Definition:Recurring Decimal

Definition

A recurring decimal is a nonterminating decimal whose decimal expansion ends with a finite number of digits which repeats itself ad infinitum.

Examples

Example: $0 \cdotp 333 \ldots$

The number $\dfrac 1 3$ can be expressed as a recurring decimal:

\(\ds \dfrac 1 3\)

\(=\)

\(\ds 0 \cdotp 333 \ldots\)

\(\ds \)

\(=\)

\(\ds 0 \cdotp \dot 3\)

using recurrence notation

This would be voiced:

nought point three recurring

or:

zero point three recurring

and so on.

Example: $0 \cdotp 714 \, 285 \, 714 \ldots$

The number $\dfrac 5 7$ can be expressed as a recurring decimal:

\(\ds \dfrac 5 7\)

\(=\)

\(\ds 0 \cdotp 714 \, 285 \, 714 \ldots\)

\(\ds \)

\(=\)

\(\ds 0 \cdotp \dot 714 \, 28 \dot 5\)

using recurrence notation

Also known as

A recurring decimal is also known as:

a repeating decimal

a repeated decimal

a periodic decimal.

Also see

• Definition:Terminating Decimal
• Definition:Nonterminating Decimal
• Definition:Nonperiodic Decimal

• Results about recurring decimals can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): decimal
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): recurring decimal
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): decimal
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): recurring decimal