# 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:

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

## Historical Note

The earliest known discussion of the system of decimal notation is found in a work by Bhaskara II Acharya from the $12$th century.

However, there is evidence to suggest that some form of decimal notation was known about for some $500$ years before that.