WikiJournal Preprints/On operations with zero considered indeterminate or undefined

On operations with zero considered indeterminate or undefined

Author information

Author: Charles Ewan Milner

Author email address: charles.milner@outlook.com

Author ORCID iD link: https://orcid.org/0009-0001-5580-1345

Abstract

This paper explores correct and meaningful answers to multiple operations relating to zero that have been considered indeterminate or undefined in mathematics.

Main Paper

There are many operations which involve the number zero that are considered indeterminate or undefined in mathematics. However, there are some answers to some of these operations which make them more meaningful in a mathematical sense.

$\frac{0}{0}$ is considered indeterminate because it is equal to, by definition, the value $x$ such that $0x=0$. However, as most numbers have this property of $x$, they should all be valid answers for this operation. The problem is that many mathematicians do not accept an operation being equal to more than one value. However, truthfully, using logic and the axioms of mathematics, it can be concluded simply that it is a valid answer that $\frac{0}{0}=x$ for any value of $x$ such that $0x=0$, but because normal methods of algebra do not always work with this answer, they should not be used with it.

While $\frac{0}{0}$ can be given an infinite number of values, division of a number other than zero by zero is unable be given a value of any real number or even of a well-known non-real number. However, it is important to note that from a mathematical standpoint, these operations, in the form $\frac{x}{0}$ where $x\ne 0$, give legitimate values that are numbers, but just not in the set of the real numbers. Another notable fact about these values can be found with some simple mathematics. Because $-1\cdot -\frac{1}{0}=\frac{1}{0}$, it is true that $\frac{\frac{1}{0}}{-1}=-\frac{1}{0}$. However, it can also be found that $\frac{\frac{1}{0}}{-1}=\frac{1}{0}\cdot \frac{1}{-1}=\frac{1}{0}$. This shows that $\frac{1}{0}=-\frac{1}{0}$.

There is much debate about what the value of $0^0$ is. It has been claimed that it is $1$, which is partially correct, but that is not the best definition. It can actually be found that $0^0=0^{1-1}=\frac{0^1}{0^1}=\frac{0}{0}$. Because $\frac{0}{0}$ is equal to any number $x$ such that $0x=0$ and $1$ fills this property of $x$, it is true that $0^0=1$, but there are an infinite number of other values that $0^0$ is also equal to.

These are truthful and meaningful answers to a few key operations relating to zero that are considered indeterminate or undefined in mathematics.

Additional information

Acknowledgements

This paper was authored by Charles Ewan Milner.

Competing interests

There are no competing interests relating to this paper held by its authors.