Definition:Decimal Notation

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Definition

Decimal notation is the quotidian technique of expressing numbers in base $10$.

Every number $x \in \R$ is expressed in the form:

$\ds x = \sum_{j \mathop \in \Z} r_j 10^j$

where:

$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$


Examples

Decimal Number $209$

In decimal notation $209$ means:

\(\ds 209\) \(=\) \(\ds 2 \times 10^2 + 0 \times 10^1 + 9 \times 10^0\)
\(\ds \) \(=\) \(\ds 200 + 0 + 9\)


Decimal Number $4129$

In decimal notation $4129$ means:

\(\ds 4129\) \(=\) \(\ds 4 \times 10^3 + 1 \times 10^2 + 2 \times 10^1 + 9 \times 10^0\)
\(\ds \) \(=\) \(\ds 4000 + 100 + 20 + 9\)


Also known as

Some sources use the term denary for decimal.


Also see

  • Results about decimal notation can be found here.


Historical Note

The earliest recorded occurrence of decimal notation appears in $\text {595}$.

The earliest record of decimal notation including the zero digit is from $\text {876}$.

Some early discussion of decimal notation is found in a work by Bhaskara II Acharya from the $12$th century.

It was adopted by the Arabs and introduced into Europe mainly through translations of Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala by Muhammad ibn Musa al-Khwarizmi ("The Compendious Book on Calculation by Completion and Balancing").


Sources

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