Definition:Zero (Number)

From ProofWiki
Jump to navigation Jump to search


This page has been identified as a candidate for refactoring of advanced complexity.
In particular: Establish the relationship between this definition and that for the number fields as introduced below.
Until this has been finished, please leave {{Refactor}} in the code.

New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only.

Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


Definition

The number zero is defined as being the cardinal of the empty set.


Naturally Ordered Semigroup

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

Then from Naturally Ordered Semigroup Axiom $\text {NO} 1$: Well-Ordered, $\struct {S, \circ, \preceq}$ has a smallest element.


This smallest element of $\struct {S, \circ, \preceq}$ is called zero and has the symbol $0$.

That is:

$\forall n \in S: 0 \preceq n$


Natural Numbers

Integers

Rational Numbers

Real Numbers

Complex Numbers

Let $\C$ denote the set of complex numbers.

The zero of $\C$ is the complex number:

$0 + 0 i$



This article is incomplete.
In particular: Define this in the context of the standard number fields.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Stub}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



Also known as

The somewhat outdated term cipher or cypher can on occasion be seen for the number zero, especially when used in the context of a zero digit in a basis representation.

The words nought or naught can also be seen.

Younger children often use the word nothing.


Also see


Historical Note

The Babylonians from the $2$nd century BCE used a number base system of arithmetic, with a placeholder to indicate that a particular place within a number was empty, but its use was inconsistent. However, they had no actual recognition of zero as a mathematical concept in its own right.


The Ancient Greeks had no conception of zero as a number.


The concept of zero was invented by the mathematicians of India. The Bakhshali Manuscript from the $3$rd century CE contains the first reference to it.


However, even then there were reservations about its existence, and misunderstanding about how it behaved.

In Ganita Sara Samgraha of Mahaviracharya, c. $850$ CE appears:

A number multiplied by zero is zero and that number remains unchanged which is divided by, added to or diminished by zero.


It was not until the propagation of Arabic numbers, where its use as a placeholder made it important, that it became commonplace.


Linguistic Note

The Sanskrit word used by the early Indian mathematicians for zero was sunya, which means empty, or blank.

In Arabic this was translated as sifr.

This was translated via the Latin zephirum into various European languages as zero, cifre, cifra, and into English as zero and cipher.


Note that the plural of zero is either zeros or zeroes. On $\mathsf{Pr} \infty \mathsf{fWiki}$, zeroes is preferred.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Zero_(Number)&oldid=582668"