Repeating decimals are rational

Builds on Decimal expansion Rational numbers Sum of a finite geometric progression

Statement

Let $x \ge 0$ be a real number with a decimal expansion that is eventually periodic. That is, there are $N \ge 0$ and $p \ge 1$ with $d_{k + p} = d_k$ for each $k > N$ . Then $x$ is a rational number.

Remarks

Proof: $10^{N+p} x - 10^N x$ is an integer, because the periodic tails cancel. Thus $x$ is a quotient of an integer by $10^N (10^p - 1)$ . For example, $x = 0.\overline{36}$ gives $100 x - x = 36$ , thus $x = 36/99 = 4/11$ . The converse is also true: the decimal expansion of each rational number is eventually periodic.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Mathematical induction concept
Rational numbers object
Real numbers object
Set bounded above definition
Sum of a finite geometric progression statement
Supremum definition
Completeness axiom statement
Decimal expansion definition
Repeating decimals are rational statement

Tag	`065S5B9HF1`
URI	`m2://065S5B9HF1`
Kind	statement, theorem
Writing	W0 stub
MSC	11A63
Status	This copy of the site has no build data. Run `m2 status 065S5B9HF1`.
Uses	Decimal expansion, Rational numbers, Sum of a finite geometric progression

Repeating decimals are rational

Statement

Remarks

Reading path

Further reading

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