Definition:Decimal Notation

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 see

• Definition:Decimal Expansion

• Results about decimal notation can be found here.

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.

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {1-2}$ The Basis Representation Theorem
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $10$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $10$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): decimal notation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): decimal notation