 \cdot -∞)$	Rule II
$-1 \cdot [-∞, 0]in$	Multiplication for interval number
$[-1 \cdot -∞, -1 \cdot 0]in$	Operation in $\overline {\mathbb R}$
$[0, ∞]in$	Rule I
$0 \cdot ∞$

Implementation

In the test folder is a C++ implementation of the interval number and the unit tests.

Some indeterminate forms as interval numbers

At point of writing, the expression $\frac{0}{0}$ is undefined, also in $\overline {\mathbb R}$ [5].
However, with interval numbers, the expression can be defined:

$\frac{0}{0} = [-∞, ∞]in$

Because the limits of the given example formula results in +∞ and -∞:

$\lim_{n\to0^+}\frac{n}{n^2} = +∞$

$\lim_{n\to0^-}\frac{n}{n^2} = -∞$

Conclusion

Using this approach, other indeterminate forms could be expressed as an interval and solved to equations as well. Especially the usage and current definition in measure theory should be further evaluated [6].

For now, it is shown, that the algebraic structure of the interval number is a Magma. However, including the other mathematical operations, the algebraic structure could be further investigated.

Furthermore, other intervals for indeterminate forms could be estimated and defined.

References

1 Indeterminate form

https://en.wikipedia.org/wiki/Indeterminate_form

2 Extended real number line

https://en.wikipedia.org/wiki/Extended_real_number_line

3 Algebraic structure

https://en.wikipedia.org/wiki/Algebraic_structure

4 Magma (algebra)

https://en.wikipedia.org/wiki/Magma_(algebra)

5 Affinely Extended Real Numbers

https://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html

6 extended real numbers

https://planetmath.org/extendedrealnumbers