Definition:Zero (Number)
Establish the equivalence of this definition and that for the number fields as introduced below.
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Contents
Definition
The number zero is defined as being the cardinal of the empty set.
Naturally Ordered Semigroup
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.
Then from axiom $(NO 1)$, $\left({S, \circ, \preceq}\right)$ has a smallest element.
This smallest element of $\left({S, \circ, \preceq}\right)$ 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:
- $0 + 0 i$
Define this in the context of the standard number fields.
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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 Bahkshali 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 Mahāvīrāchārya, 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.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $2$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra: $\S 8$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): Chapter $1$: Complex Numbers: The Real Number System: $2$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $-1$ and $i$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $-1$ and $i$
